Anisotropic x16 LOD (LOD)

Percentage Accurate: 75.8% → 75.8%

Time: 21.9s

Alternatives: 8

Speedup: N/A×

Specification

\[\left(\left(\left(\left(\left(\left(1 \leq w \land w \leq 16384\right) \land \left(1 \leq h \land h \leq 16384\right)\right) \land \left(10^{-20} \leq \left|dX.u\right| \land \left|dX.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dX.v\right| \land \left|dX.v\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.u\right| \land \left|dY.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.v\right| \land \left|dY.v\right| \leq 10^{+20}\right)\right) \land maxAniso = 16\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_1 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_2 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_3 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_4 := \mathsf{max}\left(t\_3 \cdot t\_3 + t\_0 \cdot t\_0, t\_1 \cdot t\_1 + t\_2 \cdot t\_2\right)\\ t_5 := \sqrt{t\_4}\\ t_6 := \left|t\_3 \cdot t\_2 - t\_0 \cdot t\_1\right|\\ \log_{2} \begin{array}{l} \mathbf{if}\;\frac{t\_4}{t\_6} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{t\_5}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_6}{t\_5}\\ \end{array} \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (* (floor w) dY.u))
        (t_2 (* (floor h) dY.v))
        (t_3 (* (floor w) dX.u))
        (t_4 (fmax (+ (* t_3 t_3) (* t_0 t_0)) (+ (* t_1 t_1) (* t_2 t_2))))
        (t_5 (sqrt t_4))
        (t_6 (fabs (- (* t_3 t_2) (* t_0 t_1)))))
   (log2
    (if (> (/ t_4 t_6) (floor maxAniso))
      (/ t_5 (floor maxAniso))
      (/ t_6 t_5)))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = floorf(h) * dX_46_v;
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = floorf(h) * dY_46_v;
	float t_3 = floorf(w) * dX_46_u;
	float t_4 = fmaxf(((t_3 * t_3) + (t_0 * t_0)), ((t_1 * t_1) + (t_2 * t_2)));
	float t_5 = sqrtf(t_4);
	float t_6 = fabsf(((t_3 * t_2) - (t_0 * t_1)));
	float tmp;
	if ((t_4 / t_6) > floorf(maxAniso)) {
		tmp = t_5 / floorf(maxAniso);
	} else {
		tmp = t_6 / t_5;
	}
	return log2f(tmp);
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(h) * dY_46_v)
	t_3 = Float32(floor(w) * dX_46_u)
	t_4 = (Float32(Float32(t_3 * t_3) + Float32(t_0 * t_0)) != Float32(Float32(t_3 * t_3) + Float32(t_0 * t_0))) ? Float32(Float32(t_1 * t_1) + Float32(t_2 * t_2)) : ((Float32(Float32(t_1 * t_1) + Float32(t_2 * t_2)) != Float32(Float32(t_1 * t_1) + Float32(t_2 * t_2))) ? Float32(Float32(t_3 * t_3) + Float32(t_0 * t_0)) : max(Float32(Float32(t_3 * t_3) + Float32(t_0 * t_0)), Float32(Float32(t_1 * t_1) + Float32(t_2 * t_2))))
	t_5 = sqrt(t_4)
	t_6 = abs(Float32(Float32(t_3 * t_2) - Float32(t_0 * t_1)))
	tmp = Float32(0.0)
	if (Float32(t_4 / t_6) > floor(maxAniso))
		tmp = Float32(t_5 / floor(maxAniso));
	else
		tmp = Float32(t_6 / t_5);
	end
	return log2(tmp)
end

MATLAB

function tmp_2 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(h) * dX_46_v;
	t_1 = floor(w) * dY_46_u;
	t_2 = floor(h) * dY_46_v;
	t_3 = floor(w) * dX_46_u;
	t_4 = max(((t_3 * t_3) + (t_0 * t_0)), ((t_1 * t_1) + (t_2 * t_2)));
	t_5 = sqrt(t_4);
	t_6 = abs(((t_3 * t_2) - (t_0 * t_1)));
	tmp = single(0.0);
	if ((t_4 / t_6) > floor(maxAniso))
		tmp = t_5 / floor(maxAniso);
	else
		tmp = t_6 / t_5;
	end
	tmp_2 = log2(tmp);
end

Initial Program: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_1 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_2 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_3 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_4 := \mathsf{max}\left(t\_3 \cdot t\_3 + t\_0 \cdot t\_0, t\_1 \cdot t\_1 + t\_2 \cdot t\_2\right)\\ t_5 := \sqrt{t\_4}\\ t_6 := \left|t\_3 \cdot t\_2 - t\_0 \cdot t\_1\right|\\ \log_{2} \begin{array}{l} \mathbf{if}\;\frac{t\_4}{t\_6} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{t\_5}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_6}{t\_5}\\ \end{array} \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (* (floor w) dY.u))
        (t_2 (* (floor h) dY.v))
        (t_3 (* (floor w) dX.u))
        (t_4 (fmax (+ (* t_3 t_3) (* t_0 t_0)) (+ (* t_1 t_1) (* t_2 t_2))))
        (t_5 (sqrt t_4))
        (t_6 (fabs (- (* t_3 t_2) (* t_0 t_1)))))
   (log2
    (if (> (/ t_4 t_6) (floor maxAniso))
      (/ t_5 (floor maxAniso))
      (/ t_6 t_5)))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = floorf(h) * dX_46_v;
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = floorf(h) * dY_46_v;
	float t_3 = floorf(w) * dX_46_u;
	float t_4 = fmaxf(((t_3 * t_3) + (t_0 * t_0)), ((t_1 * t_1) + (t_2 * t_2)));
	float t_5 = sqrtf(t_4);
	float t_6 = fabsf(((t_3 * t_2) - (t_0 * t_1)));
	float tmp;
	if ((t_4 / t_6) > floorf(maxAniso)) {
		tmp = t_5 / floorf(maxAniso);
	} else {
		tmp = t_6 / t_5;
	}
	return log2f(tmp);
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(h) * dY_46_v)
	t_3 = Float32(floor(w) * dX_46_u)
	t_4 = (Float32(Float32(t_3 * t_3) + Float32(t_0 * t_0)) != Float32(Float32(t_3 * t_3) + Float32(t_0 * t_0))) ? Float32(Float32(t_1 * t_1) + Float32(t_2 * t_2)) : ((Float32(Float32(t_1 * t_1) + Float32(t_2 * t_2)) != Float32(Float32(t_1 * t_1) + Float32(t_2 * t_2))) ? Float32(Float32(t_3 * t_3) + Float32(t_0 * t_0)) : max(Float32(Float32(t_3 * t_3) + Float32(t_0 * t_0)), Float32(Float32(t_1 * t_1) + Float32(t_2 * t_2))))
	t_5 = sqrt(t_4)
	t_6 = abs(Float32(Float32(t_3 * t_2) - Float32(t_0 * t_1)))
	tmp = Float32(0.0)
	if (Float32(t_4 / t_6) > floor(maxAniso))
		tmp = Float32(t_5 / floor(maxAniso));
	else
		tmp = Float32(t_6 / t_5);
	end
	return log2(tmp)
end

MATLAB

function tmp_2 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(h) * dX_46_v;
	t_1 = floor(w) * dY_46_u;
	t_2 = floor(h) * dY_46_v;
	t_3 = floor(w) * dX_46_u;
	t_4 = max(((t_3 * t_3) + (t_0 * t_0)), ((t_1 * t_1) + (t_2 * t_2)));
	t_5 = sqrt(t_4);
	t_6 = abs(((t_3 * t_2) - (t_0 * t_1)));
	tmp = single(0.0);
	if ((t_4 / t_6) > floor(maxAniso))
		tmp = t_5 / floor(maxAniso);
	else
		tmp = t_6 / t_5;
	end
	tmp_2 = log2(tmp);
end

Alternative 1: 75.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2}\right)\\ t_1 := \sqrt{t\_0}\\ t_2 := \left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left|\mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right)\right|\\ \log_{2} \begin{array}{l} \mathbf{if}\;\frac{t\_0}{t\_2} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{t\_1}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{t\_1}\\ \end{array} \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0
         (fmax
          (+ (pow (* dX.u (floor w)) 2.0) (pow (* (floor h) dX.v) 2.0))
          (+ (pow (* (floor h) dY.v) 2.0) (pow (* (floor w) dY.u) 2.0))))
        (t_1 (sqrt t_0))
        (t_2
         (* (* (floor w) (floor h)) (fabs (fma dX.u dY.v (* dX.v (- dY.u)))))))
   (log2
    (if (> (/ t_0 t_2) (floor maxAniso))
      (/ t_1 (floor maxAniso))
      (/ t_2 t_1)))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = fmaxf((powf((dX_46_u * floorf(w)), 2.0f) + powf((floorf(h) * dX_46_v), 2.0f)), (powf((floorf(h) * dY_46_v), 2.0f) + powf((floorf(w) * dY_46_u), 2.0f)));
	float t_1 = sqrtf(t_0);
	float t_2 = (floorf(w) * floorf(h)) * fabsf(fmaf(dX_46_u, dY_46_v, (dX_46_v * -dY_46_u)));
	float tmp;
	if ((t_0 / t_2) > floorf(maxAniso)) {
		tmp = t_1 / floorf(maxAniso);
	} else {
		tmp = t_2 / t_1;
	}
	return log2f(tmp);
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = (Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) != Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(floor(h) * dX_46_v) ^ Float32(2.0)))) ? Float32((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + (Float32(floor(w) * dY_46_u) ^ Float32(2.0))) : ((Float32((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + (Float32(floor(w) * dY_46_u) ^ Float32(2.0))) != Float32((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + (Float32(floor(w) * dY_46_u) ^ Float32(2.0)))) ? Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) : max(Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(floor(h) * dX_46_v) ^ Float32(2.0))), Float32((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + (Float32(floor(w) * dY_46_u) ^ Float32(2.0)))))
	t_1 = sqrt(t_0)
	t_2 = Float32(Float32(floor(w) * floor(h)) * abs(fma(dX_46_u, dY_46_v, Float32(dX_46_v * Float32(-dY_46_u)))))
	tmp = Float32(0.0)
	if (Float32(t_0 / t_2) > floor(maxAniso))
		tmp = Float32(t_1 / floor(maxAniso));
	else
		tmp = Float32(t_2 / t_1);
	end
	return log2(tmp)
end

Derivation

1. Initial program 76.9%

   \[\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
2. Add Preprocessing

4. Applied egg-rr 76.9%

   \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}{\left|\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, \left\lfloor h\right\rfloor \cdot dY.v, -\left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, \left\lfloor h\right\rfloor \cdot dY.v, -\left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}}\\ \end{array}} \]

6. Applied egg-rr 76.9%

   \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}{\left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left|\mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left|\mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right)\right|}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}}\\ \end{array}} \]

7. Final simplification 76.9%

   \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2}\right)}{\left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left|\mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2}\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left\lfloor w\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left|\mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right)\right|}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2}\right)}}\\ \end{array} \]
8. Add Preprocessing

Alternative 2: 75.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_1 := t\_0 \cdot \left(dY.u \cdot dY.u\right)\\ t_2 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_3 := \mathsf{fma}\left(dX.v, dX.v \cdot t\_2, \left(dX.u \cdot dX.u\right) \cdot t\_0\right)\\ t_4 := \mathsf{max}\left(t\_3, \mathsf{fma}\left(dY.v, dY.v \cdot t\_2, t\_1\right)\right)\\ t_5 := \left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|\\ \log_{2} \begin{array}{l} \mathbf{if}\;\frac{t\_4}{t\_5} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{t\_4}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;t\_5 \cdot \sqrt{\frac{1}{\mathsf{max}\left(t\_3, t\_1\right)}}\\ \end{array} \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (pow (floor w) 2.0))
        (t_1 (* t_0 (* dY.u dY.u)))
        (t_2 (pow (floor h) 2.0))
        (t_3 (fma dX.v (* dX.v t_2) (* (* dX.u dX.u) t_0)))
        (t_4 (fmax t_3 (fma dY.v (* dY.v t_2) t_1)))
        (t_5
         (fabs (* (floor h) (* (floor w) (fma dX.v (- dY.u) (* dX.u dY.v)))))))
   (log2
    (if (> (/ t_4 t_5) (floor maxAniso))
      (/ (sqrt t_4) (floor maxAniso))
      (* t_5 (sqrt (/ 1.0 (fmax t_3 t_1))))))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = powf(floorf(w), 2.0f);
	float t_1 = t_0 * (dY_46_u * dY_46_u);
	float t_2 = powf(floorf(h), 2.0f);
	float t_3 = fmaf(dX_46_v, (dX_46_v * t_2), ((dX_46_u * dX_46_u) * t_0));
	float t_4 = fmaxf(t_3, fmaf(dY_46_v, (dY_46_v * t_2), t_1));
	float t_5 = fabsf((floorf(h) * (floorf(w) * fmaf(dX_46_v, -dY_46_u, (dX_46_u * dY_46_v)))));
	float tmp;
	if ((t_4 / t_5) > floorf(maxAniso)) {
		tmp = sqrtf(t_4) / floorf(maxAniso);
	} else {
		tmp = t_5 * sqrtf((1.0f / fmaxf(t_3, t_1)));
	}
	return log2f(tmp);
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(w) ^ Float32(2.0)
	t_1 = Float32(t_0 * Float32(dY_46_u * dY_46_u))
	t_2 = floor(h) ^ Float32(2.0)
	t_3 = fma(dX_46_v, Float32(dX_46_v * t_2), Float32(Float32(dX_46_u * dX_46_u) * t_0))
	t_4 = (t_3 != t_3) ? fma(dY_46_v, Float32(dY_46_v * t_2), t_1) : ((fma(dY_46_v, Float32(dY_46_v * t_2), t_1) != fma(dY_46_v, Float32(dY_46_v * t_2), t_1)) ? t_3 : max(t_3, fma(dY_46_v, Float32(dY_46_v * t_2), t_1)))
	t_5 = abs(Float32(floor(h) * Float32(floor(w) * fma(dX_46_v, Float32(-dY_46_u), Float32(dX_46_u * dY_46_v)))))
	tmp = Float32(0.0)
	if (Float32(t_4 / t_5) > floor(maxAniso))
		tmp = Float32(sqrt(t_4) / floor(maxAniso));
	else
		tmp = Float32(t_5 * sqrt(Float32(Float32(1.0) / ((t_3 != t_3) ? t_1 : ((t_1 != t_1) ? t_3 : max(t_3, t_1))))));
	end
	return log2(tmp)
end

Derivation

1. Initial program 76.9%

   \[\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
2. Add Preprocessing

3. Taylor expanded in w around 0

   \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}{\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \end{array}} \]

5. Simplified 76.9%

   \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array}} \]

6. Taylor expanded in dY.v around 0

   \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   2. lower-*.f32 N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   3. lower-pow.f32 N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   4. lower-floor.f32 N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   5. unpow2 N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   6. lower-*.f32 76.7

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

8. Simplified 76.7%

   \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]
9. Add Preprocessing

Alternative 3: 68.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_1 := \left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|\\ t_2 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_3 := \mathsf{fma}\left(dY.v, dY.v \cdot t\_0, t\_2 \cdot \left(dY.u \cdot dY.u\right)\right)\\ t_4 := \left(dX.u \cdot dX.u\right) \cdot t\_2\\ \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot t\_2\right), t\_3\right)}{t\_1} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot t\_0, t\_4\right), t\_3\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{\frac{1}{\mathsf{max}\left(t\_4, t\_3\right)}}\\ \end{array} \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (pow (floor h) 2.0))
        (t_1
         (fabs (* (floor h) (* (floor w) (fma dX.v (- dY.u) (* dX.u dY.v))))))
        (t_2 (pow (floor w) 2.0))
        (t_3 (fma dY.v (* dY.v t_0) (* t_2 (* dY.u dY.u))))
        (t_4 (* (* dX.u dX.u) t_2)))
   (log2
    (if (> (/ (fmax (* dX.u (* dX.u t_2)) t_3) t_1) (floor maxAniso))
      (/ (sqrt (fmax (fma dX.v (* dX.v t_0) t_4) t_3)) (floor maxAniso))
      (* t_1 (sqrt (/ 1.0 (fmax t_4 t_3))))))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = powf(floorf(h), 2.0f);
	float t_1 = fabsf((floorf(h) * (floorf(w) * fmaf(dX_46_v, -dY_46_u, (dX_46_u * dY_46_v)))));
	float t_2 = powf(floorf(w), 2.0f);
	float t_3 = fmaf(dY_46_v, (dY_46_v * t_0), (t_2 * (dY_46_u * dY_46_u)));
	float t_4 = (dX_46_u * dX_46_u) * t_2;
	float tmp;
	if ((fmaxf((dX_46_u * (dX_46_u * t_2)), t_3) / t_1) > floorf(maxAniso)) {
		tmp = sqrtf(fmaxf(fmaf(dX_46_v, (dX_46_v * t_0), t_4), t_3)) / floorf(maxAniso);
	} else {
		tmp = t_1 * sqrtf((1.0f / fmaxf(t_4, t_3)));
	}
	return log2f(tmp);
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(h) ^ Float32(2.0)
	t_1 = abs(Float32(floor(h) * Float32(floor(w) * fma(dX_46_v, Float32(-dY_46_u), Float32(dX_46_u * dY_46_v)))))
	t_2 = floor(w) ^ Float32(2.0)
	t_3 = fma(dY_46_v, Float32(dY_46_v * t_0), Float32(t_2 * Float32(dY_46_u * dY_46_u)))
	t_4 = Float32(Float32(dX_46_u * dX_46_u) * t_2)
	tmp = Float32(0.0)
	if (Float32(((Float32(dX_46_u * Float32(dX_46_u * t_2)) != Float32(dX_46_u * Float32(dX_46_u * t_2))) ? t_3 : ((t_3 != t_3) ? Float32(dX_46_u * Float32(dX_46_u * t_2)) : max(Float32(dX_46_u * Float32(dX_46_u * t_2)), t_3))) / t_1) > floor(maxAniso))
		tmp = Float32(sqrt(((fma(dX_46_v, Float32(dX_46_v * t_0), t_4) != fma(dX_46_v, Float32(dX_46_v * t_0), t_4)) ? t_3 : ((t_3 != t_3) ? fma(dX_46_v, Float32(dX_46_v * t_0), t_4) : max(fma(dX_46_v, Float32(dX_46_v * t_0), t_4), t_3)))) / floor(maxAniso));
	else
		tmp = Float32(t_1 * sqrt(Float32(Float32(1.0) / ((t_4 != t_4) ? t_3 : ((t_3 != t_3) ? t_4 : max(t_4, t_3))))));
	end
	return log2(tmp)
end

Derivation

1. Initial program 76.9%

   \[\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
2. Add Preprocessing

3. Taylor expanded in w around 0

   \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}{\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \end{array}} \]

5. Simplified 76.9%

   \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array}} \]

6. Taylor expanded in dX.v around 0

   \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]
7. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   2. associate-*l* N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   3. *-commutative N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   4. lower-*.f32 N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   5. *-commutative N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   6. lower-*.f32 N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   7. lower-pow.f32 N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   8. lower-floor.f32 69.5

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array} \]

8. Simplified 69.5%

   \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array} \]

9. Taylor expanded in dX.v around 0

   \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]
10. Step-by-step derivation

    1. lower-*.f32 N/A

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

    2. unpow2 N/A

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

    3. lower-*.f32 N/A

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

    4. lower-pow.f32 N/A

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

    5. lower-floor.f32 70.6

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array} \]

11. Simplified 70.6%

    \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array} \]
12. Add Preprocessing

Alternative 4: 61.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_1 := dX.u \cdot \left(dX.u \cdot t\_0\right)\\ t_2 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_3 := \left(dX.u \cdot dX.u\right) \cdot t\_0\\ t_4 := \mathsf{fma}\left(dX.v, dX.v \cdot t\_2, t\_3\right)\\ t_5 := t\_0 \cdot \left(dY.u \cdot dY.u\right)\\ t_6 := \left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|\\ t_7 := \mathsf{fma}\left(dY.v, dY.v \cdot t\_2, t\_5\right)\\ \mathbf{if}\;dY.u \leq 50:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(t\_1, t\_2 \cdot \left(dY.v \cdot dY.v\right)\right)}{t\_6} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{{\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2}\right)\right)}^{0.5}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;t\_6 \cdot \sqrt{\frac{1}{\mathsf{max}\left(t\_4, t\_7\right)}}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(t\_1, t\_7\right)}{t\_6} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(t\_3, t\_7\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;t\_6 \cdot \sqrt{\frac{1}{\mathsf{max}\left(t\_4, t\_5\right)}}\\ \end{array}\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (pow (floor w) 2.0))
        (t_1 (* dX.u (* dX.u t_0)))
        (t_2 (pow (floor h) 2.0))
        (t_3 (* (* dX.u dX.u) t_0))
        (t_4 (fma dX.v (* dX.v t_2) t_3))
        (t_5 (* t_0 (* dY.u dY.u)))
        (t_6
         (fabs (* (floor h) (* (floor w) (fma dX.v (- dY.u) (* dX.u dY.v))))))
        (t_7 (fma dY.v (* dY.v t_2) t_5)))
   (if (<= dY.u 50.0)
     (log2
      (if (> (/ (fmax t_1 (* t_2 (* dY.v dY.v))) t_6) (floor maxAniso))
        (/
         (pow
          (fmax
           (+ (pow (* dX.u (floor w)) 2.0) (pow (* (floor h) dX.v) 2.0))
           (+ (pow (* (floor h) dY.v) 2.0) (pow (* (floor w) dY.u) 2.0)))
          0.5)
         (floor maxAniso))
        (* t_6 (sqrt (/ 1.0 (fmax t_4 t_7))))))
     (log2
      (if (> (/ (fmax t_1 t_7) t_6) (floor maxAniso))
        (/ (sqrt (fmax t_3 t_7)) (floor maxAniso))
        (* t_6 (sqrt (/ 1.0 (fmax t_4 t_5)))))))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = powf(floorf(w), 2.0f);
	float t_1 = dX_46_u * (dX_46_u * t_0);
	float t_2 = powf(floorf(h), 2.0f);
	float t_3 = (dX_46_u * dX_46_u) * t_0;
	float t_4 = fmaf(dX_46_v, (dX_46_v * t_2), t_3);
	float t_5 = t_0 * (dY_46_u * dY_46_u);
	float t_6 = fabsf((floorf(h) * (floorf(w) * fmaf(dX_46_v, -dY_46_u, (dX_46_u * dY_46_v)))));
	float t_7 = fmaf(dY_46_v, (dY_46_v * t_2), t_5);
	float tmp_1;
	if (dY_46_u <= 50.0f) {
		float tmp_2;
		if ((fmaxf(t_1, (t_2 * (dY_46_v * dY_46_v))) / t_6) > floorf(maxAniso)) {
			tmp_2 = powf(fmaxf((powf((dX_46_u * floorf(w)), 2.0f) + powf((floorf(h) * dX_46_v), 2.0f)), (powf((floorf(h) * dY_46_v), 2.0f) + powf((floorf(w) * dY_46_u), 2.0f))), 0.5f) / floorf(maxAniso);
		} else {
			tmp_2 = t_6 * sqrtf((1.0f / fmaxf(t_4, t_7)));
		}
		tmp_1 = log2f(tmp_2);
	} else {
		float tmp_3;
		if ((fmaxf(t_1, t_7) / t_6) > floorf(maxAniso)) {
			tmp_3 = sqrtf(fmaxf(t_3, t_7)) / floorf(maxAniso);
		} else {
			tmp_3 = t_6 * sqrtf((1.0f / fmaxf(t_4, t_5)));
		}
		tmp_1 = log2f(tmp_3);
	}
	return tmp_1;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(w) ^ Float32(2.0)
	t_1 = Float32(dX_46_u * Float32(dX_46_u * t_0))
	t_2 = floor(h) ^ Float32(2.0)
	t_3 = Float32(Float32(dX_46_u * dX_46_u) * t_0)
	t_4 = fma(dX_46_v, Float32(dX_46_v * t_2), t_3)
	t_5 = Float32(t_0 * Float32(dY_46_u * dY_46_u))
	t_6 = abs(Float32(floor(h) * Float32(floor(w) * fma(dX_46_v, Float32(-dY_46_u), Float32(dX_46_u * dY_46_v)))))
	t_7 = fma(dY_46_v, Float32(dY_46_v * t_2), t_5)
	tmp_1 = Float32(0.0)
	if (dY_46_u <= Float32(50.0))
		tmp_2 = Float32(0.0)
		if (Float32(((t_1 != t_1) ? Float32(t_2 * Float32(dY_46_v * dY_46_v)) : ((Float32(t_2 * Float32(dY_46_v * dY_46_v)) != Float32(t_2 * Float32(dY_46_v * dY_46_v))) ? t_1 : max(t_1, Float32(t_2 * Float32(dY_46_v * dY_46_v))))) / t_6) > floor(maxAniso))
			tmp_2 = Float32((((Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) != Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(floor(h) * dX_46_v) ^ Float32(2.0)))) ? Float32((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + (Float32(floor(w) * dY_46_u) ^ Float32(2.0))) : ((Float32((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + (Float32(floor(w) * dY_46_u) ^ Float32(2.0))) != Float32((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + (Float32(floor(w) * dY_46_u) ^ Float32(2.0)))) ? Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) : max(Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(floor(h) * dX_46_v) ^ Float32(2.0))), Float32((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + (Float32(floor(w) * dY_46_u) ^ Float32(2.0)))))) ^ Float32(0.5)) / floor(maxAniso));
		else
			tmp_2 = Float32(t_6 * sqrt(Float32(Float32(1.0) / ((t_4 != t_4) ? t_7 : ((t_7 != t_7) ? t_4 : max(t_4, t_7))))));
		end
		tmp_1 = log2(tmp_2);
	else
		tmp_3 = Float32(0.0)
		if (Float32(((t_1 != t_1) ? t_7 : ((t_7 != t_7) ? t_1 : max(t_1, t_7))) / t_6) > floor(maxAniso))
			tmp_3 = Float32(sqrt(((t_3 != t_3) ? t_7 : ((t_7 != t_7) ? t_3 : max(t_3, t_7)))) / floor(maxAniso));
		else
			tmp_3 = Float32(t_6 * sqrt(Float32(Float32(1.0) / ((t_4 != t_4) ? t_5 : ((t_5 != t_5) ? t_4 : max(t_4, t_5))))));
		end
		tmp_1 = log2(tmp_3);
	end
	return tmp_1
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 50

   1. Initial program 78.1%

      \[\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
   2. Add Preprocessing

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}{\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \end{array}} \]

   5. Simplified 78.1%

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array}} \]

   6. Taylor expanded in dX.v around 0

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      2. associate-*l* N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      3. *-commutative N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      5. *-commutative N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      8. lower-floor.f32 68.9

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array} \]

   8. Simplified 68.9%

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array} \]

   10. Applied egg-rr 68.9%

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\color{blue}{{\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{0.5}}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array} \]

   11. Taylor expanded in dY.v around inf

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{{\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{\frac{1}{2}}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{\frac{1}{2}}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dY.v}^{2}}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{{\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{\frac{1}{2}}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{\frac{1}{2}}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       2. lower-*.f32 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dY.v}^{2}}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{{\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{\frac{1}{2}}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{\frac{1}{2}}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       3. lower-pow.f32 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2}} \cdot {dY.v}^{2}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{{\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{\frac{1}{2}}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{\frac{1}{2}}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       4. lower-floor.f32 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), {\color{blue}{\left(\left\lfloor h\right\rfloor \right)}}^{2} \cdot {dY.v}^{2}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{{\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{\frac{1}{2}}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{\frac{1}{2}}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       5. unpow2 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.v \cdot dY.v\right)}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{{\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{\frac{1}{2}}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{\frac{1}{2}}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       6. lower-*.f32 64.6

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.v \cdot dY.v\right)}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{{\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{0.5}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array} \]

   13. Simplified 64.6%

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{{\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{0.5}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array} \]

   if 50 < dY.u

   1. Initial program 73.3%

      \[\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
   2. Add Preprocessing

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}{\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \end{array}} \]

   5. Simplified 73.4%

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array}} \]

   6. Taylor expanded in dX.v around 0

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      2. associate-*l* N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      3. *-commutative N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      5. *-commutative N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      8. lower-floor.f32 71.3

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array} \]

   8. Simplified 71.3%

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array} \]

   9. Taylor expanded in dX.v around 0

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]
   10. Step-by-step derivation

       1. lower-*.f32 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       2. unpow2 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       3. lower-*.f32 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       4. lower-pow.f32 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       5. lower-floor.f32 71.3

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array} \]

   11. Simplified 71.3%

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array} \]

   12. Taylor expanded in dY.v around 0

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       2. lower-*.f32 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       3. lower-pow.f32 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       4. lower-floor.f32 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       5. unpow2 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       6. lower-*.f32 71.3

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   14. Simplified 71.3%

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.u \leq 50:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{{\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2}\right)\right)}^{0.5}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_1 := dX.u \cdot \left(dX.u \cdot t\_0\right)\\ t_2 := \left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|\\ t_3 := t\_0 \cdot \left(dY.u \cdot dY.u\right)\\ t_4 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_5 := \mathsf{fma}\left(dX.v, dX.v \cdot t\_4, \left(dX.u \cdot dX.u\right) \cdot t\_0\right)\\ t_6 := \mathsf{max}\left(t\_5, \mathsf{fma}\left(dY.v, dY.v \cdot t\_4, t\_3\right)\right)\\ \mathbf{if}\;dY.u \leq 100000000:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(t\_1, t\_4 \cdot \left(dY.v \cdot dY.v\right)\right)}{t\_2} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{{\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2}\right)\right)}^{0.5}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \sqrt{\frac{1}{t\_6}}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(t\_1, dY.u \cdot \left(dY.u \cdot t\_0\right)\right)}{t\_2} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{t\_6}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \sqrt{\frac{1}{\mathsf{max}\left(t\_5, t\_3\right)}}\\ \end{array}\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (pow (floor w) 2.0))
        (t_1 (* dX.u (* dX.u t_0)))
        (t_2
         (fabs (* (floor h) (* (floor w) (fma dX.v (- dY.u) (* dX.u dY.v))))))
        (t_3 (* t_0 (* dY.u dY.u)))
        (t_4 (pow (floor h) 2.0))
        (t_5 (fma dX.v (* dX.v t_4) (* (* dX.u dX.u) t_0)))
        (t_6 (fmax t_5 (fma dY.v (* dY.v t_4) t_3))))
   (if (<= dY.u 100000000.0)
     (log2
      (if (> (/ (fmax t_1 (* t_4 (* dY.v dY.v))) t_2) (floor maxAniso))
        (/
         (pow
          (fmax
           (+ (pow (* dX.u (floor w)) 2.0) (pow (* (floor h) dX.v) 2.0))
           (+ (pow (* (floor h) dY.v) 2.0) (pow (* (floor w) dY.u) 2.0)))
          0.5)
         (floor maxAniso))
        (* t_2 (sqrt (/ 1.0 t_6)))))
     (log2
      (if (> (/ (fmax t_1 (* dY.u (* dY.u t_0))) t_2) (floor maxAniso))
        (/ (sqrt t_6) (floor maxAniso))
        (* t_2 (sqrt (/ 1.0 (fmax t_5 t_3)))))))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = powf(floorf(w), 2.0f);
	float t_1 = dX_46_u * (dX_46_u * t_0);
	float t_2 = fabsf((floorf(h) * (floorf(w) * fmaf(dX_46_v, -dY_46_u, (dX_46_u * dY_46_v)))));
	float t_3 = t_0 * (dY_46_u * dY_46_u);
	float t_4 = powf(floorf(h), 2.0f);
	float t_5 = fmaf(dX_46_v, (dX_46_v * t_4), ((dX_46_u * dX_46_u) * t_0));
	float t_6 = fmaxf(t_5, fmaf(dY_46_v, (dY_46_v * t_4), t_3));
	float tmp_1;
	if (dY_46_u <= 100000000.0f) {
		float tmp_2;
		if ((fmaxf(t_1, (t_4 * (dY_46_v * dY_46_v))) / t_2) > floorf(maxAniso)) {
			tmp_2 = powf(fmaxf((powf((dX_46_u * floorf(w)), 2.0f) + powf((floorf(h) * dX_46_v), 2.0f)), (powf((floorf(h) * dY_46_v), 2.0f) + powf((floorf(w) * dY_46_u), 2.0f))), 0.5f) / floorf(maxAniso);
		} else {
			tmp_2 = t_2 * sqrtf((1.0f / t_6));
		}
		tmp_1 = log2f(tmp_2);
	} else {
		float tmp_3;
		if ((fmaxf(t_1, (dY_46_u * (dY_46_u * t_0))) / t_2) > floorf(maxAniso)) {
			tmp_3 = sqrtf(t_6) / floorf(maxAniso);
		} else {
			tmp_3 = t_2 * sqrtf((1.0f / fmaxf(t_5, t_3)));
		}
		tmp_1 = log2f(tmp_3);
	}
	return tmp_1;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(w) ^ Float32(2.0)
	t_1 = Float32(dX_46_u * Float32(dX_46_u * t_0))
	t_2 = abs(Float32(floor(h) * Float32(floor(w) * fma(dX_46_v, Float32(-dY_46_u), Float32(dX_46_u * dY_46_v)))))
	t_3 = Float32(t_0 * Float32(dY_46_u * dY_46_u))
	t_4 = floor(h) ^ Float32(2.0)
	t_5 = fma(dX_46_v, Float32(dX_46_v * t_4), Float32(Float32(dX_46_u * dX_46_u) * t_0))
	t_6 = (t_5 != t_5) ? fma(dY_46_v, Float32(dY_46_v * t_4), t_3) : ((fma(dY_46_v, Float32(dY_46_v * t_4), t_3) != fma(dY_46_v, Float32(dY_46_v * t_4), t_3)) ? t_5 : max(t_5, fma(dY_46_v, Float32(dY_46_v * t_4), t_3)))
	tmp_1 = Float32(0.0)
	if (dY_46_u <= Float32(100000000.0))
		tmp_2 = Float32(0.0)
		if (Float32(((t_1 != t_1) ? Float32(t_4 * Float32(dY_46_v * dY_46_v)) : ((Float32(t_4 * Float32(dY_46_v * dY_46_v)) != Float32(t_4 * Float32(dY_46_v * dY_46_v))) ? t_1 : max(t_1, Float32(t_4 * Float32(dY_46_v * dY_46_v))))) / t_2) > floor(maxAniso))
			tmp_2 = Float32((((Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) != Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(floor(h) * dX_46_v) ^ Float32(2.0)))) ? Float32((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + (Float32(floor(w) * dY_46_u) ^ Float32(2.0))) : ((Float32((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + (Float32(floor(w) * dY_46_u) ^ Float32(2.0))) != Float32((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + (Float32(floor(w) * dY_46_u) ^ Float32(2.0)))) ? Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) : max(Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(floor(h) * dX_46_v) ^ Float32(2.0))), Float32((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + (Float32(floor(w) * dY_46_u) ^ Float32(2.0)))))) ^ Float32(0.5)) / floor(maxAniso));
		else
			tmp_2 = Float32(t_2 * sqrt(Float32(Float32(1.0) / t_6)));
		end
		tmp_1 = log2(tmp_2);
	else
		tmp_3 = Float32(0.0)
		if (Float32(((t_1 != t_1) ? Float32(dY_46_u * Float32(dY_46_u * t_0)) : ((Float32(dY_46_u * Float32(dY_46_u * t_0)) != Float32(dY_46_u * Float32(dY_46_u * t_0))) ? t_1 : max(t_1, Float32(dY_46_u * Float32(dY_46_u * t_0))))) / t_2) > floor(maxAniso))
			tmp_3 = Float32(sqrt(t_6) / floor(maxAniso));
		else
			tmp_3 = Float32(t_2 * sqrt(Float32(Float32(1.0) / ((t_5 != t_5) ? t_3 : ((t_3 != t_3) ? t_5 : max(t_5, t_3))))));
		end
		tmp_1 = log2(tmp_3);
	end
	return tmp_1
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 1e8

   1. Initial program 79.2%

      \[\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
   2. Add Preprocessing

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}{\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \end{array}} \]

   5. Simplified 79.2%

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array}} \]

   6. Taylor expanded in dX.v around 0

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      2. associate-*l* N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      3. *-commutative N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      5. *-commutative N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      8. lower-floor.f32 70.5

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array} \]

   8. Simplified 70.5%

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array} \]

   10. Applied egg-rr 70.5%

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\color{blue}{{\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{0.5}}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array} \]

   11. Taylor expanded in dY.v around inf

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{{\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{\frac{1}{2}}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{\frac{1}{2}}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dY.v}^{2}}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{{\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{\frac{1}{2}}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{\frac{1}{2}}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       2. lower-*.f32 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dY.v}^{2}}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{{\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{\frac{1}{2}}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{\frac{1}{2}}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       3. lower-pow.f32 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2}} \cdot {dY.v}^{2}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{{\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{\frac{1}{2}}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{\frac{1}{2}}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       4. lower-floor.f32 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), {\color{blue}{\left(\left\lfloor h\right\rfloor \right)}}^{2} \cdot {dY.v}^{2}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{{\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{\frac{1}{2}}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{\frac{1}{2}}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       5. unpow2 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.v \cdot dY.v\right)}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{{\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{\frac{1}{2}}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{\frac{1}{2}}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       6. lower-*.f32 66.2

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.v \cdot dY.v\right)}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{{\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{0.5}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array} \]

   13. Simplified 66.2%

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{{\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}^{0.5}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array} \]

   if 1e8 < dY.u

   1. Initial program 66.7%

      \[\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
   2. Add Preprocessing

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}{\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \end{array}} \]

   5. Simplified 66.7%

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array}} \]

   6. Taylor expanded in dX.v around 0

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      2. associate-*l* N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      3. *-commutative N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      5. *-commutative N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      8. lower-floor.f32 65.3

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array} \]

   8. Simplified 65.3%

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array} \]

   9. Taylor expanded in dY.v around 0

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{\left(dY.u \cdot dY.u\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       2. associate-*l* N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{dY.u \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       3. *-commutative N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       4. lower-*.f32 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       5. lower-*.f32 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       6. lower-pow.f32 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       7. lower-floor.f32 65.1

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left({\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array} \]

   11. Simplified 65.1%

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array} \]

   12. Taylor expanded in dY.v around 0

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       2. lower-*.f32 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       3. lower-pow.f32 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       4. lower-floor.f32 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       5. unpow2 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       6. lower-*.f32 66.3

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   14. Simplified 66.3%

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.u \leq 100000000:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{{\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2}\right)\right)}^{0.5}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_1 := dX.u \cdot \left(dX.u \cdot t\_0\right)\\ t_2 := \left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|\\ t_3 := t\_0 \cdot \left(dY.u \cdot dY.u\right)\\ t_4 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_5 := \mathsf{fma}\left(dX.v, dX.v \cdot t\_4, \left(dX.u \cdot dX.u\right) \cdot t\_0\right)\\ t_6 := \mathsf{max}\left(t\_5, \mathsf{fma}\left(dY.v, dY.v \cdot t\_4, t\_3\right)\right)\\ t_7 := \frac{\sqrt{t\_6}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{if}\;dY.u \leq 100000000:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(t\_1, t\_4 \cdot \left(dY.v \cdot dY.v\right)\right)}{t\_2} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;t\_7\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \sqrt{\frac{1}{t\_6}}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(t\_1, dY.u \cdot \left(dY.u \cdot t\_0\right)\right)}{t\_2} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;t\_7\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \sqrt{\frac{1}{\mathsf{max}\left(t\_5, t\_3\right)}}\\ \end{array}\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (pow (floor w) 2.0))
        (t_1 (* dX.u (* dX.u t_0)))
        (t_2
         (fabs (* (floor h) (* (floor w) (fma dX.v (- dY.u) (* dX.u dY.v))))))
        (t_3 (* t_0 (* dY.u dY.u)))
        (t_4 (pow (floor h) 2.0))
        (t_5 (fma dX.v (* dX.v t_4) (* (* dX.u dX.u) t_0)))
        (t_6 (fmax t_5 (fma dY.v (* dY.v t_4) t_3)))
        (t_7 (/ (sqrt t_6) (floor maxAniso))))
   (if (<= dY.u 100000000.0)
     (log2
      (if (> (/ (fmax t_1 (* t_4 (* dY.v dY.v))) t_2) (floor maxAniso))
        t_7
        (* t_2 (sqrt (/ 1.0 t_6)))))
     (log2
      (if (> (/ (fmax t_1 (* dY.u (* dY.u t_0))) t_2) (floor maxAniso))
        t_7
        (* t_2 (sqrt (/ 1.0 (fmax t_5 t_3)))))))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = powf(floorf(w), 2.0f);
	float t_1 = dX_46_u * (dX_46_u * t_0);
	float t_2 = fabsf((floorf(h) * (floorf(w) * fmaf(dX_46_v, -dY_46_u, (dX_46_u * dY_46_v)))));
	float t_3 = t_0 * (dY_46_u * dY_46_u);
	float t_4 = powf(floorf(h), 2.0f);
	float t_5 = fmaf(dX_46_v, (dX_46_v * t_4), ((dX_46_u * dX_46_u) * t_0));
	float t_6 = fmaxf(t_5, fmaf(dY_46_v, (dY_46_v * t_4), t_3));
	float t_7 = sqrtf(t_6) / floorf(maxAniso);
	float tmp_1;
	if (dY_46_u <= 100000000.0f) {
		float tmp_2;
		if ((fmaxf(t_1, (t_4 * (dY_46_v * dY_46_v))) / t_2) > floorf(maxAniso)) {
			tmp_2 = t_7;
		} else {
			tmp_2 = t_2 * sqrtf((1.0f / t_6));
		}
		tmp_1 = log2f(tmp_2);
	} else {
		float tmp_3;
		if ((fmaxf(t_1, (dY_46_u * (dY_46_u * t_0))) / t_2) > floorf(maxAniso)) {
			tmp_3 = t_7;
		} else {
			tmp_3 = t_2 * sqrtf((1.0f / fmaxf(t_5, t_3)));
		}
		tmp_1 = log2f(tmp_3);
	}
	return tmp_1;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(w) ^ Float32(2.0)
	t_1 = Float32(dX_46_u * Float32(dX_46_u * t_0))
	t_2 = abs(Float32(floor(h) * Float32(floor(w) * fma(dX_46_v, Float32(-dY_46_u), Float32(dX_46_u * dY_46_v)))))
	t_3 = Float32(t_0 * Float32(dY_46_u * dY_46_u))
	t_4 = floor(h) ^ Float32(2.0)
	t_5 = fma(dX_46_v, Float32(dX_46_v * t_4), Float32(Float32(dX_46_u * dX_46_u) * t_0))
	t_6 = (t_5 != t_5) ? fma(dY_46_v, Float32(dY_46_v * t_4), t_3) : ((fma(dY_46_v, Float32(dY_46_v * t_4), t_3) != fma(dY_46_v, Float32(dY_46_v * t_4), t_3)) ? t_5 : max(t_5, fma(dY_46_v, Float32(dY_46_v * t_4), t_3)))
	t_7 = Float32(sqrt(t_6) / floor(maxAniso))
	tmp_1 = Float32(0.0)
	if (dY_46_u <= Float32(100000000.0))
		tmp_2 = Float32(0.0)
		if (Float32(((t_1 != t_1) ? Float32(t_4 * Float32(dY_46_v * dY_46_v)) : ((Float32(t_4 * Float32(dY_46_v * dY_46_v)) != Float32(t_4 * Float32(dY_46_v * dY_46_v))) ? t_1 : max(t_1, Float32(t_4 * Float32(dY_46_v * dY_46_v))))) / t_2) > floor(maxAniso))
			tmp_2 = t_7;
		else
			tmp_2 = Float32(t_2 * sqrt(Float32(Float32(1.0) / t_6)));
		end
		tmp_1 = log2(tmp_2);
	else
		tmp_3 = Float32(0.0)
		if (Float32(((t_1 != t_1) ? Float32(dY_46_u * Float32(dY_46_u * t_0)) : ((Float32(dY_46_u * Float32(dY_46_u * t_0)) != Float32(dY_46_u * Float32(dY_46_u * t_0))) ? t_1 : max(t_1, Float32(dY_46_u * Float32(dY_46_u * t_0))))) / t_2) > floor(maxAniso))
			tmp_3 = t_7;
		else
			tmp_3 = Float32(t_2 * sqrt(Float32(Float32(1.0) / ((t_5 != t_5) ? t_3 : ((t_3 != t_3) ? t_5 : max(t_5, t_3))))));
		end
		tmp_1 = log2(tmp_3);
	end
	return tmp_1
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 1e8

   1. Initial program 79.2%

      \[\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
   2. Add Preprocessing

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}{\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \end{array}} \]

   5. Simplified 79.2%

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array}} \]

   6. Taylor expanded in dX.v around 0

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      2. associate-*l* N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      3. *-commutative N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      5. *-commutative N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      8. lower-floor.f32 70.5

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array} \]

   8. Simplified 70.5%

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array} \]

   9. Taylor expanded in dY.v around inf

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dY.v}^{2}}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       2. lower-*.f32 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dY.v}^{2}}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       3. lower-pow.f32 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2}} \cdot {dY.v}^{2}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       4. lower-floor.f32 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), {\color{blue}{\left(\left\lfloor h\right\rfloor \right)}}^{2} \cdot {dY.v}^{2}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       5. unpow2 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.v \cdot dY.v\right)}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       6. lower-*.f32 66.2

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.v \cdot dY.v\right)}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array} \]

   11. Simplified 66.2%

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array} \]

   if 1e8 < dY.u

   1. Initial program 66.7%

      \[\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
   2. Add Preprocessing

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}{\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \end{array}} \]

   5. Simplified 66.7%

      \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array}} \]

   6. Taylor expanded in dX.v around 0

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      2. associate-*l* N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      3. *-commutative N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      5. *-commutative N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      8. lower-floor.f32 65.3

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array} \]

   8. Simplified 65.3%

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array} \]

   9. Taylor expanded in dY.v around 0

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{\left(dY.u \cdot dY.u\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       2. associate-*l* N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{dY.u \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       3. *-commutative N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       4. lower-*.f32 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       5. lower-*.f32 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       6. lower-pow.f32 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       7. lower-floor.f32 65.1

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left({\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array} \]

   11. Simplified 65.1%

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array} \]

   12. Taylor expanded in dY.v around 0

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       2. lower-*.f32 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       3. lower-pow.f32 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       4. lower-floor.f32 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       5. unpow2 N/A

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

       6. lower-*.f32 66.3

          \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   14. Simplified 66.3%

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.u \leq 100000000:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|\\ t_1 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_2 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_3 := \mathsf{fma}\left(dX.v, dX.v \cdot t\_1, \left(dX.u \cdot dX.u\right) \cdot t\_2\right)\\ t_4 := t\_2 \cdot \left(dY.u \cdot dY.u\right)\\ \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot t\_2\right), dY.u \cdot \left(dY.u \cdot t\_2\right)\right)}{t\_0} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(t\_3, \mathsf{fma}\left(dY.v, dY.v \cdot t\_1, t\_4\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{\frac{1}{\mathsf{max}\left(t\_3, t\_4\right)}}\\ \end{array} \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0
         (fabs (* (floor h) (* (floor w) (fma dX.v (- dY.u) (* dX.u dY.v))))))
        (t_1 (pow (floor h) 2.0))
        (t_2 (pow (floor w) 2.0))
        (t_3 (fma dX.v (* dX.v t_1) (* (* dX.u dX.u) t_2)))
        (t_4 (* t_2 (* dY.u dY.u))))
   (log2
    (if (>
         (/ (fmax (* dX.u (* dX.u t_2)) (* dY.u (* dY.u t_2))) t_0)
         (floor maxAniso))
      (/ (sqrt (fmax t_3 (fma dY.v (* dY.v t_1) t_4))) (floor maxAniso))
      (* t_0 (sqrt (/ 1.0 (fmax t_3 t_4))))))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = fabsf((floorf(h) * (floorf(w) * fmaf(dX_46_v, -dY_46_u, (dX_46_u * dY_46_v)))));
	float t_1 = powf(floorf(h), 2.0f);
	float t_2 = powf(floorf(w), 2.0f);
	float t_3 = fmaf(dX_46_v, (dX_46_v * t_1), ((dX_46_u * dX_46_u) * t_2));
	float t_4 = t_2 * (dY_46_u * dY_46_u);
	float tmp;
	if ((fmaxf((dX_46_u * (dX_46_u * t_2)), (dY_46_u * (dY_46_u * t_2))) / t_0) > floorf(maxAniso)) {
		tmp = sqrtf(fmaxf(t_3, fmaf(dY_46_v, (dY_46_v * t_1), t_4))) / floorf(maxAniso);
	} else {
		tmp = t_0 * sqrtf((1.0f / fmaxf(t_3, t_4)));
	}
	return log2f(tmp);
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = abs(Float32(floor(h) * Float32(floor(w) * fma(dX_46_v, Float32(-dY_46_u), Float32(dX_46_u * dY_46_v)))))
	t_1 = floor(h) ^ Float32(2.0)
	t_2 = floor(w) ^ Float32(2.0)
	t_3 = fma(dX_46_v, Float32(dX_46_v * t_1), Float32(Float32(dX_46_u * dX_46_u) * t_2))
	t_4 = Float32(t_2 * Float32(dY_46_u * dY_46_u))
	tmp = Float32(0.0)
	if (Float32(((Float32(dX_46_u * Float32(dX_46_u * t_2)) != Float32(dX_46_u * Float32(dX_46_u * t_2))) ? Float32(dY_46_u * Float32(dY_46_u * t_2)) : ((Float32(dY_46_u * Float32(dY_46_u * t_2)) != Float32(dY_46_u * Float32(dY_46_u * t_2))) ? Float32(dX_46_u * Float32(dX_46_u * t_2)) : max(Float32(dX_46_u * Float32(dX_46_u * t_2)), Float32(dY_46_u * Float32(dY_46_u * t_2))))) / t_0) > floor(maxAniso))
		tmp = Float32(sqrt(((t_3 != t_3) ? fma(dY_46_v, Float32(dY_46_v * t_1), t_4) : ((fma(dY_46_v, Float32(dY_46_v * t_1), t_4) != fma(dY_46_v, Float32(dY_46_v * t_1), t_4)) ? t_3 : max(t_3, fma(dY_46_v, Float32(dY_46_v * t_1), t_4))))) / floor(maxAniso));
	else
		tmp = Float32(t_0 * sqrt(Float32(Float32(1.0) / ((t_3 != t_3) ? t_4 : ((t_4 != t_4) ? t_3 : max(t_3, t_4))))));
	end
	return log2(tmp)
end

Derivation

1. Initial program 76.9%

   \[\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
2. Add Preprocessing

3. Taylor expanded in w around 0

   \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}{\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \end{array}} \]

5. Simplified 76.9%

   \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array}} \]

6. Taylor expanded in dX.v around 0

   \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]
7. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   2. associate-*l* N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   3. *-commutative N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   4. lower-*.f32 N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   5. *-commutative N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   6. lower-*.f32 N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   7. lower-pow.f32 N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   8. lower-floor.f32 69.5

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array} \]

8. Simplified 69.5%

   \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array} \]

9. Taylor expanded in dY.v around 0

   \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]
10. Step-by-step derivation

    1. unpow2 N/A

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{\left(dY.u \cdot dY.u\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

    2. associate-*l* N/A

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{dY.u \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

    3. *-commutative N/A

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

    4. lower-*.f32 N/A

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

    5. lower-*.f32 N/A

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

    6. lower-pow.f32 N/A

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

    7. lower-floor.f32 54.2

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left({\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array} \]

11. Simplified 54.2%

    \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array} \]

12. Taylor expanded in dY.v around 0

    \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]
13. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

    2. lower-*.f32 N/A

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

    3. lower-pow.f32 N/A

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

    4. lower-floor.f32 N/A

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

    5. unpow2 N/A

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

    6. lower-*.f32 56.5

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

14. Simplified 56.5%

    \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

15. Final simplification 56.5%

    \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]
16. Add Preprocessing

Alternative 8: 50.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|\\ t_1 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_2 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_3 := \mathsf{fma}\left(dX.v, dX.v \cdot t\_1, \left(dX.u \cdot dX.u\right) \cdot t\_2\right)\\ \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot t\_2\right), dY.u \cdot \left(dY.u \cdot t\_2\right)\right)}{t\_0} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(t\_3, \mathsf{fma}\left(dY.v, dY.v \cdot t\_1, t\_2 \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{\frac{1}{\mathsf{max}\left(t\_3, t\_1 \cdot \left(dY.v \cdot dY.v\right)\right)}}\\ \end{array} \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0
         (fabs (* (floor h) (* (floor w) (fma dX.v (- dY.u) (* dX.u dY.v))))))
        (t_1 (pow (floor h) 2.0))
        (t_2 (pow (floor w) 2.0))
        (t_3 (fma dX.v (* dX.v t_1) (* (* dX.u dX.u) t_2))))
   (log2
    (if (>
         (/ (fmax (* dX.u (* dX.u t_2)) (* dY.u (* dY.u t_2))) t_0)
         (floor maxAniso))
      (/
       (sqrt (fmax t_3 (fma dY.v (* dY.v t_1) (* t_2 (* dY.u dY.u)))))
       (floor maxAniso))
      (* t_0 (sqrt (/ 1.0 (fmax t_3 (* t_1 (* dY.v dY.v))))))))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = fabsf((floorf(h) * (floorf(w) * fmaf(dX_46_v, -dY_46_u, (dX_46_u * dY_46_v)))));
	float t_1 = powf(floorf(h), 2.0f);
	float t_2 = powf(floorf(w), 2.0f);
	float t_3 = fmaf(dX_46_v, (dX_46_v * t_1), ((dX_46_u * dX_46_u) * t_2));
	float tmp;
	if ((fmaxf((dX_46_u * (dX_46_u * t_2)), (dY_46_u * (dY_46_u * t_2))) / t_0) > floorf(maxAniso)) {
		tmp = sqrtf(fmaxf(t_3, fmaf(dY_46_v, (dY_46_v * t_1), (t_2 * (dY_46_u * dY_46_u))))) / floorf(maxAniso);
	} else {
		tmp = t_0 * sqrtf((1.0f / fmaxf(t_3, (t_1 * (dY_46_v * dY_46_v)))));
	}
	return log2f(tmp);
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = abs(Float32(floor(h) * Float32(floor(w) * fma(dX_46_v, Float32(-dY_46_u), Float32(dX_46_u * dY_46_v)))))
	t_1 = floor(h) ^ Float32(2.0)
	t_2 = floor(w) ^ Float32(2.0)
	t_3 = fma(dX_46_v, Float32(dX_46_v * t_1), Float32(Float32(dX_46_u * dX_46_u) * t_2))
	tmp = Float32(0.0)
	if (Float32(((Float32(dX_46_u * Float32(dX_46_u * t_2)) != Float32(dX_46_u * Float32(dX_46_u * t_2))) ? Float32(dY_46_u * Float32(dY_46_u * t_2)) : ((Float32(dY_46_u * Float32(dY_46_u * t_2)) != Float32(dY_46_u * Float32(dY_46_u * t_2))) ? Float32(dX_46_u * Float32(dX_46_u * t_2)) : max(Float32(dX_46_u * Float32(dX_46_u * t_2)), Float32(dY_46_u * Float32(dY_46_u * t_2))))) / t_0) > floor(maxAniso))
		tmp = Float32(sqrt(((t_3 != t_3) ? fma(dY_46_v, Float32(dY_46_v * t_1), Float32(t_2 * Float32(dY_46_u * dY_46_u))) : ((fma(dY_46_v, Float32(dY_46_v * t_1), Float32(t_2 * Float32(dY_46_u * dY_46_u))) != fma(dY_46_v, Float32(dY_46_v * t_1), Float32(t_2 * Float32(dY_46_u * dY_46_u)))) ? t_3 : max(t_3, fma(dY_46_v, Float32(dY_46_v * t_1), Float32(t_2 * Float32(dY_46_u * dY_46_u))))))) / floor(maxAniso));
	else
		tmp = Float32(t_0 * sqrt(Float32(Float32(1.0) / ((t_3 != t_3) ? Float32(t_1 * Float32(dY_46_v * dY_46_v)) : ((Float32(t_1 * Float32(dY_46_v * dY_46_v)) != Float32(t_1 * Float32(dY_46_v * dY_46_v))) ? t_3 : max(t_3, Float32(t_1 * Float32(dY_46_v * dY_46_v))))))));
	end
	return log2(tmp)
end

Derivation

1. Initial program 76.9%

   \[\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right) - \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\right|}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}\\ \end{array} \]
2. Add Preprocessing

3. Taylor expanded in w around 0

   \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}{\left|dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) - dX.v \cdot \left(dY.u \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left\lfloor maxAniso\right\rfloor } \cdot \sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\\ \end{array}} \]

5. Simplified 76.9%

   \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array}} \]

6. Taylor expanded in dX.v around 0

   \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]
7. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   2. associate-*l* N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   3. *-commutative N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   4. lower-*.f32 N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   5. *-commutative N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   6. lower-*.f32 N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   7. lower-pow.f32 N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   8. lower-floor.f32 69.5

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array} \]

8. Simplified 69.5%

   \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array} \]

9. Taylor expanded in dY.v around 0

   \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]
10. Step-by-step derivation

    1. unpow2 N/A

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{\left(dY.u \cdot dY.u\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

    2. associate-*l* N/A

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{dY.u \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

    3. *-commutative N/A

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

    4. lower-*.f32 N/A

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

    5. lower-*.f32 N/A

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

    6. lower-pow.f32 N/A

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

    7. lower-floor.f32 54.2

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left({\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array} \]

11. Simplified 54.2%

    \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)}\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}\\ \end{array} \]

12. Taylor expanded in dY.v around inf

    \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]
13. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

    2. lower-*.f32 N/A

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

    3. lower-pow.f32 N/A

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

    4. lower-floor.f32 N/A

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

    5. unpow2 N/A

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, \mathsf{neg}\left(dY.u\right), dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

    6. lower-*.f32 54.0

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)}}\\ \end{array} \]

14. Simplified 54.0%

    \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)}}\\ \end{array} \]

15. Final simplification 54.0%

    \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)}{\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\left\lfloor h\right\rfloor \cdot \left(\left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, dX.u \cdot dY.v\right)\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)}}\\ \end{array} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024208 
(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
  :name "Anisotropic x16 LOD (LOD)"
  :precision binary32
  :pre (and (and (and (and (and (and (and (<= 1.0 w) (<= w 16384.0)) (and (<= 1.0 h) (<= h 16384.0))) (and (<= 1e-20 (fabs dX.u)) (<= (fabs dX.u) 1e+20))) (and (<= 1e-20 (fabs dX.v)) (<= (fabs dX.v) 1e+20))) (and (<= 1e-20 (fabs dY.u)) (<= (fabs dY.u) 1e+20))) (and (<= 1e-20 (fabs dY.v)) (<= (fabs dY.v) 1e+20))) (== maxAniso 16.0))
  (log2 (if (> (/ (fmax (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v)))) (fabs (- (* (* (floor w) dX.u) (* (floor h) dY.v)) (* (* (floor h) dX.v) (* (floor w) dY.u))))) (floor maxAniso)) (/ (sqrt (fmax (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v))))) (floor maxAniso)) (/ (fabs (- (* (* (floor w) dX.u) (* (floor h) dY.v)) (* (* (floor h) dX.v) (* (floor w) dY.u)))) (sqrt (fmax (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v)))))))))