Anisotropic x16 LOD (LOD)

Percentage Accurate: 76.6% → 76.6%

Time: 25.8s

Alternatives: 5

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: 76.6% 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: 76.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_1 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_2 := \mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot t\_1, dX.v \cdot \left(dX.v \cdot t\_0\right)\right), \mathsf{fma}\left(dY.u, t\_1 \cdot dY.u, t\_0 \cdot \left(dY.v \cdot dY.v\right)\right)\right)\\ t_3 := \left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dY.u \cdot \left(-dX.v\right)\right), 0\right)\right|\\ \log_{2} \begin{array}{l} \mathbf{if}\;\frac{t\_2}{t\_3} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{t\_2}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;t\_3 \cdot \frac{1}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}}\\ \end{array} \end{array} \end{array} \]

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 (pow (floor w) 2.0))
        (t_2
         (fmax
          (fma dX.u (* dX.u t_1) (* dX.v (* dX.v t_0)))
          (fma dY.u (* t_1 dY.u) (* t_0 (* dY.v dY.v)))))
        (t_3
         (fabs
          (fma
           (floor h)
           (* (floor w) (fma dX.u dY.v (* dY.u (- dX.v))))
           0.0))))
   (log2
    (if (> (/ t_2 t_3) (floor maxAniso))
      (/ (sqrt t_2) (floor maxAniso))
      (*
       t_3
       (/
        1.0
        (sqrt
         (fmax
          (+ (pow (* dX.u (floor w)) 2.0) (pow (* dX.v (floor h)) 2.0))
          (+ (pow (* (floor w) dY.u) 2.0) (pow (* (floor h) dY.v) 2.0))))))))))

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 = powf(floorf(w), 2.0f);
	float t_2 = fmaxf(fmaf(dX_46_u, (dX_46_u * t_1), (dX_46_v * (dX_46_v * t_0))), fmaf(dY_46_u, (t_1 * dY_46_u), (t_0 * (dY_46_v * dY_46_v))));
	float t_3 = fabsf(fmaf(floorf(h), (floorf(w) * fmaf(dX_46_u, dY_46_v, (dY_46_u * -dX_46_v))), 0.0f));
	float tmp;
	if ((t_2 / t_3) > floorf(maxAniso)) {
		tmp = sqrtf(t_2) / floorf(maxAniso);
	} else {
		tmp = t_3 * (1.0f / sqrtf(fmaxf((powf((dX_46_u * floorf(w)), 2.0f) + powf((dX_46_v * floorf(h)), 2.0f)), (powf((floorf(w) * dY_46_u), 2.0f) + powf((floorf(h) * dY_46_v), 2.0f)))));
	}
	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 = floor(w) ^ Float32(2.0)
	t_2 = (fma(dX_46_u, Float32(dX_46_u * t_1), Float32(dX_46_v * Float32(dX_46_v * t_0))) != fma(dX_46_u, Float32(dX_46_u * t_1), Float32(dX_46_v * Float32(dX_46_v * t_0)))) ? fma(dY_46_u, Float32(t_1 * dY_46_u), Float32(t_0 * Float32(dY_46_v * dY_46_v))) : ((fma(dY_46_u, Float32(t_1 * dY_46_u), Float32(t_0 * Float32(dY_46_v * dY_46_v))) != fma(dY_46_u, Float32(t_1 * dY_46_u), Float32(t_0 * Float32(dY_46_v * dY_46_v)))) ? fma(dX_46_u, Float32(dX_46_u * t_1), Float32(dX_46_v * Float32(dX_46_v * t_0))) : max(fma(dX_46_u, Float32(dX_46_u * t_1), Float32(dX_46_v * Float32(dX_46_v * t_0))), fma(dY_46_u, Float32(t_1 * dY_46_u), Float32(t_0 * Float32(dY_46_v * dY_46_v)))))
	t_3 = abs(fma(floor(h), Float32(floor(w) * fma(dX_46_u, dY_46_v, Float32(dY_46_u * Float32(-dX_46_v)))), Float32(0.0)))
	tmp = Float32(0.0)
	if (Float32(t_2 / t_3) > floor(maxAniso))
		tmp = Float32(sqrt(t_2) / floor(maxAniso));
	else
		tmp = Float32(t_3 * Float32(Float32(1.0) / sqrt(((Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dX_46_v * floor(h)) ^ Float32(2.0))) != Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dX_46_v * floor(h)) ^ Float32(2.0)))) ? Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + (Float32(floor(h) * dY_46_v) ^ Float32(2.0))) : ((Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + (Float32(floor(h) * dY_46_v) ^ Float32(2.0))) != Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + (Float32(floor(h) * dY_46_v) ^ Float32(2.0)))) ? Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dX_46_v * floor(h)) ^ Float32(2.0))) : max(Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dX_46_v * floor(h)) ^ Float32(2.0))), Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + (Float32(floor(h) * dY_46_v) ^ Float32(2.0)))))))));
	end
	return log2(tmp)
end

Derivation

1. Initial program 79.0%

   \[\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

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

5. Simplified 79.0%

   \[\leadsto \log_{2} \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}\\ } \end{array}} \]
6. Step-by-step derivation

   1. sqrt-div N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }}\\ \end{array} \]

   2. metadata-eval N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }}\\ \end{array} \]

   4. sqrt-lowering-sqrt.f32 N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\color{blue}{\left\lfloor maxAniso\right\rfloor }}\\ \end{array} \]

   5. fmax-lowering-fmax.f32 N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor \color{blue}{maxAniso}\right\rfloor }\\ \end{array} \]

7. Applied egg-rr 79.1%

   \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right| \cdot \frac{1}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}}}\\ \end{array} \]

8. Final simplification 79.1%

   \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dY.u \cdot \left(-dX.v\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dY.u \cdot \left(-dX.v\right)\right), 0\right)\right| \cdot \frac{1}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}}\\ \end{array} \]
9. Add Preprocessing

Alternative 2: 74.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := dX.v \cdot \left\lfloor h\right\rfloor \\ t_1 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_2 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_3 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_4 := \mathsf{max}\left({t\_3}^{2} + {t\_0}^{2}, {t\_1}^{2} + {t\_2}^{2}\right)\\ \log_{2} \begin{array}{l} \mathbf{if}\;\frac{t\_4}{\left|\mathsf{fma}\left(\left\lfloor w\right\rfloor , dX.u \cdot t\_2, 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(t\_3 \cdot t\_3 + t\_0 \cdot t\_0, t\_1 \cdot t\_1 + t\_2 \cdot t\_2\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dY.u \cdot \left(-dX.v\right)\right), 0\right)}{\sqrt{t\_4}}\\ \end{array} \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* dX.v (floor h)))
        (t_1 (* (floor w) dY.u))
        (t_2 (* (floor h) dY.v))
        (t_3 (* dX.u (floor w)))
        (t_4
         (fmax
          (+ (pow t_3 2.0) (pow t_0 2.0))
          (+ (pow t_1 2.0) (pow t_2 2.0)))))
   (log2
    (if (> (/ t_4 (fabs (fma (floor w) (* dX.u t_2) 0.0))) (floor maxAniso))
      (/
       (sqrt (fmax (+ (* t_3 t_3) (* t_0 t_0)) (+ (* t_1 t_1) (* t_2 t_2))))
       (floor maxAniso))
      (/
       (fma (floor h) (* (floor w) (fma dX.u dY.v (* dY.u (- dX.v)))) 0.0)
       (sqrt 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 = dX_46_v * floorf(h);
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = floorf(h) * dY_46_v;
	float t_3 = dX_46_u * floorf(w);
	float t_4 = fmaxf((powf(t_3, 2.0f) + powf(t_0, 2.0f)), (powf(t_1, 2.0f) + powf(t_2, 2.0f)));
	float tmp;
	if ((t_4 / fabsf(fmaf(floorf(w), (dX_46_u * t_2), 0.0f))) > floorf(maxAniso)) {
		tmp = sqrtf(fmaxf(((t_3 * t_3) + (t_0 * t_0)), ((t_1 * t_1) + (t_2 * t_2)))) / floorf(maxAniso);
	} else {
		tmp = fmaf(floorf(h), (floorf(w) * fmaf(dX_46_u, dY_46_v, (dY_46_u * -dX_46_v))), 0.0f) / sqrtf(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 = Float32(dX_46_v * floor(h))
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(h) * dY_46_v)
	t_3 = Float32(dX_46_u * floor(w))
	t_4 = (Float32((t_3 ^ Float32(2.0)) + (t_0 ^ Float32(2.0))) != Float32((t_3 ^ Float32(2.0)) + (t_0 ^ Float32(2.0)))) ? Float32((t_1 ^ Float32(2.0)) + (t_2 ^ Float32(2.0))) : ((Float32((t_1 ^ Float32(2.0)) + (t_2 ^ Float32(2.0))) != Float32((t_1 ^ Float32(2.0)) + (t_2 ^ Float32(2.0)))) ? Float32((t_3 ^ Float32(2.0)) + (t_0 ^ Float32(2.0))) : max(Float32((t_3 ^ Float32(2.0)) + (t_0 ^ Float32(2.0))), Float32((t_1 ^ Float32(2.0)) + (t_2 ^ Float32(2.0)))))
	tmp = Float32(0.0)
	if (Float32(t_4 / abs(fma(floor(w), Float32(dX_46_u * t_2), Float32(0.0)))) > floor(maxAniso))
		tmp = Float32(sqrt(((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)))))) / floor(maxAniso));
	else
		tmp = Float32(fma(floor(h), Float32(floor(w) * fma(dX_46_u, dY_46_v, Float32(dY_46_u * Float32(-dX_46_v)))), Float32(0.0)) / sqrt(t_4));
	end
	return log2(tmp)
end

Derivation

1. Initial program 79.0%

   \[\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 dX.u around inf

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

   1. +-rgt-identity N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\color{blue}{dX.u \cdot \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right) + 0}\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   2. accelerator-lowering-fma.f32 N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\color{blue}{\mathsf{fma}\left(dX.u, dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right), 0\right)}\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   3. +-rgt-identity N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\mathsf{fma}\left(dX.u, \color{blue}{dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor w\right\rfloor \right) + 0}, 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   4. associate-*r* N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\mathsf{fma}\left(dX.u, \color{blue}{\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor w\right\rfloor } + 0, 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   5. *-commutative N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\mathsf{fma}\left(dX.u, \color{blue}{\left\lfloor w\right\rfloor \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right)} + 0, 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   6. accelerator-lowering-fma.f32 N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\mathsf{fma}\left(dX.u, \color{blue}{\mathsf{fma}\left(\left\lfloor w\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , 0\right)}, 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   7. floor-lowering-floor.f32 N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\mathsf{fma}\left(dX.u, \mathsf{fma}\left(\color{blue}{\left\lfloor w\right\rfloor }, dY.v \cdot \left\lfloor h\right\rfloor , 0\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\mathsf{fma}\left(dX.u, \mathsf{fma}\left(\left\lfloor w\right\rfloor , \color{blue}{dY.v \cdot \left\lfloor h\right\rfloor }, 0\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   9. floor-lowering-floor.f32 77.4

      \[\leadsto \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|\mathsf{fma}\left(dX.u, \mathsf{fma}\left(\left\lfloor w\right\rfloor , dY.v \cdot \color{blue}{\left\lfloor h\right\rfloor }, 0\right), 0\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} \]

5. Simplified 77.4%

   \[\leadsto \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|\color{blue}{\mathsf{fma}\left(dX.u, \mathsf{fma}\left(\left\lfloor w\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , 0\right), 0\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} \]

7. Applied egg-rr 76.7%

   \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}{\left|\mathsf{fma}\left(dX.u, \mathsf{fma}\left(\left\lfloor w\right\rfloor , dY.v \cdot \left\lfloor h\right\rfloor , 0\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, 0 - dX.v \cdot dY.u\right), 0\right)}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}}\\ \end{array} \]

9. Applied egg-rr 76.7%

   \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\color{blue}{\frac{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}{\left|\mathsf{fma}\left(\left\lfloor w\right\rfloor , dX.u \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right), 0\right)\right|}} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, 0 - dX.v \cdot dY.u\right), 0\right)}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}}\\ \end{array} \]

10. Final simplification 76.7%

    \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}{\left|\mathsf{fma}\left(\left\lfloor w\right\rfloor , dX.u \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right) + \left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dY.u \cdot \left(-dX.v\right)\right), 0\right)}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}}\\ \end{array} \]
11. Add Preprocessing

Alternative 3: 74.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}\\ t_1 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_2 := dX.v \cdot \left(dX.v \cdot t\_1\right)\\ t_3 := \left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dY.u \cdot \left(-dX.v\right)\right), 0\right)\right|\\ t_4 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_5 := \mathsf{fma}\left(dY.u, t\_4 \cdot dY.u, t\_1 \cdot \left(dY.v \cdot dY.v\right)\right)\\ t_6 := dX.u \cdot t\_4\\ t_7 := \frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, t\_6, t\_2\right), t\_5\right)}}{\left\lfloor maxAniso\right\rfloor }\\ t_8 := \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(t\_2, t\_5\right)}{t\_3} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;t\_7\\ \mathbf{else}:\\ \;\;\;\;t\_3 \cdot \frac{1}{t\_0}\\ \end{array}\\ \mathbf{if}\;dX.v \leq -20:\\ \;\;\;\;t\_8\\ \mathbf{elif}\;dX.v \leq 1500:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot t\_6, t\_5\right)}{t\_3} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;t\_7\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\mathsf{fma}\left(\left\lfloor w\right\rfloor , \left\lfloor h\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, \mathsf{fma}\left(dX.u, dY.v, 0\right)\right), 0\right)\right|}{t\_0}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_8\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0
         (sqrt
          (fmax
           (+ (pow (* dX.u (floor w)) 2.0) (pow (* dX.v (floor h)) 2.0))
           (+ (pow (* (floor w) dY.u) 2.0) (pow (* (floor h) dY.v) 2.0)))))
        (t_1 (pow (floor h) 2.0))
        (t_2 (* dX.v (* dX.v t_1)))
        (t_3
         (fabs
          (fma (floor h) (* (floor w) (fma dX.u dY.v (* dY.u (- dX.v)))) 0.0)))
        (t_4 (pow (floor w) 2.0))
        (t_5 (fma dY.u (* t_4 dY.u) (* t_1 (* dY.v dY.v))))
        (t_6 (* dX.u t_4))
        (t_7 (/ (sqrt (fmax (fma dX.u t_6 t_2) t_5)) (floor maxAniso)))
        (t_8
         (log2
          (if (> (/ (fmax t_2 t_5) t_3) (floor maxAniso))
            t_7
            (* t_3 (/ 1.0 t_0))))))
   (if (<= dX.v -20.0)
     t_8
     (if (<= dX.v 1500.0)
       (log2
        (if (> (/ (fmax (* dX.u t_6) t_5) t_3) (floor maxAniso))
          t_7
          (/
           (fabs
            (fma
             (floor w)
             (* (floor h) (fma dX.v (- dY.u) (fma dX.u dY.v 0.0)))
             0.0))
           t_0)))
       t_8))))

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 = sqrtf(fmaxf((powf((dX_46_u * floorf(w)), 2.0f) + powf((dX_46_v * floorf(h)), 2.0f)), (powf((floorf(w) * dY_46_u), 2.0f) + powf((floorf(h) * dY_46_v), 2.0f))));
	float t_1 = powf(floorf(h), 2.0f);
	float t_2 = dX_46_v * (dX_46_v * t_1);
	float t_3 = fabsf(fmaf(floorf(h), (floorf(w) * fmaf(dX_46_u, dY_46_v, (dY_46_u * -dX_46_v))), 0.0f));
	float t_4 = powf(floorf(w), 2.0f);
	float t_5 = fmaf(dY_46_u, (t_4 * dY_46_u), (t_1 * (dY_46_v * dY_46_v)));
	float t_6 = dX_46_u * t_4;
	float t_7 = sqrtf(fmaxf(fmaf(dX_46_u, t_6, t_2), t_5)) / floorf(maxAniso);
	float tmp;
	if ((fmaxf(t_2, t_5) / t_3) > floorf(maxAniso)) {
		tmp = t_7;
	} else {
		tmp = t_3 * (1.0f / t_0);
	}
	float t_8 = log2f(tmp);
	float tmp_1;
	if (dX_46_v <= -20.0f) {
		tmp_1 = t_8;
	} else if (dX_46_v <= 1500.0f) {
		float tmp_2;
		if ((fmaxf((dX_46_u * t_6), t_5) / t_3) > floorf(maxAniso)) {
			tmp_2 = t_7;
		} else {
			tmp_2 = fabsf(fmaf(floorf(w), (floorf(h) * fmaf(dX_46_v, -dY_46_u, fmaf(dX_46_u, dY_46_v, 0.0f))), 0.0f)) / t_0;
		}
		tmp_1 = log2f(tmp_2);
	} else {
		tmp_1 = t_8;
	}
	return tmp_1;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = sqrt(((Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dX_46_v * floor(h)) ^ Float32(2.0))) != Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dX_46_v * floor(h)) ^ Float32(2.0)))) ? Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + (Float32(floor(h) * dY_46_v) ^ Float32(2.0))) : ((Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + (Float32(floor(h) * dY_46_v) ^ Float32(2.0))) != Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + (Float32(floor(h) * dY_46_v) ^ Float32(2.0)))) ? Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dX_46_v * floor(h)) ^ Float32(2.0))) : max(Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dX_46_v * floor(h)) ^ Float32(2.0))), Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + (Float32(floor(h) * dY_46_v) ^ Float32(2.0)))))))
	t_1 = floor(h) ^ Float32(2.0)
	t_2 = Float32(dX_46_v * Float32(dX_46_v * t_1))
	t_3 = abs(fma(floor(h), Float32(floor(w) * fma(dX_46_u, dY_46_v, Float32(dY_46_u * Float32(-dX_46_v)))), Float32(0.0)))
	t_4 = floor(w) ^ Float32(2.0)
	t_5 = fma(dY_46_u, Float32(t_4 * dY_46_u), Float32(t_1 * Float32(dY_46_v * dY_46_v)))
	t_6 = Float32(dX_46_u * t_4)
	t_7 = Float32(sqrt(((fma(dX_46_u, t_6, t_2) != fma(dX_46_u, t_6, t_2)) ? t_5 : ((t_5 != t_5) ? fma(dX_46_u, t_6, t_2) : max(fma(dX_46_u, t_6, t_2), t_5)))) / floor(maxAniso))
	tmp = Float32(0.0)
	if (Float32(((t_2 != t_2) ? t_5 : ((t_5 != t_5) ? t_2 : max(t_2, t_5))) / t_3) > floor(maxAniso))
		tmp = t_7;
	else
		tmp = Float32(t_3 * Float32(Float32(1.0) / t_0));
	end
	t_8 = log2(tmp)
	tmp_1 = Float32(0.0)
	if (dX_46_v <= Float32(-20.0))
		tmp_1 = t_8;
	elseif (dX_46_v <= Float32(1500.0))
		tmp_2 = Float32(0.0)
		if (Float32(((Float32(dX_46_u * t_6) != Float32(dX_46_u * t_6)) ? t_5 : ((t_5 != t_5) ? Float32(dX_46_u * t_6) : max(Float32(dX_46_u * t_6), t_5))) / t_3) > floor(maxAniso))
			tmp_2 = t_7;
		else
			tmp_2 = Float32(abs(fma(floor(w), Float32(floor(h) * fma(dX_46_v, Float32(-dY_46_u), fma(dX_46_u, dY_46_v, Float32(0.0)))), Float32(0.0))) / t_0);
		end
		tmp_1 = log2(tmp_2);
	else
		tmp_1 = t_8;
	end
	return tmp_1
end

Derivation

1. Split input into 2 regimes

2. if dX.v < -20 or 1500 < dX.v

   1. Initial program 70.4%

      \[\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

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

   5. Simplified 70.4%

      \[\leadsto \log_{2} \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. sqrt-div N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }}\\ \end{array} \]

      4. sqrt-lowering-sqrt.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\color{blue}{\left\lfloor maxAniso\right\rfloor }}\\ \end{array} \]

      5. fmax-lowering-fmax.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor \color{blue}{maxAniso}\right\rfloor }\\ \end{array} \]

   7. Applied egg-rr 70.4%

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right| \cdot \frac{1}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}}}\\ \end{array} \]

   8. Taylor expanded in dX.u around 0

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{\left(dX.v \cdot dX.v\right)} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      2. associate-*l* N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.v \cdot \color{blue}{\left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      5. pow-lowering-pow.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.v \cdot \left(dX.v \cdot \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2}}\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      6. floor-lowering-floor.f32 67.9

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.v \cdot \left(dX.v \cdot {\color{blue}{\left(\left\lfloor h\right\rfloor \right)}}^{2}\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right| \cdot \frac{1}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}}\\ \end{array} \]

   10. Simplified 67.9%

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right| \cdot \frac{1}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}}\\ \end{array} \]

   if -20 < dX.v < 1500

   1. Initial program 85.0%

      \[\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

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

   5. Simplified 85.1%

      \[\leadsto \log_{2} \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. sqrt-div N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }}\\ \end{array} \]

      4. sqrt-lowering-sqrt.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\color{blue}{\left\lfloor maxAniso\right\rfloor }}\\ \end{array} \]

      5. fmax-lowering-fmax.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor \color{blue}{maxAniso}\right\rfloor }\\ \end{array} \]

   7. Applied egg-rr 85.1%

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right| \cdot \frac{1}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}}}\\ \end{array} \]

   8. Taylor expanded in dX.u around inf

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      2. associate-*l* N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      5. pow-lowering-pow.f32 N/A

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

      6. floor-lowering-floor.f32 84.6

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right| \cdot \frac{1}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}}\\ \end{array} \]

   10. Simplified 84.6%

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right| \cdot \frac{1}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}}\\ \end{array} \]

   12. Applied egg-rr 84.6%

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\mathsf{fma}\left(\left\lfloor w\right\rfloor , \mathsf{fma}\left(dX.v, -dY.u, \mathsf{fma}\left(dX.u, dY.v, 0\right)\right) \cdot \left\lfloor h\right\rfloor , 0\right)\right|}{\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}}\\ \end{array} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq -20:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dY.u \cdot \left(-dX.v\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dY.u \cdot \left(-dX.v\right)\right), 0\right)\right| \cdot \frac{1}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \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}\\ \mathbf{elif}\;dX.v \leq 1500:\\ \;\;\;\;\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.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dY.u \cdot \left(-dX.v\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\mathsf{fma}\left(\left\lfloor w\right\rfloor , \left\lfloor h\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, \mathsf{fma}\left(dX.u, dY.v, 0\right)\right), 0\right)\right|}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \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}\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dY.u \cdot \left(-dX.v\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dY.u \cdot \left(-dX.v\right)\right), 0\right)\right| \cdot \frac{1}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \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}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_1 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_2 := \mathsf{fma}\left(dY.u, t\_1 \cdot dY.u, t\_0 \cdot \left(dY.v \cdot dY.v\right)\right)\\ t_3 := dX.u \cdot t\_1\\ \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot t\_3, t\_2\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dY.u \cdot \left(-dX.v\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, t\_3, dX.v \cdot \left(dX.v \cdot t\_0\right)\right), t\_2\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\mathsf{fma}\left(\left\lfloor w\right\rfloor , \left\lfloor h\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, \mathsf{fma}\left(dX.u, dY.v, 0\right)\right), 0\right)\right|}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}}\\ \end{array} \end{array} \end{array} \]

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 (pow (floor w) 2.0))
        (t_2 (fma dY.u (* t_1 dY.u) (* t_0 (* dY.v dY.v))))
        (t_3 (* dX.u t_1)))
   (log2
    (if (>
         (/
          (fmax (* dX.u t_3) t_2)
          (fabs
           (fma
            (floor h)
            (* (floor w) (fma dX.u dY.v (* dY.u (- dX.v))))
            0.0)))
         (floor maxAniso))
      (/
       (sqrt (fmax (fma dX.u t_3 (* dX.v (* dX.v t_0))) t_2))
       (floor maxAniso))
      (/
       (fabs
        (fma
         (floor w)
         (* (floor h) (fma dX.v (- dY.u) (fma dX.u dY.v 0.0)))
         0.0))
       (sqrt
        (fmax
         (+ (pow (* dX.u (floor w)) 2.0) (pow (* dX.v (floor h)) 2.0))
         (+ (pow (* (floor w) dY.u) 2.0) (pow (* (floor h) dY.v) 2.0)))))))))

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 = powf(floorf(w), 2.0f);
	float t_2 = fmaf(dY_46_u, (t_1 * dY_46_u), (t_0 * (dY_46_v * dY_46_v)));
	float t_3 = dX_46_u * t_1;
	float tmp;
	if ((fmaxf((dX_46_u * t_3), t_2) / fabsf(fmaf(floorf(h), (floorf(w) * fmaf(dX_46_u, dY_46_v, (dY_46_u * -dX_46_v))), 0.0f))) > floorf(maxAniso)) {
		tmp = sqrtf(fmaxf(fmaf(dX_46_u, t_3, (dX_46_v * (dX_46_v * t_0))), t_2)) / floorf(maxAniso);
	} else {
		tmp = fabsf(fmaf(floorf(w), (floorf(h) * fmaf(dX_46_v, -dY_46_u, fmaf(dX_46_u, dY_46_v, 0.0f))), 0.0f)) / sqrtf(fmaxf((powf((dX_46_u * floorf(w)), 2.0f) + powf((dX_46_v * floorf(h)), 2.0f)), (powf((floorf(w) * dY_46_u), 2.0f) + powf((floorf(h) * dY_46_v), 2.0f))));
	}
	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 = floor(w) ^ Float32(2.0)
	t_2 = fma(dY_46_u, Float32(t_1 * dY_46_u), Float32(t_0 * Float32(dY_46_v * dY_46_v)))
	t_3 = Float32(dX_46_u * t_1)
	tmp = Float32(0.0)
	if (Float32(((Float32(dX_46_u * t_3) != Float32(dX_46_u * t_3)) ? t_2 : ((t_2 != t_2) ? Float32(dX_46_u * t_3) : max(Float32(dX_46_u * t_3), t_2))) / abs(fma(floor(h), Float32(floor(w) * fma(dX_46_u, dY_46_v, Float32(dY_46_u * Float32(-dX_46_v)))), Float32(0.0)))) > floor(maxAniso))
		tmp = Float32(sqrt(((fma(dX_46_u, t_3, Float32(dX_46_v * Float32(dX_46_v * t_0))) != fma(dX_46_u, t_3, Float32(dX_46_v * Float32(dX_46_v * t_0)))) ? t_2 : ((t_2 != t_2) ? fma(dX_46_u, t_3, Float32(dX_46_v * Float32(dX_46_v * t_0))) : max(fma(dX_46_u, t_3, Float32(dX_46_v * Float32(dX_46_v * t_0))), t_2)))) / floor(maxAniso));
	else
		tmp = Float32(abs(fma(floor(w), Float32(floor(h) * fma(dX_46_v, Float32(-dY_46_u), fma(dX_46_u, dY_46_v, Float32(0.0)))), Float32(0.0))) / sqrt(((Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dX_46_v * floor(h)) ^ Float32(2.0))) != Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dX_46_v * floor(h)) ^ Float32(2.0)))) ? Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + (Float32(floor(h) * dY_46_v) ^ Float32(2.0))) : ((Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + (Float32(floor(h) * dY_46_v) ^ Float32(2.0))) != Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + (Float32(floor(h) * dY_46_v) ^ Float32(2.0)))) ? Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dX_46_v * floor(h)) ^ Float32(2.0))) : max(Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dX_46_v * floor(h)) ^ Float32(2.0))), Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + (Float32(floor(h) * dY_46_v) ^ Float32(2.0))))))));
	end
	return log2(tmp)
end

Derivation

1. Initial program 79.0%

   \[\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

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

5. Simplified 79.0%

   \[\leadsto \log_{2} \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}\\ } \end{array}} \]
6. Step-by-step derivation

   1. sqrt-div N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }}\\ \end{array} \]

   2. metadata-eval N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }}\\ \end{array} \]

   4. sqrt-lowering-sqrt.f32 N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\color{blue}{\left\lfloor maxAniso\right\rfloor }}\\ \end{array} \]

   5. fmax-lowering-fmax.f32 N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor \color{blue}{maxAniso}\right\rfloor }\\ \end{array} \]

7. Applied egg-rr 79.1%

   \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right| \cdot \frac{1}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}}}\\ \end{array} \]

8. Taylor expanded in dX.u around inf

   \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]
9. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   2. associate-*l* N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   5. pow-lowering-pow.f32 N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   6. floor-lowering-floor.f32 70.6

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right| \cdot \frac{1}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}}\\ \end{array} \]

10. Simplified 70.6%

    \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right| \cdot \frac{1}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}}\\ \end{array} \]

12. Applied egg-rr 70.6%

    \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\mathsf{fma}\left(\left\lfloor w\right\rfloor , \mathsf{fma}\left(dX.v, -dY.u, \mathsf{fma}\left(dX.u, dY.v, 0\right)\right) \cdot \left\lfloor h\right\rfloor , 0\right)\right|}{\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}}\\ \end{array} \]

13. Final simplification 70.6%

    \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dY.u \cdot \left(-dX.v\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\mathsf{fma}\left(\left\lfloor w\right\rfloor , \left\lfloor h\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, \mathsf{fma}\left(dX.u, dY.v, 0\right)\right), 0\right)\right|}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}}\\ \end{array} \]
14. Add Preprocessing

Alternative 5: 67.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\\ t_1 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_2 := {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\\ t_3 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(t\_2, t\_0\right)}{\left|\mathsf{fma}\left(\left\lfloor w\right\rfloor , \left\lfloor h\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, \mathsf{fma}\left(dX.u, dY.v, 0\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot t\_1, dX.v \cdot \left(dX.v \cdot t\_3\right)\right), \mathsf{fma}\left(dY.u, t\_1 \cdot dY.u, t\_3 \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dY.u \cdot \left(-dX.v\right)\right), 0\right)\right| \cdot \frac{1}{\sqrt{\mathsf{max}\left(t\_2 + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, t\_0\right)}}\\ \end{array} \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (+ (pow (* (floor w) dY.u) 2.0) (pow (* (floor h) dY.v) 2.0)))
        (t_1 (pow (floor w) 2.0))
        (t_2 (pow (* dX.u (floor w)) 2.0))
        (t_3 (pow (floor h) 2.0)))
   (log2
    (if (>
         (/
          (fmax t_2 t_0)
          (fabs
           (fma
            (floor w)
            (* (floor h) (fma dX.v (- dY.u) (fma dX.u dY.v 0.0)))
            0.0)))
         (floor maxAniso))
      (/
       (sqrt
        (fmax
         (fma dX.u (* dX.u t_1) (* dX.v (* dX.v t_3)))
         (fma dY.u (* t_1 dY.u) (* t_3 (* dY.v dY.v)))))
       (floor maxAniso))
      (*
       (fabs
        (fma (floor h) (* (floor w) (fma dX.u dY.v (* dY.u (- dX.v)))) 0.0))
       (/ 1.0 (sqrt (fmax (+ t_2 (pow (* dX.v (floor h)) 2.0)) t_0))))))))

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) * dY_46_u), 2.0f) + powf((floorf(h) * dY_46_v), 2.0f);
	float t_1 = powf(floorf(w), 2.0f);
	float t_2 = powf((dX_46_u * floorf(w)), 2.0f);
	float t_3 = powf(floorf(h), 2.0f);
	float tmp;
	if ((fmaxf(t_2, t_0) / fabsf(fmaf(floorf(w), (floorf(h) * fmaf(dX_46_v, -dY_46_u, fmaf(dX_46_u, dY_46_v, 0.0f))), 0.0f))) > floorf(maxAniso)) {
		tmp = sqrtf(fmaxf(fmaf(dX_46_u, (dX_46_u * t_1), (dX_46_v * (dX_46_v * t_3))), fmaf(dY_46_u, (t_1 * dY_46_u), (t_3 * (dY_46_v * dY_46_v))))) / floorf(maxAniso);
	} else {
		tmp = fabsf(fmaf(floorf(h), (floorf(w) * fmaf(dX_46_u, dY_46_v, (dY_46_u * -dX_46_v))), 0.0f)) * (1.0f / sqrtf(fmaxf((t_2 + powf((dX_46_v * floorf(h)), 2.0f)), t_0)));
	}
	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(floor(w) * dY_46_u) ^ Float32(2.0)) + (Float32(floor(h) * dY_46_v) ^ Float32(2.0)))
	t_1 = floor(w) ^ Float32(2.0)
	t_2 = Float32(dX_46_u * floor(w)) ^ Float32(2.0)
	t_3 = floor(h) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (Float32(((t_2 != t_2) ? t_0 : ((t_0 != t_0) ? t_2 : max(t_2, t_0))) / abs(fma(floor(w), Float32(floor(h) * fma(dX_46_v, Float32(-dY_46_u), fma(dX_46_u, dY_46_v, Float32(0.0)))), Float32(0.0)))) > floor(maxAniso))
		tmp = Float32(sqrt(((fma(dX_46_u, Float32(dX_46_u * t_1), Float32(dX_46_v * Float32(dX_46_v * t_3))) != fma(dX_46_u, Float32(dX_46_u * t_1), Float32(dX_46_v * Float32(dX_46_v * t_3)))) ? fma(dY_46_u, Float32(t_1 * dY_46_u), Float32(t_3 * Float32(dY_46_v * dY_46_v))) : ((fma(dY_46_u, Float32(t_1 * dY_46_u), Float32(t_3 * Float32(dY_46_v * dY_46_v))) != fma(dY_46_u, Float32(t_1 * dY_46_u), Float32(t_3 * Float32(dY_46_v * dY_46_v)))) ? fma(dX_46_u, Float32(dX_46_u * t_1), Float32(dX_46_v * Float32(dX_46_v * t_3))) : max(fma(dX_46_u, Float32(dX_46_u * t_1), Float32(dX_46_v * Float32(dX_46_v * t_3))), fma(dY_46_u, Float32(t_1 * dY_46_u), Float32(t_3 * Float32(dY_46_v * dY_46_v))))))) / floor(maxAniso));
	else
		tmp = Float32(abs(fma(floor(h), Float32(floor(w) * fma(dX_46_u, dY_46_v, Float32(dY_46_u * Float32(-dX_46_v)))), Float32(0.0))) * Float32(Float32(1.0) / sqrt(((Float32(t_2 + (Float32(dX_46_v * floor(h)) ^ Float32(2.0))) != Float32(t_2 + (Float32(dX_46_v * floor(h)) ^ Float32(2.0)))) ? t_0 : ((t_0 != t_0) ? Float32(t_2 + (Float32(dX_46_v * floor(h)) ^ Float32(2.0))) : max(Float32(t_2 + (Float32(dX_46_v * floor(h)) ^ Float32(2.0))), t_0))))));
	end
	return log2(tmp)
end

Derivation

1. Initial program 79.0%

   \[\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

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

5. Simplified 79.0%

   \[\leadsto \log_{2} \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right| \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}\\ } \end{array}} \]
6. Step-by-step derivation

   1. sqrt-div N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }}\\ \end{array} \]

   2. metadata-eval N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }}\\ \end{array} \]

   4. sqrt-lowering-sqrt.f32 N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\color{blue}{\left\lfloor maxAniso\right\rfloor }}\\ \end{array} \]

   5. fmax-lowering-fmax.f32 N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor \color{blue}{maxAniso}\right\rfloor }\\ \end{array} \]

7. Applied egg-rr 79.1%

   \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right| \cdot \frac{1}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}}}\\ \end{array} \]

8. Taylor expanded in dX.u around inf

   \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]
9. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   2. associate-*l* N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   5. pow-lowering-pow.f32 N/A

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(\mathsf{neg}\left(dY.u\right)\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \end{array} \]

   6. floor-lowering-floor.f32 70.6

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right| \cdot \frac{1}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}}\\ \end{array} \]

10. Simplified 70.6%

    \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}{\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right| \cdot \frac{1}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}}\\ \end{array} \]

12. Applied egg-rr 70.6%

    \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\color{blue}{\frac{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}{\left|\mathsf{fma}\left(\left\lfloor w\right\rfloor , \mathsf{fma}\left(dX.v, -dY.u, \mathsf{fma}\left(dX.u, dY.v, 0\right)\right) \cdot \left\lfloor h\right\rfloor , 0\right)\right|}} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dX.v \cdot \left(-dY.u\right)\right), 0\right)\right| \cdot \frac{1}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}}\\ \end{array} \]

13. Final simplification 70.6%

    \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}{\left|\mathsf{fma}\left(\left\lfloor w\right\rfloor , \left\lfloor h\right\rfloor \cdot \mathsf{fma}\left(dX.v, -dY.u, \mathsf{fma}\left(dX.u, dY.v, 0\right)\right), 0\right)\right|} > \left\lfloor maxAniso\right\rfloor :\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}{\left\lfloor maxAniso\right\rfloor }\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor w\right\rfloor \cdot \mathsf{fma}\left(dX.u, dY.v, dY.u \cdot \left(-dX.v\right)\right), 0\right)\right| \cdot \frac{1}{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}}\\ \end{array} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024197 
(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)))))))))