Anisotropic x16 LOD (LOD)

Percentage Accurate: 76.6% → 76.5%

Time: 31.6s

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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_1 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_2 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_3 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_4 := \mathsf{max}\left(t_3 \cdot t_3 + t_0 \cdot t_0, t_1 \cdot t_1 + t_2 \cdot t_2\right)\\ t_5 := \sqrt{t_4}\\ t_6 := \left|t_3 \cdot t_2 - t_0 \cdot t_1\right|\\ \log_{2} \begin{array}{l} \mathbf{if}\;\frac{t_4}{t_6} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{t_5}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_6}{t_5}\\ \end{array} \end{array} \end{array} \]

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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_1 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_2 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_3 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_4 := \mathsf{max}\left(t_3 \cdot t_3 + t_0 \cdot t_0, t_1 \cdot t_1 + t_2 \cdot t_2\right)\\ t_5 := \sqrt{t_4}\\ t_6 := \left|t_3 \cdot t_2 - t_0 \cdot t_1\right|\\ \log_{2} \begin{array}{l} \mathbf{if}\;\frac{t_4}{t_6} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{t_5}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_6}{t_5}\\ \end{array} \end{array} \end{array} \]

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: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|\\ t_1 := \mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dX.u \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)\\ t_2 := \sqrt{t_1}\\ \log_{2} \begin{array}{l} \mathbf{if}\;\frac{t_1}{t_0} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{t_2}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{t_2}\\ \end{array} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 73.5%

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

3. Simplified 73.6%

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

4. Final simplification 73.6%

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

Alternative 2: 74.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\\ t_1 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_2 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_3 := \mathsf{max}\left({t_1}^{2} + \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}, \mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot dY.u\right), {t_2}^{2}\right)\right)\\ \log_{2} \begin{array}{l} \mathbf{if}\;\frac{t_3}{\left|\left\lfloorw\right\rfloor \cdot t_0\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dX.u \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot t_1\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot t_2\right)\right)\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\frac{\sqrt{t_3}}{\left\lfloorw\right\rfloor}}\\ \end{array} \end{array} \end{array} \]

FPCore

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

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

Derivation

1. Initial program 73.5%

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

3. Simplified 73.6%

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

   1. expm1-log1p-u 73.5%

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dX.u - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dX.u - dX.v \cdot dY.u\right)\right)\right|}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}\right)\right)\\ \end{array} \]

   2. expm1-udef 73.5%

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dX.u - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dX.u - dX.v \cdot dY.u\right)\right)\right|}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}\right)} - 1\\ \end{array} \]

5. Applied egg-rr 70.4%

   \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dX.u - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dX.u \cdot dX.u\right), {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}}\right)} - 1\\ \end{array} \]

7. Simplified 70.4%

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

8. Taylor expanded in w around 0 70.4%

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

10. Simplified 70.4%

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

11. Final simplification 70.4%

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

Alternative 3: 44.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_1 := {t_0}^{2}\\ t_2 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_3 := {t_2}^{2}\\ t_4 := {\left(\left\lfloorw\right\rfloor\right)}^{2}\\ \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u \cdot dX.u, t_4, t_1\right), t_3 + t_4 \cdot \left(dY.u \cdot dY.u\right)\right)}{dX.u \cdot \left(\left\lfloorh\right\rfloor \cdot \left(\left\lfloorw\right\rfloor \cdot dY.v\right)\right)} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dX.u \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot t_0\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot t_2\right)\right)\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|}{\sqrt{\mathsf{max}\left(t_1 + \left(dX.u \cdot dX.u\right) \cdot t_4, \mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot dY.u\right), t_3\right)\right)}}\\ \end{array} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 73.5%

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

3. Simplified 73.6%

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

   1. frac-2neg 70.4%

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

   2. div-inv 70.4%

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

5. Applied egg-rr 45.7%

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

   1. associate-*r/ 37.0%

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

7. Simplified 45.7%

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

8. Taylor expanded in dX.v around 0 43.4%

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

   1. fma-def 36.8%

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

   2. unpow2 36.8%

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

   3. fma-udef 36.8%

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

   4. *-commutative 36.8%

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

   5. +-commutative 36.8%

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

   6. *-commutative 36.8%

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

   7. associate-*r* 36.8%

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

   8. unpow2 36.8%

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

   9. unpow2 36.8%

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

10. Simplified 43.4%

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

11. Taylor expanded in w around 0 43.4%

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

13. Simplified 43.4%

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

14. Final simplification 43.4%

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

Alternative 4: 38.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_1 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_2 := {t_1}^{2}\\ t_3 := {\left(\left\lfloorw\right\rfloor\right)}^{2}\\ t_4 := {t_0}^{2}\\ t_5 := \mathsf{max}\left(\mathsf{fma}\left(dX.u \cdot dX.u, t_3, t_4\right), t_2 + t_3 \cdot \left(dY.u \cdot dY.u\right)\right)\\ t_6 := \frac{\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)}{\frac{\sqrt{\mathsf{max}\left(t_4 + \left(dX.u \cdot dX.u\right) \cdot t_3, \mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot dY.u\right), t_2\right)\right)}}{\left\lfloorw\right\rfloor}}\\ t_7 := \frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dX.u \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot t_0\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot t_1\right)\right)\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{if}\;dX.v \leq 1.2000000104706032 \cdot 10^{-8}:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{t_5}{dX.u \cdot \left(\left\lfloorh\right\rfloor \cdot \left(\left\lfloorw\right\rfloor \cdot dY.v\right)\right)} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;t_7\\ \mathbf{else}:\\ \;\;\;\;t_6\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{-t_5}{dX.v \cdot \left(\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot dY.u\right)\right)} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;t_7\\ \mathbf{else}:\\ \;\;\;\;t_6\\ \end{array}\\ \end{array} \end{array} \]

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 h) dY.v))
        (t_2 (pow t_1 2.0))
        (t_3 (pow (floor w) 2.0))
        (t_4 (pow t_0 2.0))
        (t_5 (fmax (fma (* dX.u dX.u) t_3 t_4) (+ t_2 (* t_3 (* dY.u dY.u)))))
        (t_6
         (/
          (* (floor h) (- (* dX.u dY.v) (* dX.v dY.u)))
          (/
           (sqrt
            (fmax
             (+ t_4 (* (* dX.u dX.u) t_3))
             (fma (floor w) (* (floor w) (* dY.u dY.u)) t_2)))
           (floor w))))
        (t_7
         (/
          (sqrt
           (fmax
            (fma
             (floor w)
             (* (floor w) (* dX.u dX.u))
             (* (floor h) (* dX.v t_0)))
            (fma
             (floor w)
             (* dY.u (* (floor w) dY.u))
             (* (floor h) (* dY.v t_1)))))
          (floor maxAniso))))
   (if (<= dX.v 1.2000000104706032e-8)
     (log2
      (if (>
           (/ t_5 (* dX.u (* (floor h) (* (floor w) dY.v))))
           (floor maxAniso))
        t_7
        t_6))
     (log2
      (if (>
           (/ (- t_5) (* dX.v (* (floor w) (* (floor h) dY.u))))
           (floor maxAniso))
        t_7
        t_6)))))

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(h) * dY_46_v;
	float t_2 = powf(t_1, 2.0f);
	float t_3 = powf(floorf(w), 2.0f);
	float t_4 = powf(t_0, 2.0f);
	float t_5 = fmaxf(fmaf((dX_46_u * dX_46_u), t_3, t_4), (t_2 + (t_3 * (dY_46_u * dY_46_u))));
	float t_6 = (floorf(h) * ((dX_46_u * dY_46_v) - (dX_46_v * dY_46_u))) / (sqrtf(fmaxf((t_4 + ((dX_46_u * dX_46_u) * t_3)), fmaf(floorf(w), (floorf(w) * (dY_46_u * dY_46_u)), t_2))) / floorf(w));
	float t_7 = sqrtf(fmaxf(fmaf(floorf(w), (floorf(w) * (dX_46_u * dX_46_u)), (floorf(h) * (dX_46_v * t_0))), fmaf(floorf(w), (dY_46_u * (floorf(w) * dY_46_u)), (floorf(h) * (dY_46_v * t_1))))) / floorf(maxAniso);
	float tmp_1;
	if (dX_46_v <= 1.2000000104706032e-8f) {
		float tmp_2;
		if ((t_5 / (dX_46_u * (floorf(h) * (floorf(w) * dY_46_v)))) > floorf(maxAniso)) {
			tmp_2 = t_7;
		} else {
			tmp_2 = t_6;
		}
		tmp_1 = log2f(tmp_2);
	} else {
		float tmp_3;
		if ((-t_5 / (dX_46_v * (floorf(w) * (floorf(h) * dY_46_u)))) > floorf(maxAniso)) {
			tmp_3 = t_7;
		} else {
			tmp_3 = t_6;
		}
		tmp_1 = log2f(tmp_3);
	}
	return tmp_1;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(floor(h) * dY_46_v)
	t_2 = t_1 ^ Float32(2.0)
	t_3 = floor(w) ^ Float32(2.0)
	t_4 = t_0 ^ Float32(2.0)
	t_5 = (fma(Float32(dX_46_u * dX_46_u), t_3, t_4) != fma(Float32(dX_46_u * dX_46_u), t_3, t_4)) ? Float32(t_2 + Float32(t_3 * Float32(dY_46_u * dY_46_u))) : ((Float32(t_2 + Float32(t_3 * Float32(dY_46_u * dY_46_u))) != Float32(t_2 + Float32(t_3 * Float32(dY_46_u * dY_46_u)))) ? fma(Float32(dX_46_u * dX_46_u), t_3, t_4) : max(fma(Float32(dX_46_u * dX_46_u), t_3, t_4), Float32(t_2 + Float32(t_3 * Float32(dY_46_u * dY_46_u)))))
	t_6 = Float32(Float32(floor(h) * Float32(Float32(dX_46_u * dY_46_v) - Float32(dX_46_v * dY_46_u))) / Float32(sqrt(((Float32(t_4 + Float32(Float32(dX_46_u * dX_46_u) * t_3)) != Float32(t_4 + Float32(Float32(dX_46_u * dX_46_u) * t_3))) ? fma(floor(w), Float32(floor(w) * Float32(dY_46_u * dY_46_u)), t_2) : ((fma(floor(w), Float32(floor(w) * Float32(dY_46_u * dY_46_u)), t_2) != fma(floor(w), Float32(floor(w) * Float32(dY_46_u * dY_46_u)), t_2)) ? Float32(t_4 + Float32(Float32(dX_46_u * dX_46_u) * t_3)) : max(Float32(t_4 + Float32(Float32(dX_46_u * dX_46_u) * t_3)), fma(floor(w), Float32(floor(w) * Float32(dY_46_u * dY_46_u)), t_2))))) / floor(w)))
	t_7 = Float32(sqrt(((fma(floor(w), Float32(floor(w) * Float32(dX_46_u * dX_46_u)), Float32(floor(h) * Float32(dX_46_v * t_0))) != fma(floor(w), Float32(floor(w) * Float32(dX_46_u * dX_46_u)), Float32(floor(h) * Float32(dX_46_v * t_0)))) ? fma(floor(w), Float32(dY_46_u * Float32(floor(w) * dY_46_u)), Float32(floor(h) * Float32(dY_46_v * t_1))) : ((fma(floor(w), Float32(dY_46_u * Float32(floor(w) * dY_46_u)), Float32(floor(h) * Float32(dY_46_v * t_1))) != fma(floor(w), Float32(dY_46_u * Float32(floor(w) * dY_46_u)), Float32(floor(h) * Float32(dY_46_v * t_1)))) ? fma(floor(w), Float32(floor(w) * Float32(dX_46_u * dX_46_u)), Float32(floor(h) * Float32(dX_46_v * t_0))) : max(fma(floor(w), Float32(floor(w) * Float32(dX_46_u * dX_46_u)), Float32(floor(h) * Float32(dX_46_v * t_0))), fma(floor(w), Float32(dY_46_u * Float32(floor(w) * dY_46_u)), Float32(floor(h) * Float32(dY_46_v * t_1))))))) / floor(maxAniso))
	tmp_1 = Float32(0.0)
	if (dX_46_v <= Float32(1.2000000104706032e-8))
		tmp_2 = Float32(0.0)
		if (Float32(t_5 / Float32(dX_46_u * Float32(floor(h) * Float32(floor(w) * dY_46_v)))) > floor(maxAniso))
			tmp_2 = t_7;
		else
			tmp_2 = t_6;
		end
		tmp_1 = log2(tmp_2);
	else
		tmp_3 = Float32(0.0)
		if (Float32(Float32(-t_5) / Float32(dX_46_v * Float32(floor(w) * Float32(floor(h) * dY_46_u)))) > floor(maxAniso))
			tmp_3 = t_7;
		else
			tmp_3 = t_6;
		end
		tmp_1 = log2(tmp_3);
	end
	return tmp_1
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 1.20000001e-8

   1. Initial program 72.4%

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

   3. Simplified 72.4%

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

      1. expm1-log1p-u 72.4%

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dX.u - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dX.u - dX.v \cdot dY.u\right)\right)\right|}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}\right)\right)\\ \end{array} \]

      2. expm1-udef 72.4%

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dX.u - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dX.u - dX.v \cdot dY.u\right)\right)\right|}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}\right)} - 1\\ \end{array} \]

   5. Applied egg-rr 69.4%

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dX.u - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dX.u \cdot dX.u\right), {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}}\right)} - 1\\ \end{array} \]

   7. Simplified 69.5%

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dX.u - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot \left(dX.u \cdot dX.u\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot dY.u\right), {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}}{\left\lfloorw\right\rfloor}}\\ \end{array} \]
   8. Step-by-step derivation

      1. frac-2neg 69.5%

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

      2. div-inv 69.5%

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

   9. Applied egg-rr 38.8%

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\color{blue}{\left(-\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dX.u \cdot dX.u\right), {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}\right)\right)\right) \cdot \frac{1}{\left(-\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right) \cdot \left\lfloorw\right\rfloor}} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot \left(dX.u \cdot dX.u\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot dY.u\right), {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}}{\left\lfloorw\right\rfloor}}\\ \end{array} \]
   10. Step-by-step derivation

       1. associate-*r/ 38.8%

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

   11. Simplified 38.8%

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

   12. Taylor expanded in dX.v around 0 41.6%

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\color{blue}{\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2} + {\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, \mathsf{fma}\left(\left\lfloorw\right\rfloor, {dY.u}^{2} \cdot \left\lfloorw\right\rfloor, {\left(dY.v \cdot \left\lfloorh\right\rfloor\right)}^{2}\right)\right)}{dX.u \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot \left\lfloorw\right\rfloor\right)\right)}} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot \left(dX.u \cdot dX.u\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot dY.u\right), {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}}{\left\lfloorw\right\rfloor}}\\ \end{array} \]
   13. Step-by-step derivation

       1. fma-def 41.6%

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

       2. unpow2 41.6%

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

       3. fma-udef 41.6%

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

       4. *-commutative 41.6%

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

       5. +-commutative 41.6%

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

       6. *-commutative 41.6%

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

       7. associate-*r* 41.6%

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

       8. unpow2 41.6%

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

       9. unpow2 41.6%

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

   14. Simplified 41.6%

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

   if 1.20000001e-8 < dX.v

   1. Initial program 75.8%

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

   3. Simplified 75.9%

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

      1. expm1-log1p-u 75.9%

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dX.u - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dX.u - dX.v \cdot dY.u\right)\right)\right|}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}\right)\right)\\ \end{array} \]

      2. expm1-udef 75.7%

         \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dX.u - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dX.u - dX.v \cdot dY.u\right)\right)\right|}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}\right)} - 1\\ \end{array} \]

   5. Applied egg-rr 72.2%

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dX.u - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dX.u \cdot dX.u\right), {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}}\right)} - 1\\ \end{array} \]

   7. Simplified 72.4%

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dX.u - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot \left(dX.u \cdot dX.u\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot dY.u\right), {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}}{\left\lfloorw\right\rfloor}}\\ \end{array} \]
   8. Step-by-step derivation

      1. frac-2neg 72.4%

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

      2. div-inv 72.4%

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

   9. Applied egg-rr 33.5%

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\color{blue}{\left(-\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dX.u \cdot dX.u\right), {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}\right)\right)\right) \cdot \frac{1}{\left(-\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right) \cdot \left\lfloorw\right\rfloor}} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot \left(dX.u \cdot dX.u\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot dY.u\right), {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}}{\left\lfloorw\right\rfloor}}\\ \end{array} \]
   10. Step-by-step derivation

       1. associate-*r/ 33.5%

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

   11. Simplified 33.5%

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

   12. Taylor expanded in dX.v around inf 37.5%

       \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\color{blue}{-1 \cdot \frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2} + {\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, \mathsf{fma}\left(\left\lfloorw\right\rfloor, {dY.u}^{2} \cdot \left\lfloorw\right\rfloor, {\left(dY.v \cdot \left\lfloorh\right\rfloor\right)}^{2}\right)\right)}{dX.v \cdot \left(dY.u \cdot \left(\left\lfloorh\right\rfloor \cdot \left\lfloorw\right\rfloor\right)\right)}} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot \left(dX.u \cdot dX.u\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot dY.u\right), {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}}{\left\lfloorw\right\rfloor}}\\ \end{array} \]
   13. Step-by-step derivation

       1. associate-*r/ 37.5%

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

   14. Simplified 37.5%

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

4. Final simplification 40.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 1.2000000104706032 \cdot 10^{-8}:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u \cdot dX.u, {\left(\left\lfloorw\right\rfloor\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}\right), {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)}{dX.u \cdot \left(\left\lfloorh\right\rfloor \cdot \left(\left\lfloorw\right\rfloor \cdot dY.v\right)\right)} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dX.u \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)}{\frac{\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2} + \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}, \mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot dY.u\right), {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}}{\left\lfloorw\right\rfloor}}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{-\mathsf{max}\left(\mathsf{fma}\left(dX.u \cdot dX.u, {\left(\left\lfloorw\right\rfloor\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}\right), {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)}{dX.v \cdot \left(\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot dY.u\right)\right)} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dX.u \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)}{\frac{\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2} + \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}, \mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot dY.u\right), {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}}{\left\lfloorw\right\rfloor}}\\ \end{array}\\ \end{array} \]

Alternative 5: 39.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_1 := {t_0}^{2}\\ t_2 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_3 := {t_2}^{2}\\ t_4 := {\left(\left\lfloorw\right\rfloor\right)}^{2}\\ \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(dX.u \cdot dX.u, t_4, t_1\right), t_3 + t_4 \cdot \left(dY.u \cdot dY.u\right)\right)}{dX.u \cdot \left(\left\lfloorh\right\rfloor \cdot \left(\left\lfloorw\right\rfloor \cdot dY.v\right)\right)} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dX.u \cdot dX.u\right), \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot t_0\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot t_2\right)\right)\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)}{\frac{\sqrt{\mathsf{max}\left(t_1 + \left(dX.u \cdot dX.u\right) \cdot t_4, \mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot dY.u\right), t_3\right)\right)}}{\left\lfloorw\right\rfloor}}\\ \end{array} \end{array} \end{array} \]

FPCore

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

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 = powf(t_0, 2.0f);
	float t_2 = floorf(h) * dY_46_v;
	float t_3 = powf(t_2, 2.0f);
	float t_4 = powf(floorf(w), 2.0f);
	float tmp;
	if ((fmaxf(fmaf((dX_46_u * dX_46_u), t_4, t_1), (t_3 + (t_4 * (dY_46_u * dY_46_u)))) / (dX_46_u * (floorf(h) * (floorf(w) * dY_46_v)))) > floorf(maxAniso)) {
		tmp = sqrtf(fmaxf(fmaf(floorf(w), (floorf(w) * (dX_46_u * dX_46_u)), (floorf(h) * (dX_46_v * t_0))), fmaf(floorf(w), (dY_46_u * (floorf(w) * dY_46_u)), (floorf(h) * (dY_46_v * t_2))))) / floorf(maxAniso);
	} else {
		tmp = (floorf(h) * ((dX_46_u * dY_46_v) - (dX_46_v * dY_46_u))) / (sqrtf(fmaxf((t_1 + ((dX_46_u * dX_46_u) * t_4)), fmaf(floorf(w), (floorf(w) * (dY_46_u * dY_46_u)), t_3))) / floorf(w));
	}
	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 = t_0 ^ Float32(2.0)
	t_2 = Float32(floor(h) * dY_46_v)
	t_3 = t_2 ^ Float32(2.0)
	t_4 = floor(w) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (Float32(((fma(Float32(dX_46_u * dX_46_u), t_4, t_1) != fma(Float32(dX_46_u * dX_46_u), t_4, t_1)) ? Float32(t_3 + Float32(t_4 * Float32(dY_46_u * dY_46_u))) : ((Float32(t_3 + Float32(t_4 * Float32(dY_46_u * dY_46_u))) != Float32(t_3 + Float32(t_4 * Float32(dY_46_u * dY_46_u)))) ? fma(Float32(dX_46_u * dX_46_u), t_4, t_1) : max(fma(Float32(dX_46_u * dX_46_u), t_4, t_1), Float32(t_3 + Float32(t_4 * Float32(dY_46_u * dY_46_u)))))) / Float32(dX_46_u * Float32(floor(h) * Float32(floor(w) * dY_46_v)))) > floor(maxAniso))
		tmp = Float32(sqrt(((fma(floor(w), Float32(floor(w) * Float32(dX_46_u * dX_46_u)), Float32(floor(h) * Float32(dX_46_v * t_0))) != fma(floor(w), Float32(floor(w) * Float32(dX_46_u * dX_46_u)), Float32(floor(h) * Float32(dX_46_v * t_0)))) ? fma(floor(w), Float32(dY_46_u * Float32(floor(w) * dY_46_u)), Float32(floor(h) * Float32(dY_46_v * t_2))) : ((fma(floor(w), Float32(dY_46_u * Float32(floor(w) * dY_46_u)), Float32(floor(h) * Float32(dY_46_v * t_2))) != fma(floor(w), Float32(dY_46_u * Float32(floor(w) * dY_46_u)), Float32(floor(h) * Float32(dY_46_v * t_2)))) ? fma(floor(w), Float32(floor(w) * Float32(dX_46_u * dX_46_u)), Float32(floor(h) * Float32(dX_46_v * t_0))) : max(fma(floor(w), Float32(floor(w) * Float32(dX_46_u * dX_46_u)), Float32(floor(h) * Float32(dX_46_v * t_0))), fma(floor(w), Float32(dY_46_u * Float32(floor(w) * dY_46_u)), Float32(floor(h) * Float32(dY_46_v * t_2))))))) / floor(maxAniso));
	else
		tmp = Float32(Float32(floor(h) * Float32(Float32(dX_46_u * dY_46_v) - Float32(dX_46_v * dY_46_u))) / Float32(sqrt(((Float32(t_1 + Float32(Float32(dX_46_u * dX_46_u) * t_4)) != Float32(t_1 + Float32(Float32(dX_46_u * dX_46_u) * t_4))) ? fma(floor(w), Float32(floor(w) * Float32(dY_46_u * dY_46_u)), t_3) : ((fma(floor(w), Float32(floor(w) * Float32(dY_46_u * dY_46_u)), t_3) != fma(floor(w), Float32(floor(w) * Float32(dY_46_u * dY_46_u)), t_3)) ? Float32(t_1 + Float32(Float32(dX_46_u * dX_46_u) * t_4)) : max(Float32(t_1 + Float32(Float32(dX_46_u * dX_46_u) * t_4)), fma(floor(w), Float32(floor(w) * Float32(dY_46_u * dY_46_u)), t_3))))) / floor(w)));
	end
	return log2(tmp)
end

Derivation

1. Initial program 73.5%

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

3. Simplified 73.6%

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

   1. expm1-log1p-u 73.5%

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dX.u - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dX.u - dX.v \cdot dY.u\right)\right)\right|}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}\right)\right)\\ \end{array} \]

   2. expm1-udef 73.5%

      \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dX.u - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dX.u - dX.v \cdot dY.u\right)\right)\right|}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}\right)} - 1\\ \end{array} \]

5. Applied egg-rr 70.4%

   \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dX.u - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dX.u \cdot dX.u\right), {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}}\right)} - 1\\ \end{array} \]

7. Simplified 70.4%

   \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dY.v \cdot dX.u - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot \left(dX.u \cdot dX.u\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot dY.u\right), {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}}{\left\lfloorw\right\rfloor}}\\ \end{array} \]
8. Step-by-step derivation

   1. frac-2neg 70.4%

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

   2. div-inv 70.4%

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

9. Applied egg-rr 37.0%

   \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\color{blue}{\left(-\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dX.u \cdot dX.u\right), {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}\right)\right)\right) \cdot \frac{1}{\left(-\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right) \cdot \left\lfloorw\right\rfloor}} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot \left(dX.u \cdot dX.u\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot dY.u\right), {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}}{\left\lfloorw\right\rfloor}}\\ \end{array} \]
10. Step-by-step derivation

    1. associate-*r/ 37.0%

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

11. Simplified 37.0%

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

12. Taylor expanded in dX.v around 0 36.8%

    \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\color{blue}{\frac{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2} + {\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, \mathsf{fma}\left(\left\lfloorw\right\rfloor, {dY.u}^{2} \cdot \left\lfloorw\right\rfloor, {\left(dY.v \cdot \left\lfloorh\right\rfloor\right)}^{2}\right)\right)}{dX.u \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot \left\lfloorw\right\rfloor\right)\right)}} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot \left(dX.u \cdot dX.u\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(dY.u \cdot dY.u\right), {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}}{\left\lfloorw\right\rfloor}}\\ \end{array} \]
13. Step-by-step derivation

    1. fma-def 36.8%

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

    2. unpow2 36.8%

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

    3. fma-udef 36.8%

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

    4. *-commutative 36.8%

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

    5. +-commutative 36.8%

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

    6. *-commutative 36.8%

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

    7. associate-*r* 36.8%

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

    8. unpow2 36.8%

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

    9. unpow2 36.8%

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

14. Simplified 36.8%

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

15. Final simplification 36.8%

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

Reproduce

herbie shell --seed 2023271 
(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)))))))))