Anisotropic x16 LOD (LOD)

Percentage Accurate: 75.6% → 75.6%

Time: 45.5s

Alternatives: 8

Speedup: N/A×

Specification

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

Math

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

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: 75.6% 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: 75.6% accurate, 1.0× speedup

Math

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

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) (* (floor h) (* dX.v dX.v))))
          (fma
           (floor w)
           (* (floor w) (* dY.u dY.u))
           (* (floor h) (* (floor h) (* dY.v 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) * (floorf(h) * (dX_46_v * dX_46_v)))), fmaf(floorf(w), (floorf(w) * (dY_46_u * dY_46_u)), (floorf(h) * (floorf(h) * (dY_46_v * 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(floor(h) * Float32(dX_46_v * dX_46_v)))) != fma(floor(w), Float32(floor(w) * Float32(dX_46_u * dX_46_u)), Float32(floor(h) * Float32(floor(h) * Float32(dX_46_v * dX_46_v))))) ? fma(floor(w), Float32(floor(w) * Float32(dY_46_u * dY_46_u)), Float32(floor(h) * Float32(floor(h) * Float32(dY_46_v * dY_46_v)))) : ((fma(floor(w), Float32(floor(w) * Float32(dY_46_u * dY_46_u)), Float32(floor(h) * Float32(floor(h) * Float32(dY_46_v * dY_46_v)))) != fma(floor(w), Float32(floor(w) * Float32(dY_46_u * dY_46_u)), Float32(floor(h) * Float32(floor(h) * Float32(dY_46_v * dY_46_v))))) ? fma(floor(w), Float32(floor(w) * Float32(dX_46_u * dX_46_u)), Float32(floor(h) * Float32(floor(h) * Float32(dX_46_v * dX_46_v)))) : max(fma(floor(w), Float32(floor(w) * Float32(dX_46_u * dX_46_u)), Float32(floor(h) * Float32(floor(h) * Float32(dX_46_v * dX_46_v)))), fma(floor(w), Float32(floor(w) * Float32(dY_46_u * dY_46_u)), Float32(floor(h) * Float32(floor(h) * Float32(dY_46_v * 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 80.0%

   \[\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 80.1%

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

5. Final simplification 80.1%

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

Alternative 2: 61.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|\\ t_1 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_2 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_3 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_4 := {\left(\mathsf{hypot}\left(t\_3, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\\ t_5 := \mathsf{max}\left({\left(\mathsf{hypot}\left(t\_2, t\_1\right)\right)}^{2}, t\_4\right)\\ t_6 := \left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(t\_3 \cdot {t\_5}^{-0.5}\right)\right)\\ t_7 := \frac{\sqrt{t\_5}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{if}\;dX.u \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({t\_1}^{2}, t\_4\right)}{t\_0} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;t\_7\\ \mathbf{else}:\\ \;\;\;\;t\_6\\ \end{array}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({t\_2}^{2}, t\_4\right)}{t\_0} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;t\_7\\ \mathbf{else}:\\ \;\;\;\;t\_6\\ \end{array}\right)\right)\\ \end{array} \end{array} \]

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 (* (floor h) dX.v))
        (t_2 (* (floor w) dX.u))
        (t_3 (* (floor h) dY.v))
        (t_4 (pow (hypot t_3 (* (floor w) dY.u)) 2.0))
        (t_5 (fmax (pow (hypot t_2 t_1) 2.0) t_4))
        (t_6 (* (floor w) (* dX.u (* t_3 (pow t_5 -0.5)))))
        (t_7 (/ (sqrt t_5) (floor maxAniso))))
   (if (<= dX.u 0.20000000298023224)
     (expm1
      (log1p
       (log2
        (if (> (/ (fmax (pow t_1 2.0) t_4) t_0) (floor maxAniso)) t_7 t_6))))
     (expm1
      (log1p
       (log2
        (if (> (/ (fmax (pow t_2 2.0) t_4) t_0) (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 = fabsf((floorf(w) * (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(w) * dX_46_u;
	float t_3 = floorf(h) * dY_46_v;
	float t_4 = powf(hypotf(t_3, (floorf(w) * dY_46_u)), 2.0f);
	float t_5 = fmaxf(powf(hypotf(t_2, t_1), 2.0f), t_4);
	float t_6 = floorf(w) * (dX_46_u * (t_3 * powf(t_5, -0.5f)));
	float t_7 = sqrtf(t_5) / floorf(maxAniso);
	float tmp_1;
	if (dX_46_u <= 0.20000000298023224f) {
		float tmp_2;
		if ((fmaxf(powf(t_1, 2.0f), t_4) / t_0) > floorf(maxAniso)) {
			tmp_2 = t_7;
		} else {
			tmp_2 = t_6;
		}
		tmp_1 = expm1f(log1pf(log2f(tmp_2)));
	} else {
		float tmp_3;
		if ((fmaxf(powf(t_2, 2.0f), t_4) / t_0) > floorf(maxAniso)) {
			tmp_3 = t_7;
		} else {
			tmp_3 = t_6;
		}
		tmp_1 = expm1f(log1pf(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 = abs(Float32(floor(w) * 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(w) * dX_46_u)
	t_3 = Float32(floor(h) * dY_46_v)
	t_4 = hypot(t_3, Float32(floor(w) * dY_46_u)) ^ Float32(2.0)
	t_5 = ((hypot(t_2, t_1) ^ Float32(2.0)) != (hypot(t_2, t_1) ^ Float32(2.0))) ? t_4 : ((t_4 != t_4) ? (hypot(t_2, t_1) ^ Float32(2.0)) : max((hypot(t_2, t_1) ^ Float32(2.0)), t_4))
	t_6 = Float32(floor(w) * Float32(dX_46_u * Float32(t_3 * (t_5 ^ Float32(-0.5)))))
	t_7 = Float32(sqrt(t_5) / floor(maxAniso))
	tmp_1 = Float32(0.0)
	if (dX_46_u <= Float32(0.20000000298023224))
		tmp_2 = Float32(0.0)
		if (Float32((((t_1 ^ Float32(2.0)) != (t_1 ^ Float32(2.0))) ? t_4 : ((t_4 != t_4) ? (t_1 ^ Float32(2.0)) : max((t_1 ^ Float32(2.0)), t_4))) / t_0) > floor(maxAniso))
			tmp_2 = t_7;
		else
			tmp_2 = t_6;
		end
		tmp_1 = expm1(log1p(log2(tmp_2)));
	else
		tmp_3 = Float32(0.0)
		if (Float32((((t_2 ^ Float32(2.0)) != (t_2 ^ Float32(2.0))) ? t_4 : ((t_4 != t_4) ? (t_2 ^ Float32(2.0)) : max((t_2 ^ Float32(2.0)), t_4))) / t_0) > floor(maxAniso))
			tmp_3 = t_7;
		else
			tmp_3 = t_6;
		end
		tmp_1 = expm1(log1p(log2(tmp_3)));
	end
	return tmp_1
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 0.200000003

   1. Initial program 85.3%

      \[\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 85.3%

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

   6. Applied egg-rr 83.7%

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

   7. Taylor expanded in dX.u around inf 83.7%

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

   9. Simplified 83.7%

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

   11. Applied egg-rr 76.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot {dX.u}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot {dY.u}^{2}, \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot {dY.v}^{2}\right)\right)\right)}{\left|\left(\left\lfloorw\right\rfloor \cdot \left\lfloorh\right\rfloor\right) \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot {dX.u}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot {dY.u}^{2}, \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot {dY.v}^{2}\right)\right)\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\left(\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right) \cdot {\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{-0.5}\\ \end{array}\right)} - 1} \]

   13. Simplified 76.0%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot {\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{-0.5}\right)\right)\\ \end{array}\right)\right)} \]

   14. Taylor expanded in dX.u around 0 71.0%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot {\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{-0.5}\right)\right)\\ \end{array}\right)\right) \]
   15. Step-by-step derivation

       1. unpow2 71.0%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{\left(dX.v \cdot dX.v\right)} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot {\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{-0.5}\right)\right)\\ \end{array}\right)\right) \]

       2. unpow2 71.0%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot \color{blue}{\left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot {\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{-0.5}\right)\right)\\ \end{array}\right)\right) \]

       3. swap-sqr 71.0%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{\left(dX.v \cdot \left\lfloorh\right\rfloor\right) \cdot \left(dX.v \cdot \left\lfloorh\right\rfloor\right)}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot {\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{-0.5}\right)\right)\\ \end{array}\right)\right) \]

       4. unpow2 71.0%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{{\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot {\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{-0.5}\right)\right)\\ \end{array}\right)\right) \]

   16. Simplified 71.0%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{{\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot {\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{-0.5}\right)\right)\\ \end{array}\right)\right) \]

   if 0.200000003 < dX.u

   1. Initial program 65.9%

      \[\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 65.9%

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

   6. Applied egg-rr 64.5%

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

   7. Taylor expanded in dX.u around inf 64.5%

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

   9. Simplified 64.5%

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

   11. Applied egg-rr 61.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot {dX.u}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot {dY.u}^{2}, \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot {dY.v}^{2}\right)\right)\right)}{\left|\left(\left\lfloorw\right\rfloor \cdot \left\lfloorh\right\rfloor\right) \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot {dX.u}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot {dY.u}^{2}, \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot {dY.v}^{2}\right)\right)\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\left(\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right) \cdot {\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{-0.5}\\ \end{array}\right)} - 1} \]

   13. Simplified 61.2%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot {\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{-0.5}\right)\right)\\ \end{array}\right)\right)} \]

   14. Taylor expanded in dX.u around inf 59.8%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot {\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{-0.5}\right)\right)\\ \end{array}\right)\right) \]
   15. Step-by-step derivation

       1. unpow2 59.8%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right)} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot {\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{-0.5}\right)\right)\\ \end{array}\right)\right) \]

       2. unpow2 59.8%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \color{blue}{\left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right)}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot {\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{-0.5}\right)\right)\\ \end{array}\right)\right) \]

       3. swap-sqr 59.8%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot \left\lfloorw\right\rfloor\right) \cdot \left(dX.u \cdot \left\lfloorw\right\rfloor\right)}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot {\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{-0.5}\right)\right)\\ \end{array}\right)\right) \]

       4. unpow2 59.8%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{{\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot {\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{-0.5}\right)\right)\\ \end{array}\right)\right) \]

   16. Simplified 59.8%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{{\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot {\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{-0.5}\right)\right)\\ \end{array}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.u \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\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({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot {\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{-0.5}\right)\right)\\ \end{array}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\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({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot {\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{-0.5}\right)\right)\\ \end{array}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_1 := \mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(t\_0, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\\ \log_{2} \begin{array}{l} \mathbf{if}\;\frac{t\_1}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{t\_1}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(t\_0 \cdot {t\_1}^{-0.5}\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dY.v))
        (t_1
         (fmax
          (pow (hypot (* (floor w) dX.u) (* (floor h) dX.v)) 2.0)
          (pow (hypot t_0 (* (floor w) dY.u)) 2.0))))
   (log2
    (if (>
         (/
          t_1
          (fabs (* (floor w) (* (floor h) (- (* dX.u dY.v) (* dX.v dY.u))))))
         (floor maxAniso))
      (/ (sqrt t_1) (floor maxAniso))
      (* (floor w) (* dX.u (* t_0 (pow t_1 -0.5))))))))

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) * dY_46_v;
	float t_1 = fmaxf(powf(hypotf((floorf(w) * dX_46_u), (floorf(h) * dX_46_v)), 2.0f), powf(hypotf(t_0, (floorf(w) * dY_46_u)), 2.0f));
	float tmp;
	if ((t_1 / fabsf((floorf(w) * (floorf(h) * ((dX_46_u * dY_46_v) - (dX_46_v * dY_46_u)))))) > floorf(maxAniso)) {
		tmp = sqrtf(t_1) / floorf(maxAniso);
	} else {
		tmp = floorf(w) * (dX_46_u * (t_0 * powf(t_1, -0.5f)));
	}
	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) * dY_46_v)
	t_1 = ((hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v)) ^ Float32(2.0)) != (hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v)) ^ Float32(2.0))) ? (hypot(t_0, Float32(floor(w) * dY_46_u)) ^ Float32(2.0)) : (((hypot(t_0, Float32(floor(w) * dY_46_u)) ^ Float32(2.0)) != (hypot(t_0, Float32(floor(w) * dY_46_u)) ^ Float32(2.0))) ? (hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v)) ^ Float32(2.0)) : max((hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v)) ^ Float32(2.0)), (hypot(t_0, Float32(floor(w) * dY_46_u)) ^ Float32(2.0))))
	tmp = Float32(0.0)
	if (Float32(t_1 / abs(Float32(floor(w) * Float32(floor(h) * Float32(Float32(dX_46_u * dY_46_v) - Float32(dX_46_v * dY_46_u)))))) > floor(maxAniso))
		tmp = Float32(sqrt(t_1) / floor(maxAniso));
	else
		tmp = Float32(floor(w) * Float32(dX_46_u * Float32(t_0 * (t_1 ^ Float32(-0.5)))));
	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) * dY_46_v;
	t_1 = max((hypot((floor(w) * dX_46_u), (floor(h) * dX_46_v)) ^ single(2.0)), (hypot(t_0, (floor(w) * dY_46_u)) ^ single(2.0)));
	tmp = single(0.0);
	if ((t_1 / abs((floor(w) * (floor(h) * ((dX_46_u * dY_46_v) - (dX_46_v * dY_46_u)))))) > floor(maxAniso))
		tmp = sqrt(t_1) / floor(maxAniso);
	else
		tmp = floor(w) * (dX_46_u * (t_0 * (t_1 ^ single(-0.5))));
	end
	tmp_2 = log2(tmp);
end

Derivation

1. Initial program 80.0%

   \[\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 80.1%

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

6. Applied egg-rr 78.5%

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

7. Taylor expanded in dX.u around inf 78.5%

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

9. Simplified 78.5%

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

11. Applied egg-rr 78.5%

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

13. Simplified 78.5%

    \[\leadsto \color{blue}{\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot {\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{-0.5}\right)\right)\\ \end{array}} \]

14. Final simplification 78.5%

    \[\leadsto \log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot {\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{-0.5}\right)\right)\\ \end{array} \]
15. Add Preprocessing

Alternative 4: 56.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_1 := {\left(\mathsf{hypot}\left(t\_0, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\\ t_2 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_3 := \mathsf{max}\left({\left(\mathsf{hypot}\left(t\_2, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, t\_1\right)\\ \mathsf{expm1}\left(\mathsf{log1p}\left(\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({t\_2}^{2}, t\_1\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{t\_3}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(t\_0 \cdot {t\_3}^{-0.5}\right)\right)\\ \end{array}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dY.v))
        (t_1 (pow (hypot t_0 (* (floor w) dY.u)) 2.0))
        (t_2 (* (floor w) dX.u))
        (t_3 (fmax (pow (hypot t_2 (* (floor h) dX.v)) 2.0) t_1)))
   (expm1
    (log1p
     (log2
      (if (>
           (/
            (fmax (pow t_2 2.0) t_1)
            (fabs (* (floor w) (* (floor h) (- (* dX.u dY.v) (* dX.v dY.u))))))
           (floor maxAniso))
        (/ (sqrt t_3) (floor maxAniso))
        (* (floor w) (* dX.u (* t_0 (pow t_3 -0.5))))))))))

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) * dY_46_v;
	float t_1 = powf(hypotf(t_0, (floorf(w) * dY_46_u)), 2.0f);
	float t_2 = floorf(w) * dX_46_u;
	float t_3 = fmaxf(powf(hypotf(t_2, (floorf(h) * dX_46_v)), 2.0f), t_1);
	float tmp;
	if ((fmaxf(powf(t_2, 2.0f), t_1) / fabsf((floorf(w) * (floorf(h) * ((dX_46_u * dY_46_v) - (dX_46_v * dY_46_u)))))) > floorf(maxAniso)) {
		tmp = sqrtf(t_3) / floorf(maxAniso);
	} else {
		tmp = floorf(w) * (dX_46_u * (t_0 * powf(t_3, -0.5f)));
	}
	return expm1f(log1pf(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) * dY_46_v)
	t_1 = hypot(t_0, Float32(floor(w) * dY_46_u)) ^ Float32(2.0)
	t_2 = Float32(floor(w) * dX_46_u)
	t_3 = ((hypot(t_2, Float32(floor(h) * dX_46_v)) ^ Float32(2.0)) != (hypot(t_2, Float32(floor(h) * dX_46_v)) ^ Float32(2.0))) ? t_1 : ((t_1 != t_1) ? (hypot(t_2, Float32(floor(h) * dX_46_v)) ^ Float32(2.0)) : max((hypot(t_2, Float32(floor(h) * dX_46_v)) ^ Float32(2.0)), t_1))
	tmp = Float32(0.0)
	if (Float32((((t_2 ^ Float32(2.0)) != (t_2 ^ Float32(2.0))) ? t_1 : ((t_1 != t_1) ? (t_2 ^ Float32(2.0)) : max((t_2 ^ Float32(2.0)), t_1))) / abs(Float32(floor(w) * Float32(floor(h) * Float32(Float32(dX_46_u * dY_46_v) - Float32(dX_46_v * dY_46_u)))))) > floor(maxAniso))
		tmp = Float32(sqrt(t_3) / floor(maxAniso));
	else
		tmp = Float32(floor(w) * Float32(dX_46_u * Float32(t_0 * (t_3 ^ Float32(-0.5)))));
	end
	return expm1(log1p(log2(tmp)))
end

Derivation

1. Initial program 80.0%

   \[\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 80.1%

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

6. Applied egg-rr 78.5%

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

7. Taylor expanded in dX.u around inf 78.5%

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

9. Simplified 78.5%

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

11. Applied egg-rr 72.1%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot {dX.u}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot {dY.u}^{2}, \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot {dY.v}^{2}\right)\right)\right)}{\left|\left(\left\lfloorw\right\rfloor \cdot \left\lfloorh\right\rfloor\right) \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot {dX.u}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot {dY.u}^{2}, \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot {dY.v}^{2}\right)\right)\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\left(\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right) \cdot {\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{-0.5}\\ \end{array}\right)} - 1} \]

13. Simplified 72.0%

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot {\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{-0.5}\right)\right)\\ \end{array}\right)\right)} \]

14. Taylor expanded in dX.u around inf 62.3%

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot {\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{-0.5}\right)\right)\\ \end{array}\right)\right) \]
15. Step-by-step derivation

    1. unpow2 62.3%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right)} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot {\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{-0.5}\right)\right)\\ \end{array}\right)\right) \]

    2. unpow2 62.3%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \color{blue}{\left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right)}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot {\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{-0.5}\right)\right)\\ \end{array}\right)\right) \]

    3. swap-sqr 62.3%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot \left\lfloorw\right\rfloor\right) \cdot \left(dX.u \cdot \left\lfloorw\right\rfloor\right)}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot {\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{-0.5}\right)\right)\\ \end{array}\right)\right) \]

    4. unpow2 62.3%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{{\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot {\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{-0.5}\right)\right)\\ \end{array}\right)\right) \]

16. Simplified 62.3%

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{{\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot {\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{-0.5}\right)\right)\\ \end{array}\right)\right) \]

17. Final simplification 62.3%

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot {\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{-0.5}\right)\right)\\ \end{array}\right)\right) \]
18. Add Preprocessing

Alternative 5: 39.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_1 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_2 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_3 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_4 := \sqrt{\mathsf{max}\left({t\_3}^{2} + {t\_0}^{2}, {t\_1}^{2} + {t\_2}^{2}\right)}\\ t_5 := \frac{t\_4}{\left\lfloormaxAniso\right\rfloor}\\ t_6 := \mathsf{max}\left({\left(\mathsf{hypot}\left(t\_0, t\_3\right)\right)}^{2}, {\left(\mathsf{hypot}\left(t\_1, t\_2\right)\right)}^{2}\right)\\ t_7 := \left(\left\lfloorw\right\rfloor \cdot \left\lfloorh\right\rfloor\right) \cdot \frac{dX.u \cdot dY.v - dX.v \cdot dY.u}{t\_4}\\ \mathbf{if}\;dY.v \leq 200:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{t\_6}{\left(dX.v \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot \left(-\left\lfloorh\right\rfloor\right)\right)} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;t\_5\\ \mathbf{else}:\\ \;\;\;\;t\_7\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{t\_6}{\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot t\_1\right)} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;t\_5\\ \mathbf{else}:\\ \;\;\;\;t\_7\\ \end{array}\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor w) dX.u))
        (t_1 (* (floor h) dY.v))
        (t_2 (* (floor w) dY.u))
        (t_3 (* (floor h) dX.v))
        (t_4
         (sqrt
          (fmax
           (+ (pow t_3 2.0) (pow t_0 2.0))
           (+ (pow t_1 2.0) (pow t_2 2.0)))))
        (t_5 (/ t_4 (floor maxAniso)))
        (t_6 (fmax (pow (hypot t_0 t_3) 2.0) (pow (hypot t_1 t_2) 2.0)))
        (t_7
         (* (* (floor w) (floor h)) (/ (- (* dX.u dY.v) (* dX.v dY.u)) t_4))))
   (if (<= dY.v 200.0)
     (log2
      (if (>
           (/ t_6 (* (* dX.v dY.u) (* (floor w) (- (floor h)))))
           (floor maxAniso))
        t_5
        t_7))
     (log2
      (if (> (/ t_6 (* (floor w) (* dX.u t_1))) (floor maxAniso)) t_5 t_7)))))

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(w) * dX_46_u;
	float t_1 = floorf(h) * dY_46_v;
	float t_2 = floorf(w) * dY_46_u;
	float t_3 = floorf(h) * dX_46_v;
	float t_4 = sqrtf(fmaxf((powf(t_3, 2.0f) + powf(t_0, 2.0f)), (powf(t_1, 2.0f) + powf(t_2, 2.0f))));
	float t_5 = t_4 / floorf(maxAniso);
	float t_6 = fmaxf(powf(hypotf(t_0, t_3), 2.0f), powf(hypotf(t_1, t_2), 2.0f));
	float t_7 = (floorf(w) * floorf(h)) * (((dX_46_u * dY_46_v) - (dX_46_v * dY_46_u)) / t_4);
	float tmp_1;
	if (dY_46_v <= 200.0f) {
		float tmp_2;
		if ((t_6 / ((dX_46_v * dY_46_u) * (floorf(w) * -floorf(h)))) > floorf(maxAniso)) {
			tmp_2 = t_5;
		} else {
			tmp_2 = t_7;
		}
		tmp_1 = log2f(tmp_2);
	} else {
		float tmp_3;
		if ((t_6 / (floorf(w) * (dX_46_u * t_1))) > floorf(maxAniso)) {
			tmp_3 = t_5;
		} else {
			tmp_3 = t_7;
		}
		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(w) * dX_46_u)
	t_1 = Float32(floor(h) * dY_46_v)
	t_2 = Float32(floor(w) * dY_46_u)
	t_3 = Float32(floor(h) * dX_46_v)
	t_4 = sqrt(((Float32((t_3 ^ Float32(2.0)) + (t_0 ^ Float32(2.0))) != Float32((t_3 ^ Float32(2.0)) + (t_0 ^ Float32(2.0)))) ? Float32((t_1 ^ Float32(2.0)) + (t_2 ^ Float32(2.0))) : ((Float32((t_1 ^ Float32(2.0)) + (t_2 ^ Float32(2.0))) != Float32((t_1 ^ Float32(2.0)) + (t_2 ^ Float32(2.0)))) ? Float32((t_3 ^ Float32(2.0)) + (t_0 ^ Float32(2.0))) : max(Float32((t_3 ^ Float32(2.0)) + (t_0 ^ Float32(2.0))), Float32((t_1 ^ Float32(2.0)) + (t_2 ^ Float32(2.0)))))))
	t_5 = Float32(t_4 / floor(maxAniso))
	t_6 = ((hypot(t_0, t_3) ^ Float32(2.0)) != (hypot(t_0, t_3) ^ Float32(2.0))) ? (hypot(t_1, t_2) ^ Float32(2.0)) : (((hypot(t_1, t_2) ^ Float32(2.0)) != (hypot(t_1, t_2) ^ Float32(2.0))) ? (hypot(t_0, t_3) ^ Float32(2.0)) : max((hypot(t_0, t_3) ^ Float32(2.0)), (hypot(t_1, t_2) ^ Float32(2.0))))
	t_7 = Float32(Float32(floor(w) * floor(h)) * Float32(Float32(Float32(dX_46_u * dY_46_v) - Float32(dX_46_v * dY_46_u)) / t_4))
	tmp_1 = Float32(0.0)
	if (dY_46_v <= Float32(200.0))
		tmp_2 = Float32(0.0)
		if (Float32(t_6 / Float32(Float32(dX_46_v * dY_46_u) * Float32(floor(w) * Float32(-floor(h))))) > floor(maxAniso))
			tmp_2 = t_5;
		else
			tmp_2 = t_7;
		end
		tmp_1 = log2(tmp_2);
	else
		tmp_3 = Float32(0.0)
		if (Float32(t_6 / Float32(floor(w) * Float32(dX_46_u * t_1))) > floor(maxAniso))
			tmp_3 = t_5;
		else
			tmp_3 = t_7;
		end
		tmp_1 = log2(tmp_3);
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(w) * dX_46_u;
	t_1 = floor(h) * dY_46_v;
	t_2 = floor(w) * dY_46_u;
	t_3 = floor(h) * dX_46_v;
	t_4 = sqrt(max(((t_3 ^ single(2.0)) + (t_0 ^ single(2.0))), ((t_1 ^ single(2.0)) + (t_2 ^ single(2.0)))));
	t_5 = t_4 / floor(maxAniso);
	t_6 = max((hypot(t_0, t_3) ^ single(2.0)), (hypot(t_1, t_2) ^ single(2.0)));
	t_7 = (floor(w) * floor(h)) * (((dX_46_u * dY_46_v) - (dX_46_v * dY_46_u)) / t_4);
	tmp_2 = single(0.0);
	if (dY_46_v <= single(200.0))
		tmp_3 = single(0.0);
		if ((t_6 / ((dX_46_v * dY_46_u) * (floor(w) * -floor(h)))) > floor(maxAniso))
			tmp_3 = t_5;
		else
			tmp_3 = t_7;
		end
		tmp_2 = log2(tmp_3);
	else
		tmp_4 = single(0.0);
		if ((t_6 / (floor(w) * (dX_46_u * t_1))) > floor(maxAniso))
			tmp_4 = t_5;
		else
			tmp_4 = t_7;
		end
		tmp_2 = log2(tmp_4);
	end
	tmp_5 = tmp_2;
end

Derivation

1. Split input into 2 regimes

2. if dY.v < 200

   1. Initial program 81.5%

      \[\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 81.5%

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

   6. Applied egg-rr 40.3%

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

   8. Simplified 40.3%

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

      1. *-un-lft-identity 40.3%

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

   10. Applied egg-rr 40.3%

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

       1. *-lft-identity 40.3%

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

   12. Simplified 40.3%

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

   13. Taylor expanded in dX.v around inf 46.4%

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

   15. Simplified 46.4%

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

   if 200 < dY.v

   1. Initial program 74.4%

      \[\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 74.4%

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

   6. Applied egg-rr 43.0%

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

   8. Simplified 43.0%

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

      1. *-un-lft-identity 43.0%

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

   10. Applied egg-rr 43.0%

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

       1. *-lft-identity 43.0%

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

   12. Simplified 43.0%

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

   13. Taylor expanded in dX.v around 0 45.9%

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

   15. Simplified 45.9%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.v \leq 200:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}{\left(dX.v \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot \left(-\left\lfloorh\right\rfloor\right)\right)} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\left(\left\lfloorw\right\rfloor \cdot \left\lfloorh\right\rfloor\right) \cdot \frac{dX.u \cdot dY.v - dX.v \cdot dY.u}{\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}{\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\left(\left\lfloorw\right\rfloor \cdot \left\lfloorh\right\rfloor\right) \cdot \frac{dX.u \cdot dY.v - dX.v \cdot dY.u}{\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}}\\ \end{array}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 38.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\\ t_1 := \sqrt{t\_0}\\ t_2 := dX.u \cdot dY.v - dX.v \cdot dY.u\\ \log_{2} \begin{array}{l} \mathbf{if}\;\frac{t\_0}{\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot t\_2\right)} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{t\_1}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \frac{t\_2}{t\_1}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

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

C

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

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = ((hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v)) ^ Float32(2.0)) != (hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v)) ^ Float32(2.0))) ? (hypot(Float32(floor(h) * dY_46_v), Float32(floor(w) * dY_46_u)) ^ Float32(2.0)) : (((hypot(Float32(floor(h) * dY_46_v), Float32(floor(w) * dY_46_u)) ^ Float32(2.0)) != (hypot(Float32(floor(h) * dY_46_v), Float32(floor(w) * dY_46_u)) ^ Float32(2.0))) ? (hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v)) ^ Float32(2.0)) : max((hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v)) ^ Float32(2.0)), (hypot(Float32(floor(h) * dY_46_v), Float32(floor(w) * dY_46_u)) ^ Float32(2.0))))
	t_1 = sqrt(t_0)
	t_2 = Float32(Float32(dX_46_u * dY_46_v) - Float32(dX_46_v * dY_46_u))
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(floor(w) * Float32(floor(h) * t_2))) > floor(maxAniso))
		tmp = Float32(t_1 / floor(maxAniso));
	else
		tmp = Float32(floor(w) * Float32(floor(h) * Float32(t_2 / t_1)));
	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 = max((hypot((floor(w) * dX_46_u), (floor(h) * dX_46_v)) ^ single(2.0)), (hypot((floor(h) * dY_46_v), (floor(w) * dY_46_u)) ^ single(2.0)));
	t_1 = sqrt(t_0);
	t_2 = (dX_46_u * dY_46_v) - (dX_46_v * dY_46_u);
	tmp = single(0.0);
	if ((t_0 / (floor(w) * (floor(h) * t_2))) > floor(maxAniso))
		tmp = t_1 / floor(maxAniso);
	else
		tmp = floor(w) * (floor(h) * (t_2 / t_1));
	end
	tmp_2 = log2(tmp);
end

Derivation

1. Initial program 80.0%

   \[\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 80.1%

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

6. Applied egg-rr 40.9%

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

8. Simplified 40.9%

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

   1. *-un-lft-identity 40.9%

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

10. Applied egg-rr 40.9%

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

    1. *-lft-identity 40.9%

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

12. Simplified 40.9%

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

14. Applied egg-rr 40.9%

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

16. Simplified 40.9%

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

17. Final simplification 40.9%

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

Alternative 7: 39.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dY_46_v)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(w) * dX_46_u)
	t_3 = Float32(floor(h) * dX_46_v)
	t_4 = sqrt(((Float32((t_3 ^ Float32(2.0)) + (t_2 ^ Float32(2.0))) != Float32((t_3 ^ Float32(2.0)) + (t_2 ^ Float32(2.0)))) ? Float32((t_0 ^ Float32(2.0)) + (t_1 ^ Float32(2.0))) : ((Float32((t_0 ^ Float32(2.0)) + (t_1 ^ Float32(2.0))) != Float32((t_0 ^ Float32(2.0)) + (t_1 ^ Float32(2.0)))) ? Float32((t_3 ^ Float32(2.0)) + (t_2 ^ Float32(2.0))) : max(Float32((t_3 ^ Float32(2.0)) + (t_2 ^ Float32(2.0))), Float32((t_0 ^ Float32(2.0)) + (t_1 ^ Float32(2.0)))))))
	tmp = Float32(0.0)
	if (Float32((((hypot(t_2, t_3) ^ Float32(2.0)) != (hypot(t_2, t_3) ^ Float32(2.0))) ? (hypot(t_0, t_1) ^ Float32(2.0)) : (((hypot(t_0, t_1) ^ Float32(2.0)) != (hypot(t_0, t_1) ^ Float32(2.0))) ? (hypot(t_2, t_3) ^ Float32(2.0)) : max((hypot(t_2, t_3) ^ Float32(2.0)), (hypot(t_0, t_1) ^ Float32(2.0))))) / Float32(floor(w) * Float32(dX_46_u * t_0))) > floor(maxAniso))
		tmp = Float32(t_4 / floor(maxAniso));
	else
		tmp = Float32(Float32(floor(w) * floor(h)) * Float32(Float32(Float32(dX_46_u * dY_46_v) - Float32(dX_46_v * dY_46_u)) / t_4));
	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) * dY_46_v;
	t_1 = floor(w) * dY_46_u;
	t_2 = floor(w) * dX_46_u;
	t_3 = floor(h) * dX_46_v;
	t_4 = sqrt(max(((t_3 ^ single(2.0)) + (t_2 ^ single(2.0))), ((t_0 ^ single(2.0)) + (t_1 ^ single(2.0)))));
	tmp = single(0.0);
	if ((max((hypot(t_2, t_3) ^ single(2.0)), (hypot(t_0, t_1) ^ single(2.0))) / (floor(w) * (dX_46_u * t_0))) > floor(maxAniso))
		tmp = t_4 / floor(maxAniso);
	else
		tmp = (floor(w) * floor(h)) * (((dX_46_u * dY_46_v) - (dX_46_v * dY_46_u)) / t_4);
	end
	tmp_2 = log2(tmp);
end

Derivation

1. Initial program 80.0%

   \[\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 80.1%

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

6. Applied egg-rr 40.9%

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

8. Simplified 40.9%

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

   1. *-un-lft-identity 40.9%

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

10. Applied egg-rr 40.9%

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

    1. *-lft-identity 40.9%

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

12. Simplified 40.9%

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

13. Taylor expanded in dX.v around 0 40.8%

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

15. Simplified 40.8%

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

16. Final simplification 40.8%

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

Alternative 8: 39.1% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dY.v))
        (t_1 (* (floor w) dY.u))
        (t_2 (pow (* (floor w) dX.u) 2.0))
        (t_3
         (sqrt
          (fmax
           (+ (pow (* (floor h) dX.v) 2.0) t_2)
           (+ (pow t_0 2.0) (pow t_1 2.0))))))
   (log2
    (if (>
         (/ (fmax t_2 (pow (hypot t_0 t_1) 2.0)) (* (floor w) (* dX.u t_0)))
         (floor maxAniso))
      (/ t_3 (floor maxAniso))
      (* (* (floor w) (floor h)) (/ (- (* dX.u dY.v) (* dX.v 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) * dY_46_v;
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = powf((floorf(w) * dX_46_u), 2.0f);
	float t_3 = sqrtf(fmaxf((powf((floorf(h) * dX_46_v), 2.0f) + t_2), (powf(t_0, 2.0f) + powf(t_1, 2.0f))));
	float tmp;
	if ((fmaxf(t_2, powf(hypotf(t_0, t_1), 2.0f)) / (floorf(w) * (dX_46_u * t_0))) > floorf(maxAniso)) {
		tmp = t_3 / floorf(maxAniso);
	} else {
		tmp = (floorf(w) * floorf(h)) * (((dX_46_u * dY_46_v) - (dX_46_v * 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) * dY_46_v)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(w) * dX_46_u) ^ Float32(2.0)
	t_3 = sqrt(((Float32((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) + t_2) != Float32((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) + t_2)) ? Float32((t_0 ^ Float32(2.0)) + (t_1 ^ Float32(2.0))) : ((Float32((t_0 ^ Float32(2.0)) + (t_1 ^ Float32(2.0))) != Float32((t_0 ^ Float32(2.0)) + (t_1 ^ Float32(2.0)))) ? Float32((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) + t_2) : max(Float32((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) + t_2), Float32((t_0 ^ Float32(2.0)) + (t_1 ^ Float32(2.0)))))))
	tmp = Float32(0.0)
	if (Float32(((t_2 != t_2) ? (hypot(t_0, t_1) ^ Float32(2.0)) : (((hypot(t_0, t_1) ^ Float32(2.0)) != (hypot(t_0, t_1) ^ Float32(2.0))) ? t_2 : max(t_2, (hypot(t_0, t_1) ^ Float32(2.0))))) / Float32(floor(w) * Float32(dX_46_u * t_0))) > floor(maxAniso))
		tmp = Float32(t_3 / floor(maxAniso));
	else
		tmp = Float32(Float32(floor(w) * floor(h)) * Float32(Float32(Float32(dX_46_u * dY_46_v) - Float32(dX_46_v * dY_46_u)) / t_3));
	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) * dY_46_v;
	t_1 = floor(w) * dY_46_u;
	t_2 = (floor(w) * dX_46_u) ^ single(2.0);
	t_3 = sqrt(max((((floor(h) * dX_46_v) ^ single(2.0)) + t_2), ((t_0 ^ single(2.0)) + (t_1 ^ single(2.0)))));
	tmp = single(0.0);
	if ((max(t_2, (hypot(t_0, t_1) ^ single(2.0))) / (floor(w) * (dX_46_u * t_0))) > floor(maxAniso))
		tmp = t_3 / floor(maxAniso);
	else
		tmp = (floor(w) * floor(h)) * (((dX_46_u * dY_46_v) - (dX_46_v * dY_46_u)) / t_3);
	end
	tmp_2 = log2(tmp);
end

Derivation

1. Initial program 80.0%

   \[\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 80.1%

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

6. Applied egg-rr 40.9%

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

8. Simplified 40.9%

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

   1. *-un-lft-identity 40.9%

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

10. Applied egg-rr 40.9%

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

    1. *-lft-identity 40.9%

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

12. Simplified 40.9%

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

13. Taylor expanded in dX.v around 0 40.8%

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

15. Simplified 40.8%

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

16. Taylor expanded in dX.u around inf 40.0%

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

    1. unpow2 62.3%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right)} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot {\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{-0.5}\right)\right)\\ \end{array}\right)\right) \]

    2. unpow2 62.3%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \color{blue}{\left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right)}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot {\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{-0.5}\right)\right)\\ \end{array}\right)\right) \]

    3. swap-sqr 62.3%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot \left\lfloorw\right\rfloor\right) \cdot \left(dX.u \cdot \left\lfloorw\right\rfloor\right)}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot {\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{-0.5}\right)\right)\\ \end{array}\right)\right) \]

    4. unpow2 62.3%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\log_{2} \begin{array}{l} \mathbf{if}\;\frac{\mathsf{max}\left(\color{blue}{{\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}{\left|\left\lfloorw\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.u \cdot dY.v - dX.v \cdot dY.u\right)\right)\right|} > \left\lfloormaxAniso\right\rfloor:\\ \;\;\;\;\frac{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}{\left\lfloormaxAniso\right\rfloor}\\ \mathbf{else}:\\ \;\;\;\;\left\lfloorw\right\rfloor \cdot \left(dX.u \cdot \left(\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot {\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{-0.5}\right)\right)\\ \end{array}\right)\right) \]

18. Simplified 40.0%

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

19. Final simplification 40.0%

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

Reproduce

herbie shell --seed 2024139 
(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)))))))))