Isotropic LOD (LOD)

Percentage Accurate: 68.0% → 68.0%

Time: 34.8s

Alternatives: 8

Speedup: 0.5×

Specification

\[\left(\left(\left(\left(\left(\left(\left(\left(1 \leq w \land w \leq 16384\right) \land \left(1 \leq h \land h \leq 16384\right)\right) \land \left(1 \leq d \land d \leq 4096\right)\right) \land \left(10^{-20} \leq \left|dX.u\right| \land \left|dX.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dX.v\right| \land \left|dX.v\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dX.w\right| \land \left|dX.w\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.u\right| \land \left|dY.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.v\right| \land \left|dY.v\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.w\right| \land \left|dY.w\right| \leq 10^{+20}\right)\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_1 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_2 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_3 := \left\lfloord\right\rfloor \cdot dY.w\\ t_4 := \left\lfloord\right\rfloor \cdot dX.w\\ t_5 := \left\lfloorw\right\rfloor \cdot dX.u\\ \log_{2} \left(\sqrt{\mathsf{max}\left(\left(t_5 \cdot t_5 + t_2 \cdot t_2\right) + t_4 \cdot t_4, \left(t_0 \cdot t_0 + t_1 \cdot t_1\right) + t_3 \cdot t_3\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor w) dY.u))
        (t_1 (* (floor h) dY.v))
        (t_2 (* (floor h) dX.v))
        (t_3 (* (floor d) dY.w))
        (t_4 (* (floor d) dX.w))
        (t_5 (* (floor w) dX.u)))
   (log2
    (sqrt
     (fmax
      (+ (+ (* t_5 t_5) (* t_2 t_2)) (* t_4 t_4))
      (+ (+ (* t_0 t_0) (* t_1 t_1)) (* t_3 t_3)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(w) * dY_46_u;
	float t_1 = floorf(h) * dY_46_v;
	float t_2 = floorf(h) * dX_46_v;
	float t_3 = floorf(d) * dY_46_w;
	float t_4 = floorf(d) * dX_46_w;
	float t_5 = floorf(w) * dX_46_u;
	return log2f(sqrtf(fmaxf((((t_5 * t_5) + (t_2 * t_2)) + (t_4 * t_4)), (((t_0 * t_0) + (t_1 * t_1)) + (t_3 * t_3)))));
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * dY_46_u)
	t_1 = Float32(floor(h) * dY_46_v)
	t_2 = Float32(floor(h) * dX_46_v)
	t_3 = Float32(floor(d) * dY_46_w)
	t_4 = Float32(floor(d) * dX_46_w)
	t_5 = Float32(floor(w) * dX_46_u)
	return log2(sqrt(((Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)) != Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4))) ? Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3)) : ((Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3)) != Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3))) ? Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)) : max(Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)), Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3)))))))
end

MATLAB

function tmp = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(w) * dY_46_u;
	t_1 = floor(h) * dY_46_v;
	t_2 = floor(h) * dX_46_v;
	t_3 = floor(d) * dY_46_w;
	t_4 = floor(d) * dX_46_w;
	t_5 = floor(w) * dX_46_u;
	tmp = log2(sqrt(max((((t_5 * t_5) + (t_2 * t_2)) + (t_4 * t_4)), (((t_0 * t_0) + (t_1 * t_1)) + (t_3 * t_3)))));
end

Initial Program: 68.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_1 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_2 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_3 := \left\lfloord\right\rfloor \cdot dY.w\\ t_4 := \left\lfloord\right\rfloor \cdot dX.w\\ t_5 := \left\lfloorw\right\rfloor \cdot dX.u\\ \log_{2} \left(\sqrt{\mathsf{max}\left(\left(t_5 \cdot t_5 + t_2 \cdot t_2\right) + t_4 \cdot t_4, \left(t_0 \cdot t_0 + t_1 \cdot t_1\right) + t_3 \cdot t_3\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor w) dY.u))
        (t_1 (* (floor h) dY.v))
        (t_2 (* (floor h) dX.v))
        (t_3 (* (floor d) dY.w))
        (t_4 (* (floor d) dX.w))
        (t_5 (* (floor w) dX.u)))
   (log2
    (sqrt
     (fmax
      (+ (+ (* t_5 t_5) (* t_2 t_2)) (* t_4 t_4))
      (+ (+ (* t_0 t_0) (* t_1 t_1)) (* t_3 t_3)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(w) * dY_46_u;
	float t_1 = floorf(h) * dY_46_v;
	float t_2 = floorf(h) * dX_46_v;
	float t_3 = floorf(d) * dY_46_w;
	float t_4 = floorf(d) * dX_46_w;
	float t_5 = floorf(w) * dX_46_u;
	return log2f(sqrtf(fmaxf((((t_5 * t_5) + (t_2 * t_2)) + (t_4 * t_4)), (((t_0 * t_0) + (t_1 * t_1)) + (t_3 * t_3)))));
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * dY_46_u)
	t_1 = Float32(floor(h) * dY_46_v)
	t_2 = Float32(floor(h) * dX_46_v)
	t_3 = Float32(floor(d) * dY_46_w)
	t_4 = Float32(floor(d) * dX_46_w)
	t_5 = Float32(floor(w) * dX_46_u)
	return log2(sqrt(((Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)) != Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4))) ? Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3)) : ((Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3)) != Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3))) ? Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)) : max(Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)), Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3)))))))
end

MATLAB

function tmp = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(w) * dY_46_u;
	t_1 = floor(h) * dY_46_v;
	t_2 = floor(h) * dX_46_v;
	t_3 = floor(d) * dY_46_w;
	t_4 = floor(d) * dX_46_w;
	t_5 = floor(w) * dX_46_u;
	tmp = log2(sqrt(max((((t_5 * t_5) + (t_2 * t_2)) + (t_4 * t_4)), (((t_0 * t_0) + (t_1 * t_1)) + (t_3 * t_3)))));
end

Alternative 1: 68.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_1 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_2 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_3 := \left\lfloord\right\rfloor \cdot dY.w\\ t_4 := \left\lfloord\right\rfloor \cdot dX.w\\ t_5 := \left\lfloorw\right\rfloor \cdot dX.u\\ \mathbf{if}\;\mathsf{max}\left(\left(t_5 \cdot t_5 + t_2 \cdot t_2\right) + t_4 \cdot t_4, \left(t_0 \cdot t_0 + t_1 \cdot t_1\right) + t_3 \cdot t_3\right) \leq \infty:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t_5, t_5, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot t_2\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot t_4\right), \mathsf{fma}\left(t_0, t_0, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot t_1\right)\right) + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot t_3\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({t_2}^{2}, {\left(\mathsf{hypot}\left(t_1, t_3\right)\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor w) dY.u))
        (t_1 (* (floor h) dY.v))
        (t_2 (* (floor h) dX.v))
        (t_3 (* (floor d) dY.w))
        (t_4 (* (floor d) dX.w))
        (t_5 (* (floor w) dX.u)))
   (if (<=
        (fmax
         (+ (+ (* t_5 t_5) (* t_2 t_2)) (* t_4 t_4))
         (+ (+ (* t_0 t_0) (* t_1 t_1)) (* t_3 t_3)))
        INFINITY)
     (log2
      (sqrt
       (fmax
        (+ (fma t_5 t_5 (* (floor h) (* dX.v t_2))) (* (floor d) (* dX.w t_4)))
        (+
         (fma t_0 t_0 (* (floor h) (* dY.v t_1)))
         (* (floor d) (* dY.w t_3))))))
     (log2 (sqrt (fmax (pow t_2 2.0) (pow (hypot t_1 t_3) 2.0)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(w) * dY_46_u;
	float t_1 = floorf(h) * dY_46_v;
	float t_2 = floorf(h) * dX_46_v;
	float t_3 = floorf(d) * dY_46_w;
	float t_4 = floorf(d) * dX_46_w;
	float t_5 = floorf(w) * dX_46_u;
	float tmp;
	if (fmaxf((((t_5 * t_5) + (t_2 * t_2)) + (t_4 * t_4)), (((t_0 * t_0) + (t_1 * t_1)) + (t_3 * t_3))) <= ((float) INFINITY)) {
		tmp = log2f(sqrtf(fmaxf((fmaf(t_5, t_5, (floorf(h) * (dX_46_v * t_2))) + (floorf(d) * (dX_46_w * t_4))), (fmaf(t_0, t_0, (floorf(h) * (dY_46_v * t_1))) + (floorf(d) * (dY_46_w * t_3))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf(t_2, 2.0f), powf(hypotf(t_1, t_3), 2.0f))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * dY_46_u)
	t_1 = Float32(floor(h) * dY_46_v)
	t_2 = Float32(floor(h) * dX_46_v)
	t_3 = Float32(floor(d) * dY_46_w)
	t_4 = Float32(floor(d) * dX_46_w)
	t_5 = Float32(floor(w) * dX_46_u)
	tmp = Float32(0.0)
	if (((Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)) != Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4))) ? Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3)) : ((Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3)) != Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3))) ? Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)) : max(Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)), Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3))))) <= Float32(Inf))
		tmp = log2(sqrt(((Float32(fma(t_5, t_5, Float32(floor(h) * Float32(dX_46_v * t_2))) + Float32(floor(d) * Float32(dX_46_w * t_4))) != Float32(fma(t_5, t_5, Float32(floor(h) * Float32(dX_46_v * t_2))) + Float32(floor(d) * Float32(dX_46_w * t_4)))) ? Float32(fma(t_0, t_0, Float32(floor(h) * Float32(dY_46_v * t_1))) + Float32(floor(d) * Float32(dY_46_w * t_3))) : ((Float32(fma(t_0, t_0, Float32(floor(h) * Float32(dY_46_v * t_1))) + Float32(floor(d) * Float32(dY_46_w * t_3))) != Float32(fma(t_0, t_0, Float32(floor(h) * Float32(dY_46_v * t_1))) + Float32(floor(d) * Float32(dY_46_w * t_3)))) ? Float32(fma(t_5, t_5, Float32(floor(h) * Float32(dX_46_v * t_2))) + Float32(floor(d) * Float32(dX_46_w * t_4))) : max(Float32(fma(t_5, t_5, Float32(floor(h) * Float32(dX_46_v * t_2))) + Float32(floor(d) * Float32(dX_46_w * t_4))), Float32(fma(t_0, t_0, Float32(floor(h) * Float32(dY_46_v * t_1))) + Float32(floor(d) * Float32(dY_46_w * t_3))))))));
	else
		tmp = log2(sqrt((((t_2 ^ Float32(2.0)) != (t_2 ^ Float32(2.0))) ? (hypot(t_1, t_3) ^ Float32(2.0)) : (((hypot(t_1, t_3) ^ Float32(2.0)) != (hypot(t_1, t_3) ^ Float32(2.0))) ? (t_2 ^ Float32(2.0)) : max((t_2 ^ Float32(2.0)), (hypot(t_1, t_3) ^ Float32(2.0)))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fmax.f32 (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (*.f32 (*.f32 (floor.f32 d) dX.w) (*.f32 (floor.f32 d) dX.w))) (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))) (*.f32 (*.f32 (floor.f32 d) dY.w) (*.f32 (floor.f32 d) dY.w)))) < +inf.0

   1. Initial program 63.0%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

   3. Simplified 63.1%

      \[\leadsto \color{blue}{\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right)} \]

   if +inf.0 < (fmax.f32 (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (*.f32 (*.f32 (floor.f32 d) dX.w) (*.f32 (floor.f32 d) dX.w))) (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))) (*.f32 (*.f32 (floor.f32 d) dY.w) (*.f32 (floor.f32 d) dY.w))))

   1. Initial program 63.0%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

   3. Simplified 63.1%

      \[\leadsto \color{blue}{\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right)} \]

   4. Taylor expanded in dY.u around 0 55.7%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]
   5. Step-by-step derivation

      1. *-commutative 55.7%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      2. unpow2 55.7%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{\left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)} \cdot {dY.v}^{2} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      3. unpow2 55.7%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right) \cdot \color{blue}{\left(dY.v \cdot dY.v\right)} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      4. swap-sqr 55.6%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      5. unpow2 55.6%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

   6. Simplified 55.6%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

   8. Applied egg-rr 55.2%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\log \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}\right) + {\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)}\right)} \]
   9. Step-by-step derivation

      1. add-exp-log 55.6%

         \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}\right) + {\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)} \]

   10. Applied egg-rr 55.6%

       \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.w \cdot \left\lfloord\right\rfloor, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right)} \]

   11. Taylor expanded in dX.v around inf 43.8%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]
   12. Step-by-step derivation

       1. *-commutative 43.8%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dX.v}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       2. unpow2 43.8%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)} \cdot {dX.v}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       3. unpow2 43.8%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right) \cdot \color{blue}{\left(dX.v \cdot dX.v\right)}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       4. swap-sqr 43.8%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       5. unpow2 43.8%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       6. *-commutative 43.8%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\color{blue}{\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

   13. Simplified 43.8%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.1%

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

Alternative 2: 68.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_1 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_2 := \left\lfloord\right\rfloor \cdot dY.w\\ t_3 := \left\lfloord\right\rfloor \cdot dX.w\\ t_4 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_5 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_6 := {t_1}^{2}\\ \mathbf{if}\;\mathsf{max}\left(\left(t_4 \cdot t_4 + t_1 \cdot t_1\right) + t_3 \cdot t_3, \left(t_5 \cdot t_5 + t_0 \cdot t_0\right) + t_2 \cdot t_2\right) \leq \infty:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({t_3}^{2} + \left(t_6 + {t_4}^{2}\right), {t_5}^{2} + \left({t_0}^{2} + {t_2}^{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t_6, {\left(\mathsf{hypot}\left(t_0, t_2\right)\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

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

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(h) * dY_46_v;
	float t_1 = floorf(h) * dX_46_v;
	float t_2 = floorf(d) * dY_46_w;
	float t_3 = floorf(d) * dX_46_w;
	float t_4 = floorf(w) * dX_46_u;
	float t_5 = floorf(w) * dY_46_u;
	float t_6 = powf(t_1, 2.0f);
	float tmp;
	if (fmaxf((((t_4 * t_4) + (t_1 * t_1)) + (t_3 * t_3)), (((t_5 * t_5) + (t_0 * t_0)) + (t_2 * t_2))) <= ((float) INFINITY)) {
		tmp = log2f(sqrtf(fmaxf((powf(t_3, 2.0f) + (t_6 + powf(t_4, 2.0f))), (powf(t_5, 2.0f) + (powf(t_0, 2.0f) + powf(t_2, 2.0f))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(t_6, powf(hypotf(t_0, t_2), 2.0f))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(h) * dY_46_v)
	t_1 = Float32(floor(h) * dX_46_v)
	t_2 = Float32(floor(d) * dY_46_w)
	t_3 = Float32(floor(d) * dX_46_w)
	t_4 = Float32(floor(w) * dX_46_u)
	t_5 = Float32(floor(w) * dY_46_u)
	t_6 = t_1 ^ Float32(2.0)
	tmp = Float32(0.0)
	if (((Float32(Float32(Float32(t_4 * t_4) + Float32(t_1 * t_1)) + Float32(t_3 * t_3)) != Float32(Float32(Float32(t_4 * t_4) + Float32(t_1 * t_1)) + Float32(t_3 * t_3))) ? Float32(Float32(Float32(t_5 * t_5) + Float32(t_0 * t_0)) + Float32(t_2 * t_2)) : ((Float32(Float32(Float32(t_5 * t_5) + Float32(t_0 * t_0)) + Float32(t_2 * t_2)) != Float32(Float32(Float32(t_5 * t_5) + Float32(t_0 * t_0)) + Float32(t_2 * t_2))) ? Float32(Float32(Float32(t_4 * t_4) + Float32(t_1 * t_1)) + Float32(t_3 * t_3)) : max(Float32(Float32(Float32(t_4 * t_4) + Float32(t_1 * t_1)) + Float32(t_3 * t_3)), Float32(Float32(Float32(t_5 * t_5) + Float32(t_0 * t_0)) + Float32(t_2 * t_2))))) <= Float32(Inf))
		tmp = log2(sqrt(((Float32((t_3 ^ Float32(2.0)) + Float32(t_6 + (t_4 ^ Float32(2.0)))) != Float32((t_3 ^ Float32(2.0)) + Float32(t_6 + (t_4 ^ Float32(2.0))))) ? Float32((t_5 ^ Float32(2.0)) + Float32((t_0 ^ Float32(2.0)) + (t_2 ^ Float32(2.0)))) : ((Float32((t_5 ^ Float32(2.0)) + Float32((t_0 ^ Float32(2.0)) + (t_2 ^ Float32(2.0)))) != Float32((t_5 ^ Float32(2.0)) + Float32((t_0 ^ Float32(2.0)) + (t_2 ^ Float32(2.0))))) ? Float32((t_3 ^ Float32(2.0)) + Float32(t_6 + (t_4 ^ Float32(2.0)))) : max(Float32((t_3 ^ Float32(2.0)) + Float32(t_6 + (t_4 ^ Float32(2.0)))), Float32((t_5 ^ Float32(2.0)) + Float32((t_0 ^ Float32(2.0)) + (t_2 ^ Float32(2.0)))))))));
	else
		tmp = log2(sqrt(((t_6 != t_6) ? (hypot(t_0, t_2) ^ Float32(2.0)) : (((hypot(t_0, t_2) ^ Float32(2.0)) != (hypot(t_0, t_2) ^ Float32(2.0))) ? t_6 : max(t_6, (hypot(t_0, t_2) ^ Float32(2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(h) * dY_46_v;
	t_1 = floor(h) * dX_46_v;
	t_2 = floor(d) * dY_46_w;
	t_3 = floor(d) * dX_46_w;
	t_4 = floor(w) * dX_46_u;
	t_5 = floor(w) * dY_46_u;
	t_6 = t_1 ^ single(2.0);
	tmp = single(0.0);
	if (max((((t_4 * t_4) + (t_1 * t_1)) + (t_3 * t_3)), (((t_5 * t_5) + (t_0 * t_0)) + (t_2 * t_2))) <= single(Inf))
		tmp = log2(sqrt(max(((t_3 ^ single(2.0)) + (t_6 + (t_4 ^ single(2.0)))), ((t_5 ^ single(2.0)) + ((t_0 ^ single(2.0)) + (t_2 ^ single(2.0)))))));
	else
		tmp = log2(sqrt(max(t_6, (hypot(t_0, t_2) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (fmax.f32 (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (*.f32 (*.f32 (floor.f32 d) dX.w) (*.f32 (floor.f32 d) dX.w))) (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))) (*.f32 (*.f32 (floor.f32 d) dY.w) (*.f32 (floor.f32 d) dY.w)))) < +inf.0

   1. Initial program 63.0%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

   2. Taylor expanded in w around 0 63.0%

      \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2} + \left({dX.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2} + {dX.w}^{2} \cdot {\left(\left\lfloord\right\rfloor\right)}^{2}\right), {dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2} + \left({dY.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloord\right\rfloor\right)}^{2}\right)\right)}\right)} \]

   4. Simplified 63.0%

      \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left({\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2} + \left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}\right), {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + \left({\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)\right)}\right)} \]

   if +inf.0 < (fmax.f32 (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (*.f32 (*.f32 (floor.f32 d) dX.w) (*.f32 (floor.f32 d) dX.w))) (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))) (*.f32 (*.f32 (floor.f32 d) dY.w) (*.f32 (floor.f32 d) dY.w))))

   1. Initial program 63.0%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

   3. Simplified 63.1%

      \[\leadsto \color{blue}{\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right)} \]

   4. Taylor expanded in dY.u around 0 55.7%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]
   5. Step-by-step derivation

      1. *-commutative 55.7%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      2. unpow2 55.7%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{\left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)} \cdot {dY.v}^{2} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      3. unpow2 55.7%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right) \cdot \color{blue}{\left(dY.v \cdot dY.v\right)} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      4. swap-sqr 55.6%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      5. unpow2 55.6%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

   6. Simplified 55.6%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

   8. Applied egg-rr 55.2%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\log \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}\right) + {\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)}\right)} \]
   9. Step-by-step derivation

      1. add-exp-log 55.6%

         \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}\right) + {\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)} \]

   10. Applied egg-rr 55.6%

       \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.w \cdot \left\lfloord\right\rfloor, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right)} \]

   11. Taylor expanded in dX.v around inf 43.8%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]
   12. Step-by-step derivation

       1. *-commutative 43.8%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dX.v}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       2. unpow2 43.8%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)} \cdot {dX.v}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       3. unpow2 43.8%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right) \cdot \color{blue}{\left(dX.v \cdot dX.v\right)}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       4. swap-sqr 43.8%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       5. unpow2 43.8%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       6. *-commutative 43.8%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\color{blue}{\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

   13. Simplified 43.8%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right) \leq \infty:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2} + \left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}\right), {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + \left({\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right)\\ \end{array} \]

Alternative 3: 60.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (log2
  (sqrt
   (fmax
    (pow
     (hypot (* (floor d) dX.w) (hypot (* (floor w) dX.u) (* (floor h) dX.v)))
     2.0)
    (pow (hypot (* (floor h) dY.v) (* (floor d) dY.w)) 2.0)))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	return log2f(sqrtf(fmaxf(powf(hypotf((floorf(d) * dX_46_w), hypotf((floorf(w) * dX_46_u), (floorf(h) * dX_46_v))), 2.0f), powf(hypotf((floorf(h) * dY_46_v), (floorf(d) * dY_46_w)), 2.0f))));
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	return log2(sqrt((((hypot(Float32(floor(d) * dX_46_w), hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dX_46_w), 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(d) * dY_46_w)) ^ Float32(2.0)) : (((hypot(Float32(floor(h) * dY_46_v), Float32(floor(d) * dY_46_w)) ^ Float32(2.0)) != (hypot(Float32(floor(h) * dY_46_v), Float32(floor(d) * dY_46_w)) ^ Float32(2.0))) ? (hypot(Float32(floor(d) * dX_46_w), hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0)) : max((hypot(Float32(floor(d) * dX_46_w), 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(d) * dY_46_w)) ^ Float32(2.0)))))))
end

MATLAB

function tmp = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	tmp = log2(sqrt(max((hypot((floor(d) * dX_46_w), hypot((floor(w) * dX_46_u), (floor(h) * dX_46_v))) ^ single(2.0)), (hypot((floor(h) * dY_46_v), (floor(d) * dY_46_w)) ^ single(2.0)))));
end

Derivation

1. Initial program 63.0%

   \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

3. Simplified 63.1%

   \[\leadsto \color{blue}{\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right)} \]

4. Taylor expanded in dY.u around 0 55.7%

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]
5. Step-by-step derivation

   1. *-commutative 55.7%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

   2. unpow2 55.7%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{\left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)} \cdot {dY.v}^{2} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

   3. unpow2 55.7%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right) \cdot \color{blue}{\left(dY.v \cdot dY.v\right)} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

   4. swap-sqr 55.6%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

   5. unpow2 55.6%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

6. Simplified 55.6%

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

8. Applied egg-rr 55.2%

   \[\leadsto \log_{2} \color{blue}{\left(e^{\log \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}\right) + {\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)}\right)} \]
9. Step-by-step derivation

   1. add-exp-log 55.6%

      \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}\right) + {\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)} \]

10. Applied egg-rr 55.6%

    \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.w \cdot \left\lfloord\right\rfloor, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right)} \]

11. Final simplification 55.6%

    \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

Alternative 4: 55.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_1 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_2 := \left\lfloorh\right\rfloor \cdot dX.v\\ \mathbf{if}\;dY.w \leq 0.0003499999875202775:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(t_1, t_2\right)\right)\right)}^{2}, {t_0}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({t_2}^{2} + {t_1}^{2}, {\left(\mathsf{hypot}\left(t_0, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor h) dY.v))
        (t_1 (* (floor w) dX.u))
        (t_2 (* (floor h) dX.v)))
   (if (<= dY.w 0.0003499999875202775)
     (log2
      (sqrt
       (fmax
        (pow (hypot (* (floor d) dX.w) (hypot t_1 t_2)) 2.0)
        (pow t_0 2.0))))
     (log2
      (sqrt
       (fmax
        (+ (pow t_2 2.0) (pow t_1 2.0))
        (pow (hypot t_0 (* (floor d) dY.w)) 2.0)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(h) * dY_46_v;
	float t_1 = floorf(w) * dX_46_u;
	float t_2 = floorf(h) * dX_46_v;
	float tmp;
	if (dY_46_w <= 0.0003499999875202775f) {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf((floorf(d) * dX_46_w), hypotf(t_1, t_2)), 2.0f), powf(t_0, 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf((powf(t_2, 2.0f) + powf(t_1, 2.0f)), powf(hypotf(t_0, (floorf(d) * dY_46_w)), 2.0f))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(h) * dY_46_v)
	t_1 = Float32(floor(w) * dX_46_u)
	t_2 = Float32(floor(h) * dX_46_v)
	tmp = Float32(0.0)
	if (dY_46_w <= Float32(0.0003499999875202775))
		tmp = log2(sqrt((((hypot(Float32(floor(d) * dX_46_w), hypot(t_1, t_2)) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dX_46_w), hypot(t_1, t_2)) ^ Float32(2.0))) ? (t_0 ^ Float32(2.0)) : (((t_0 ^ Float32(2.0)) != (t_0 ^ Float32(2.0))) ? (hypot(Float32(floor(d) * dX_46_w), hypot(t_1, t_2)) ^ Float32(2.0)) : max((hypot(Float32(floor(d) * dX_46_w), hypot(t_1, t_2)) ^ Float32(2.0)), (t_0 ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt(((Float32((t_2 ^ Float32(2.0)) + (t_1 ^ Float32(2.0))) != Float32((t_2 ^ Float32(2.0)) + (t_1 ^ Float32(2.0)))) ? (hypot(t_0, Float32(floor(d) * dY_46_w)) ^ Float32(2.0)) : (((hypot(t_0, Float32(floor(d) * dY_46_w)) ^ Float32(2.0)) != (hypot(t_0, Float32(floor(d) * dY_46_w)) ^ Float32(2.0))) ? Float32((t_2 ^ Float32(2.0)) + (t_1 ^ Float32(2.0))) : max(Float32((t_2 ^ Float32(2.0)) + (t_1 ^ Float32(2.0))), (hypot(t_0, Float32(floor(d) * dY_46_w)) ^ Float32(2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(h) * dY_46_v;
	t_1 = floor(w) * dX_46_u;
	t_2 = floor(h) * dX_46_v;
	tmp = single(0.0);
	if (dY_46_w <= single(0.0003499999875202775))
		tmp = log2(sqrt(max((hypot((floor(d) * dX_46_w), hypot(t_1, t_2)) ^ single(2.0)), (t_0 ^ single(2.0)))));
	else
		tmp = log2(sqrt(max(((t_2 ^ single(2.0)) + (t_1 ^ single(2.0))), (hypot(t_0, (floor(d) * dY_46_w)) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.w < 3.49999988e-4

   1. Initial program 65.3%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

   3. Simplified 65.3%

      \[\leadsto \color{blue}{\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right)} \]

   4. Taylor expanded in dY.u around 0 56.4%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]
   5. Step-by-step derivation

      1. *-commutative 56.4%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      2. unpow2 56.4%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{\left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)} \cdot {dY.v}^{2} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      3. unpow2 56.4%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right) \cdot \color{blue}{\left(dY.v \cdot dY.v\right)} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      4. swap-sqr 56.4%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      5. unpow2 56.4%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

   6. Simplified 56.4%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

   8. Applied egg-rr 56.0%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\log \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}\right) + {\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)}\right)} \]
   9. Step-by-step derivation

      1. add-exp-log 56.4%

         \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}\right) + {\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)} \]

   10. Applied egg-rr 56.4%

       \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.w \cdot \left\lfloord\right\rfloor, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right)} \]

   11. Taylor expanded in dY.v around inf 51.0%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.w \cdot \left\lfloord\right\rfloor, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}}\right)}\right) \]
   12. Step-by-step derivation

       1. unpow2 51.0%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.w \cdot \left\lfloord\right\rfloor, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, \color{blue}{\left(dY.v \cdot dY.v\right)} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}\right)}\right) \]

       2. unpow2 51.0%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.w \cdot \left\lfloord\right\rfloor, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, \left(dY.v \cdot dY.v\right) \cdot \color{blue}{\left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)}\right)}\right) \]

       3. swap-sqr 51.0%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.w \cdot \left\lfloord\right\rfloor, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, \color{blue}{\left(dY.v \cdot \left\lfloorh\right\rfloor\right) \cdot \left(dY.v \cdot \left\lfloorh\right\rfloor\right)}\right)}\right) \]

       4. unpow2 51.0%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.w \cdot \left\lfloord\right\rfloor, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, \color{blue}{{\left(dY.v \cdot \left\lfloorh\right\rfloor\right)}^{2}}\right)}\right) \]

       5. *-commutative 51.0%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.w \cdot \left\lfloord\right\rfloor, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\color{blue}{\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}}^{2}\right)}\right) \]

   13. Simplified 51.0%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.w \cdot \left\lfloord\right\rfloor, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, \color{blue}{{\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}}\right)}\right) \]

   if 3.49999988e-4 < dY.w

   1. Initial program 57.6%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

   3. Simplified 57.6%

      \[\leadsto \color{blue}{\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right)} \]

   4. Taylor expanded in dY.u around 0 53.8%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]
   5. Step-by-step derivation

      1. *-commutative 53.8%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      2. unpow2 53.8%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{\left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)} \cdot {dY.v}^{2} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      3. unpow2 53.8%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right) \cdot \color{blue}{\left(dY.v \cdot dY.v\right)} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      4. swap-sqr 53.7%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      5. unpow2 53.7%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

   6. Simplified 53.7%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

   8. Applied egg-rr 53.2%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\log \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}\right) + {\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)}\right)} \]
   9. Step-by-step derivation

      1. add-exp-log 53.7%

         \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}\right) + {\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)} \]

   10. Applied egg-rr 53.8%

       \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.w \cdot \left\lfloord\right\rfloor, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right)} \]

   11. Taylor expanded in dX.w around 0 53.2%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]
   12. Step-by-step derivation

       1. *-commutative 53.2%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot {dX.u}^{2}} + {dX.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       2. unpow2 53.2%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right)} \cdot {dX.u}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       3. unpow2 53.2%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right) \cdot \color{blue}{\left(dX.u \cdot dX.u\right)} + {dX.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       4. swap-sqr 53.2%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)} + {dX.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       5. unpow2 53.2%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}} + {dX.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       6. *-commutative 53.2%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\color{blue}{\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       7. *-commutative 53.2%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + \color{blue}{{\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dX.v}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       8. unpow2 53.2%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + \color{blue}{\left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)} \cdot {dX.v}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       9. unpow2 53.2%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right) \cdot \color{blue}{\left(dX.v \cdot dX.v\right)}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       10. swap-sqr 53.2%

           \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + \color{blue}{\left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       11. unpow2 53.2%

           \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + \color{blue}{{\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       12. *-commutative 53.2%

           \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\color{blue}{\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

   13. Simplified 53.2%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} + {\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.w \leq 0.0003499999875202775:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2} + {\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right)\\ \end{array} \]

Alternative 5: 55.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloord\right\rfloor \cdot dY.w\\ t_1 := \left\lfloorh\right\rfloor \cdot dX.v\\ \mathbf{if}\;dY.v \leq 250000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, t_1\right)\right)\right)}^{2}, {t_0}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({t_1}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, t_0\right)\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor d) dY.w)) (t_1 (* (floor h) dX.v)))
   (if (<= dY.v 250000.0)
     (log2
      (sqrt
       (fmax
        (pow (hypot (* (floor d) dX.w) (hypot (* (floor w) dX.u) t_1)) 2.0)
        (pow t_0 2.0))))
     (log2
      (sqrt (fmax (pow t_1 2.0) (pow (hypot (* (floor h) dY.v) t_0) 2.0)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(d) * dY_46_w;
	float t_1 = floorf(h) * dX_46_v;
	float tmp;
	if (dY_46_v <= 250000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf((floorf(d) * dX_46_w), hypotf((floorf(w) * dX_46_u), t_1)), 2.0f), powf(t_0, 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf(t_1, 2.0f), powf(hypotf((floorf(h) * dY_46_v), t_0), 2.0f))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(d) * dY_46_w)
	t_1 = Float32(floor(h) * dX_46_v)
	tmp = Float32(0.0)
	if (dY_46_v <= Float32(250000.0))
		tmp = log2(sqrt((((hypot(Float32(floor(d) * dX_46_w), hypot(Float32(floor(w) * dX_46_u), t_1)) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dX_46_w), hypot(Float32(floor(w) * dX_46_u), t_1)) ^ Float32(2.0))) ? (t_0 ^ Float32(2.0)) : (((t_0 ^ Float32(2.0)) != (t_0 ^ Float32(2.0))) ? (hypot(Float32(floor(d) * dX_46_w), hypot(Float32(floor(w) * dX_46_u), t_1)) ^ Float32(2.0)) : max((hypot(Float32(floor(d) * dX_46_w), hypot(Float32(floor(w) * dX_46_u), t_1)) ^ Float32(2.0)), (t_0 ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt((((t_1 ^ Float32(2.0)) != (t_1 ^ Float32(2.0))) ? (hypot(Float32(floor(h) * dY_46_v), t_0) ^ Float32(2.0)) : (((hypot(Float32(floor(h) * dY_46_v), t_0) ^ Float32(2.0)) != (hypot(Float32(floor(h) * dY_46_v), t_0) ^ Float32(2.0))) ? (t_1 ^ Float32(2.0)) : max((t_1 ^ Float32(2.0)), (hypot(Float32(floor(h) * dY_46_v), t_0) ^ Float32(2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(d) * dY_46_w;
	t_1 = floor(h) * dX_46_v;
	tmp = single(0.0);
	if (dY_46_v <= single(250000.0))
		tmp = log2(sqrt(max((hypot((floor(d) * dX_46_w), hypot((floor(w) * dX_46_u), t_1)) ^ single(2.0)), (t_0 ^ single(2.0)))));
	else
		tmp = log2(sqrt(max((t_1 ^ single(2.0)), (hypot((floor(h) * dY_46_v), t_0) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.v < 2.5e5

   1. Initial program 65.8%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

   3. Simplified 65.9%

      \[\leadsto \color{blue}{\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right)} \]

   4. Taylor expanded in dY.u around 0 57.5%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]
   5. Step-by-step derivation

      1. *-commutative 57.5%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      2. unpow2 57.5%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{\left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)} \cdot {dY.v}^{2} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      3. unpow2 57.5%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right) \cdot \color{blue}{\left(dY.v \cdot dY.v\right)} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      4. swap-sqr 57.4%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      5. unpow2 57.4%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

   6. Simplified 57.4%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

   8. Applied egg-rr 57.0%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\log \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}\right) + {\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)}\right)} \]
   9. Step-by-step derivation

      1. add-exp-log 57.4%

         \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}\right) + {\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)} \]

   10. Applied egg-rr 57.4%

       \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.w \cdot \left\lfloord\right\rfloor, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right)} \]

   11. Taylor expanded in dY.v around 0 52.5%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.w \cdot \left\lfloord\right\rfloor, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloord\right\rfloor\right)}^{2}}\right)}\right) \]
   12. Step-by-step derivation

       1. unpow2 52.5%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.w \cdot \left\lfloord\right\rfloor, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, \color{blue}{\left(dY.w \cdot dY.w\right)} \cdot {\left(\left\lfloord\right\rfloor\right)}^{2}\right)}\right) \]

       2. unpow2 52.5%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.w \cdot \left\lfloord\right\rfloor, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, \left(dY.w \cdot dY.w\right) \cdot \color{blue}{\left(\left\lfloord\right\rfloor \cdot \left\lfloord\right\rfloor\right)}\right)}\right) \]

       3. swap-sqr 52.5%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.w \cdot \left\lfloord\right\rfloor, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, \color{blue}{\left(dY.w \cdot \left\lfloord\right\rfloor\right) \cdot \left(dY.w \cdot \left\lfloord\right\rfloor\right)}\right)}\right) \]

       4. unpow2 52.5%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.w \cdot \left\lfloord\right\rfloor, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, \color{blue}{{\left(dY.w \cdot \left\lfloord\right\rfloor\right)}^{2}}\right)}\right) \]

       5. *-commutative 52.5%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.w \cdot \left\lfloord\right\rfloor, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\color{blue}{\left(\left\lfloord\right\rfloor \cdot dY.w\right)}}^{2}\right)}\right) \]

   13. Simplified 52.5%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.w \cdot \left\lfloord\right\rfloor, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, \color{blue}{{\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}}\right)}\right) \]

   if 2.5e5 < dY.v

   1. Initial program 52.4%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

   3. Simplified 52.4%

      \[\leadsto \color{blue}{\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right)} \]

   4. Taylor expanded in dY.u around 0 48.7%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]
   5. Step-by-step derivation

      1. *-commutative 48.7%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      2. unpow2 48.7%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{\left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)} \cdot {dY.v}^{2} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      3. unpow2 48.7%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right) \cdot \color{blue}{\left(dY.v \cdot dY.v\right)} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      4. swap-sqr 48.7%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      5. unpow2 48.7%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

   6. Simplified 48.7%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

   8. Applied egg-rr 48.3%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\log \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}\right) + {\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)}\right)} \]
   9. Step-by-step derivation

      1. add-exp-log 48.7%

         \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}\right) + {\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)} \]

   10. Applied egg-rr 48.7%

       \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.w \cdot \left\lfloord\right\rfloor, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right)} \]

   11. Taylor expanded in dX.v around inf 45.9%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]
   12. Step-by-step derivation

       1. *-commutative 45.9%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dX.v}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       2. unpow2 45.9%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)} \cdot {dX.v}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       3. unpow2 45.9%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right) \cdot \color{blue}{\left(dX.v \cdot dX.v\right)}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       4. swap-sqr 45.9%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       5. unpow2 45.9%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       6. *-commutative 45.9%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\color{blue}{\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

   13. Simplified 45.9%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.v \leq 250000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right)\\ \end{array} \]

Alternative 6: 54.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_1 := \left\lfloorh\right\rfloor \cdot dX.v\\ \mathbf{if}\;dY.w \leq 0.019999999552965164:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, t_1\right)\right)\right)}^{2}, {t_0}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({t_1}^{2}, {\left(\mathsf{hypot}\left(t_0, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor h) dY.v)) (t_1 (* (floor h) dX.v)))
   (if (<= dY.w 0.019999999552965164)
     (log2
      (sqrt
       (fmax
        (pow (hypot (* (floor d) dX.w) (hypot (* (floor w) dX.u) t_1)) 2.0)
        (pow t_0 2.0))))
     (log2
      (sqrt (fmax (pow t_1 2.0) (pow (hypot t_0 (* (floor d) dY.w)) 2.0)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(h) * dY_46_v;
	float t_1 = floorf(h) * dX_46_v;
	float tmp;
	if (dY_46_w <= 0.019999999552965164f) {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf((floorf(d) * dX_46_w), hypotf((floorf(w) * dX_46_u), t_1)), 2.0f), powf(t_0, 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf(t_1, 2.0f), powf(hypotf(t_0, (floorf(d) * dY_46_w)), 2.0f))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(h) * dY_46_v)
	t_1 = Float32(floor(h) * dX_46_v)
	tmp = Float32(0.0)
	if (dY_46_w <= Float32(0.019999999552965164))
		tmp = log2(sqrt((((hypot(Float32(floor(d) * dX_46_w), hypot(Float32(floor(w) * dX_46_u), t_1)) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dX_46_w), hypot(Float32(floor(w) * dX_46_u), t_1)) ^ Float32(2.0))) ? (t_0 ^ Float32(2.0)) : (((t_0 ^ Float32(2.0)) != (t_0 ^ Float32(2.0))) ? (hypot(Float32(floor(d) * dX_46_w), hypot(Float32(floor(w) * dX_46_u), t_1)) ^ Float32(2.0)) : max((hypot(Float32(floor(d) * dX_46_w), hypot(Float32(floor(w) * dX_46_u), t_1)) ^ Float32(2.0)), (t_0 ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt((((t_1 ^ Float32(2.0)) != (t_1 ^ Float32(2.0))) ? (hypot(t_0, Float32(floor(d) * dY_46_w)) ^ Float32(2.0)) : (((hypot(t_0, Float32(floor(d) * dY_46_w)) ^ Float32(2.0)) != (hypot(t_0, Float32(floor(d) * dY_46_w)) ^ Float32(2.0))) ? (t_1 ^ Float32(2.0)) : max((t_1 ^ Float32(2.0)), (hypot(t_0, Float32(floor(d) * dY_46_w)) ^ Float32(2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(h) * dY_46_v;
	t_1 = floor(h) * dX_46_v;
	tmp = single(0.0);
	if (dY_46_w <= single(0.019999999552965164))
		tmp = log2(sqrt(max((hypot((floor(d) * dX_46_w), hypot((floor(w) * dX_46_u), t_1)) ^ single(2.0)), (t_0 ^ single(2.0)))));
	else
		tmp = log2(sqrt(max((t_1 ^ single(2.0)), (hypot(t_0, (floor(d) * dY_46_w)) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.w < 0.0199999996

   1. Initial program 65.1%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

   3. Simplified 65.1%

      \[\leadsto \color{blue}{\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right)} \]

   4. Taylor expanded in dY.u around 0 56.7%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]
   5. Step-by-step derivation

      1. *-commutative 56.7%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      2. unpow2 56.7%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{\left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)} \cdot {dY.v}^{2} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      3. unpow2 56.7%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right) \cdot \color{blue}{\left(dY.v \cdot dY.v\right)} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      4. swap-sqr 56.7%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      5. unpow2 56.7%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

   6. Simplified 56.7%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

   8. Applied egg-rr 56.2%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\log \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}\right) + {\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)}\right)} \]
   9. Step-by-step derivation

      1. add-exp-log 56.7%

         \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}\right) + {\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)} \]

   10. Applied egg-rr 56.7%

       \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.w \cdot \left\lfloord\right\rfloor, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right)} \]

   11. Taylor expanded in dY.v around inf 51.5%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.w \cdot \left\lfloord\right\rfloor, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}}\right)}\right) \]
   12. Step-by-step derivation

       1. unpow2 51.5%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.w \cdot \left\lfloord\right\rfloor, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, \color{blue}{\left(dY.v \cdot dY.v\right)} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}\right)}\right) \]

       2. unpow2 51.5%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.w \cdot \left\lfloord\right\rfloor, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, \left(dY.v \cdot dY.v\right) \cdot \color{blue}{\left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)}\right)}\right) \]

       3. swap-sqr 51.5%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.w \cdot \left\lfloord\right\rfloor, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, \color{blue}{\left(dY.v \cdot \left\lfloorh\right\rfloor\right) \cdot \left(dY.v \cdot \left\lfloorh\right\rfloor\right)}\right)}\right) \]

       4. unpow2 51.5%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.w \cdot \left\lfloord\right\rfloor, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, \color{blue}{{\left(dY.v \cdot \left\lfloorh\right\rfloor\right)}^{2}}\right)}\right) \]

       5. *-commutative 51.5%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.w \cdot \left\lfloord\right\rfloor, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\color{blue}{\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}}^{2}\right)}\right) \]

   13. Simplified 51.5%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.w \cdot \left\lfloord\right\rfloor, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, \color{blue}{{\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}}\right)}\right) \]

   if 0.0199999996 < dY.w

   1. Initial program 57.3%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

   3. Simplified 57.4%

      \[\leadsto \color{blue}{\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right)} \]

   4. Taylor expanded in dY.u around 0 52.8%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]
   5. Step-by-step derivation

      1. *-commutative 52.8%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      2. unpow2 52.8%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{\left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)} \cdot {dY.v}^{2} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      3. unpow2 52.8%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right) \cdot \color{blue}{\left(dY.v \cdot dY.v\right)} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      4. swap-sqr 52.8%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      5. unpow2 52.8%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

   6. Simplified 52.8%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

   8. Applied egg-rr 52.2%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\log \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}\right) + {\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)}\right)} \]
   9. Step-by-step derivation

      1. add-exp-log 52.8%

         \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}\right) + {\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)} \]

   10. Applied egg-rr 52.8%

       \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.w \cdot \left\lfloord\right\rfloor, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right)} \]

   11. Taylor expanded in dX.v around inf 52.1%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]
   12. Step-by-step derivation

       1. *-commutative 52.1%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dX.v}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       2. unpow2 52.1%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)} \cdot {dX.v}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       3. unpow2 52.1%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right) \cdot \color{blue}{\left(dX.v \cdot dX.v\right)}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       4. swap-sqr 52.1%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       5. unpow2 52.1%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       6. *-commutative 52.1%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\color{blue}{\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

   13. Simplified 52.1%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.w \leq 0.019999999552965164:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right)\\ \end{array} \]

Alternative 7: 48.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\\ \mathbf{if}\;dX.v \leq 120:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, t_0\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, t_0\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (hypot (* (floor h) dY.v) (* (floor d) dY.w)) 2.0)))
   (if (<= dX.v 120.0)
     (log2 (sqrt (fmax (pow (* (floor w) dX.u) 2.0) t_0)))
     (log2 (sqrt (fmax (pow (* (floor h) dX.v) 2.0) t_0))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = powf(hypotf((floorf(h) * dY_46_v), (floorf(d) * dY_46_w)), 2.0f);
	float tmp;
	if (dX_46_v <= 120.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(w) * dX_46_u), 2.0f), t_0)));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(h) * dX_46_v), 2.0f), t_0)));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = hypot(Float32(floor(h) * dY_46_v), Float32(floor(d) * dY_46_w)) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(120.0))
		tmp = log2(sqrt((((Float32(floor(w) * dX_46_u) ^ Float32(2.0)) != (Float32(floor(w) * dX_46_u) ^ Float32(2.0))) ? t_0 : ((t_0 != t_0) ? (Float32(floor(w) * dX_46_u) ^ Float32(2.0)) : max((Float32(floor(w) * dX_46_u) ^ Float32(2.0)), t_0)))));
	else
		tmp = log2(sqrt((((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) != (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) ? t_0 : ((t_0 != t_0) ? (Float32(floor(h) * dX_46_v) ^ Float32(2.0)) : max((Float32(floor(h) * dX_46_v) ^ Float32(2.0)), t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = hypot((floor(h) * dY_46_v), (floor(d) * dY_46_w)) ^ single(2.0);
	tmp = single(0.0);
	if (dX_46_v <= single(120.0))
		tmp = log2(sqrt(max(((floor(w) * dX_46_u) ^ single(2.0)), t_0)));
	else
		tmp = log2(sqrt(max(((floor(h) * dX_46_v) ^ single(2.0)), t_0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 120

   1. Initial program 65.6%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

   3. Simplified 65.6%

      \[\leadsto \color{blue}{\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right)} \]

   4. Taylor expanded in dY.u around 0 56.0%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]
   5. Step-by-step derivation

      1. *-commutative 56.0%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      2. unpow2 56.0%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{\left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)} \cdot {dY.v}^{2} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      3. unpow2 56.0%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right) \cdot \color{blue}{\left(dY.v \cdot dY.v\right)} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      4. swap-sqr 56.0%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      5. unpow2 56.0%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

   6. Simplified 56.0%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

   8. Applied egg-rr 55.5%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\log \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}\right) + {\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)}\right)} \]
   9. Step-by-step derivation

      1. add-exp-log 56.0%

         \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}\right) + {\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)} \]

   10. Applied egg-rr 56.0%

       \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.w \cdot \left\lfloord\right\rfloor, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right)} \]

   11. Taylor expanded in dX.u around inf 46.1%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]
   12. Step-by-step derivation

       1. *-commutative 46.1%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot {dX.u}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       2. unpow2 46.1%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right)} \cdot {dX.u}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       3. unpow2 46.1%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right) \cdot \color{blue}{\left(dX.u \cdot dX.u\right)}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       4. swap-sqr 46.1%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       5. unpow2 46.1%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       6. *-commutative 46.1%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\color{blue}{\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

   13. Simplified 46.1%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

   if 120 < dX.v

   1. Initial program 56.0%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

   3. Simplified 56.0%

      \[\leadsto \color{blue}{\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right)} \]

   4. Taylor expanded in dY.u around 0 54.7%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]
   5. Step-by-step derivation

      1. *-commutative 54.7%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      2. unpow2 54.7%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{\left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)} \cdot {dY.v}^{2} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      3. unpow2 54.7%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right) \cdot \color{blue}{\left(dY.v \cdot dY.v\right)} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      4. swap-sqr 54.7%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

      5. unpow2 54.7%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

   6. Simplified 54.7%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

   8. Applied egg-rr 54.2%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\log \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}\right) + {\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)}\right)} \]
   9. Step-by-step derivation

      1. add-exp-log 54.7%

         \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}\right) + {\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)} \]

   10. Applied egg-rr 54.7%

       \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.w \cdot \left\lfloord\right\rfloor, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right)} \]

   11. Taylor expanded in dX.v around inf 50.4%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]
   12. Step-by-step derivation

       1. *-commutative 50.4%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dX.v}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       2. unpow2 50.4%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)} \cdot {dX.v}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       3. unpow2 50.4%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right) \cdot \color{blue}{\left(dX.v \cdot dX.v\right)}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       4. swap-sqr 50.4%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       5. unpow2 50.4%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

       6. *-commutative 50.4%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\color{blue}{\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

   13. Simplified 50.4%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 120:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right)\\ \end{array} \]

Alternative 8: 45.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (log2
  (sqrt
   (fmax
    (pow (* (floor w) dX.u) 2.0)
    (pow (hypot (* (floor h) dY.v) (* (floor d) dY.w)) 2.0)))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	return log2f(sqrtf(fmaxf(powf((floorf(w) * dX_46_u), 2.0f), powf(hypotf((floorf(h) * dY_46_v), (floorf(d) * dY_46_w)), 2.0f))));
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	return log2(sqrt((((Float32(floor(w) * dX_46_u) ^ Float32(2.0)) != (Float32(floor(w) * dX_46_u) ^ Float32(2.0))) ? (hypot(Float32(floor(h) * dY_46_v), Float32(floor(d) * dY_46_w)) ^ Float32(2.0)) : (((hypot(Float32(floor(h) * dY_46_v), Float32(floor(d) * dY_46_w)) ^ Float32(2.0)) != (hypot(Float32(floor(h) * dY_46_v), Float32(floor(d) * dY_46_w)) ^ Float32(2.0))) ? (Float32(floor(w) * dX_46_u) ^ Float32(2.0)) : max((Float32(floor(w) * dX_46_u) ^ Float32(2.0)), (hypot(Float32(floor(h) * dY_46_v), Float32(floor(d) * dY_46_w)) ^ Float32(2.0)))))))
end

MATLAB

function tmp = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	tmp = log2(sqrt(max(((floor(w) * dX_46_u) ^ single(2.0)), (hypot((floor(h) * dY_46_v), (floor(d) * dY_46_w)) ^ single(2.0)))));
end

Derivation

1. Initial program 63.0%

   \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

3. Simplified 63.1%

   \[\leadsto \color{blue}{\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right)} \]

4. Taylor expanded in dY.u around 0 55.7%

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]
5. Step-by-step derivation

   1. *-commutative 55.7%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

   2. unpow2 55.7%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{\left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right)} \cdot {dY.v}^{2} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

   3. unpow2 55.7%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \left(\left\lfloorh\right\rfloor \cdot \left\lfloorh\right\rfloor\right) \cdot \color{blue}{\left(dY.v \cdot dY.v\right)} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

   4. swap-sqr 55.6%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

   5. unpow2 55.6%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

6. Simplified 55.6%

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) + \left\lfloord\right\rfloor \cdot \left(dX.w \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right)\right), \color{blue}{{\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}} + \left\lfloord\right\rfloor \cdot \left(dY.w \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)\right)}\right) \]

8. Applied egg-rr 55.2%

   \[\leadsto \log_{2} \color{blue}{\left(e^{\log \left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}\right) + {\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)}\right)} \]
9. Step-by-step derivation

   1. add-exp-log 55.6%

      \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left(\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} + {\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}\right) + {\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)} \]

10. Applied egg-rr 55.6%

    \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.w \cdot \left\lfloord\right\rfloor, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right)} \]

11. Taylor expanded in dX.u around inf 42.4%

    \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]
12. Step-by-step derivation

    1. *-commutative 42.4%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot {dX.u}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

    2. unpow2 42.4%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right)} \cdot {dX.u}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

    3. unpow2 42.4%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right) \cdot \color{blue}{\left(dX.u \cdot dX.u\right)}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

    4. swap-sqr 42.4%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

    5. unpow2 42.4%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

    6. *-commutative 42.4%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\color{blue}{\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

13. Simplified 42.4%

    \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

14. Final simplification 42.4%

    \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloord\right\rfloor \cdot dY.w\right)\right)}^{2}\right)}\right) \]

Reproduce

herbie shell --seed 2023301 
(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
  :name "Isotropic LOD (LOD)"
  :precision binary32
  :pre (and (and (and (and (and (and (and (and (and (<= 1.0 w) (<= w 16384.0)) (and (<= 1.0 h) (<= h 16384.0))) (and (<= 1.0 d) (<= d 4096.0))) (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 dX.w)) (<= (fabs dX.w) 1e+20))) (and (<= 1e-20 (fabs dY.u)) (<= (fabs dY.u) 1e+20))) (and (<= 1e-20 (fabs dY.v)) (<= (fabs dY.v) 1e+20))) (and (<= 1e-20 (fabs dY.w)) (<= (fabs dY.w) 1e+20)))
  (log2 (sqrt (fmax (+ (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (* (* (floor d) dX.w) (* (floor d) dX.w))) (+ (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v))) (* (* (floor d) dY.w) (* (floor d) dY.w)))))))