Isotropic LOD (LOD)

Percentage Accurate: 67.7% → 67.7%

Time: 27.0s

Alternatives: 13

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: 67.7% 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: 67.7% accurate, 0.5× speedup

Math

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

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
        (pow (hypot t_4 (hypot t_5 t_2)) 2.0)
        (pow (hypot t_3 (hypot t_0 t_1)) 2.0))))
     (log2
      (sqrt (fmax (pow t_4 2.0) (* (* dY.u dY.u) (pow (floor 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(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(powf(hypotf(t_4, hypotf(t_5, t_2)), 2.0f), powf(hypotf(t_3, hypotf(t_0, t_1)), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf(t_4, 2.0f), ((dY_46_u * dY_46_u) * powf(floorf(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(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((((hypot(t_4, hypot(t_5, t_2)) ^ Float32(2.0)) != (hypot(t_4, hypot(t_5, t_2)) ^ Float32(2.0))) ? (hypot(t_3, hypot(t_0, t_1)) ^ Float32(2.0)) : (((hypot(t_3, hypot(t_0, t_1)) ^ Float32(2.0)) != (hypot(t_3, hypot(t_0, t_1)) ^ Float32(2.0))) ? (hypot(t_4, hypot(t_5, t_2)) ^ Float32(2.0)) : max((hypot(t_4, hypot(t_5, t_2)) ^ Float32(2.0)), (hypot(t_3, hypot(t_0, t_1)) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt((((t_4 ^ Float32(2.0)) != (t_4 ^ Float32(2.0))) ? Float32(Float32(dY_46_u * dY_46_u) * (floor(w) ^ Float32(2.0))) : ((Float32(Float32(dY_46_u * dY_46_u) * (floor(w) ^ Float32(2.0))) != Float32(Float32(dY_46_u * dY_46_u) * (floor(w) ^ Float32(2.0)))) ? (t_4 ^ Float32(2.0)) : max((t_4 ^ Float32(2.0)), Float32(Float32(dY_46_u * dY_46_u) * (floor(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(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 = single(0.0);
	if (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))) <= single(Inf))
		tmp = log2(sqrt(max((hypot(t_4, hypot(t_5, t_2)) ^ single(2.0)), (hypot(t_3, hypot(t_0, t_1)) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max((t_4 ^ single(2.0)), ((dY_46_u * dY_46_u) * (floor(w) ^ 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 68.2%

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

      1. expm1-log1p-u 67.6%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)\right)\right)} \]

      2. expm1-udef 67.4%

         \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)} - 1\right)} \]

   3. Applied egg-rr 67.4%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   4. Step-by-step derivation

      1. expm1-def 67.6%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 68.2%

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

      3. *-commutative 68.2%

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

      4. *-commutative 68.2%

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

      5. *-commutative 68.2%

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

   5. Simplified 68.2%

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

   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 68.2%

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

      1. expm1-log1p-u 67.6%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)\right)\right)} \]

      2. expm1-udef 67.4%

         \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)} - 1\right)} \]

   3. Applied egg-rr 67.4%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   4. Step-by-step derivation

      1. expm1-def 67.6%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 68.2%

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

      3. *-commutative 68.2%

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

      4. *-commutative 68.2%

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

      5. *-commutative 68.2%

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

   5. Simplified 68.2%

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

   6. Taylor expanded in dY.u around inf 56.3%

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

      1. unpow2 56.3%

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

   8. Simplified 56.3%

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

   9. Taylor expanded in dX.w around inf 38.9%

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

4. Final simplification 68.2%

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

Alternative 2: 62.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloorw\right\rfloor\right)}^{2}\\ t_1 := \left\lfloord\right\rfloor \cdot dX.w\\ \mathbf{if}\;dX.v \leq 180:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t_0, dX.u \cdot dX.u, {t_1}^{2}\right), {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(t_1, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, \left(dY.u \cdot dY.u\right) \cdot t_0\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (floor w) 2.0)) (t_1 (* (floor d) dX.w)))
   (if (<= dX.v 180.0)
     (log2
      (sqrt
       (fmax
        (fma t_0 (* dX.u dX.u) (pow t_1 2.0))
        (pow
         (hypot
          (* (floor d) dY.w)
          (hypot (* (floor w) dY.u) (* (floor h) dY.v)))
         2.0))))
     (log2
      (sqrt
       (fmax
        (pow (hypot t_1 (hypot (* (floor w) dX.u) (* (floor h) dX.v))) 2.0)
        (* (* dY.u dY.u) 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(floorf(w), 2.0f);
	float t_1 = floorf(d) * dX_46_w;
	float tmp;
	if (dX_46_v <= 180.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf(t_0, (dX_46_u * dX_46_u), powf(t_1, 2.0f)), powf(hypotf((floorf(d) * dY_46_w), hypotf((floorf(w) * dY_46_u), (floorf(h) * dY_46_v))), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf(t_1, hypotf((floorf(w) * dX_46_u), (floorf(h) * dX_46_v))), 2.0f), ((dY_46_u * dY_46_u) * 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 = floor(w) ^ Float32(2.0)
	t_1 = Float32(floor(d) * dX_46_w)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(180.0))
		tmp = log2(sqrt(((fma(t_0, Float32(dX_46_u * dX_46_u), (t_1 ^ Float32(2.0))) != fma(t_0, Float32(dX_46_u * dX_46_u), (t_1 ^ Float32(2.0)))) ? (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) : (((hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0))) ? fma(t_0, Float32(dX_46_u * dX_46_u), (t_1 ^ Float32(2.0))) : max(fma(t_0, Float32(dX_46_u * dX_46_u), (t_1 ^ Float32(2.0))), (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt((((hypot(t_1, hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0)) != (hypot(t_1, hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0))) ? Float32(Float32(dY_46_u * dY_46_u) * t_0) : ((Float32(Float32(dY_46_u * dY_46_u) * t_0) != Float32(Float32(dY_46_u * dY_46_u) * t_0)) ? (hypot(t_1, hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0)) : max((hypot(t_1, hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0)), Float32(Float32(dY_46_u * dY_46_u) * t_0))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 180

   1. Initial program 69.8%

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

      1. expm1-log1p-u 69.1%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)\right)\right)} \]

      2. expm1-udef 68.8%

         \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)} - 1\right)} \]

   3. Applied egg-rr 68.8%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   4. Step-by-step derivation

      1. expm1-def 69.1%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 69.8%

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

      3. *-commutative 69.8%

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

      4. *-commutative 69.8%

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

      5. *-commutative 69.8%

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

   5. Simplified 69.8%

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

   6. Taylor expanded in dX.v around 0 63.2%

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

      1. *-commutative 63.2%

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

      2. fma-def 63.2%

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

      3. unpow2 63.2%

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

      4. unpow2 63.2%

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

      5. unpow2 63.2%

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

      6. swap-sqr 63.2%

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

      7. unpow2 63.2%

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

   8. Simplified 63.2%

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

   if 180 < dX.v

   1. Initial program 63.5%

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

      1. expm1-log1p-u 63.1%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)\right)\right)} \]

      2. expm1-udef 63.1%

         \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)} - 1\right)} \]

   3. Applied egg-rr 63.1%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   4. Step-by-step derivation

      1. expm1-def 63.1%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 63.5%

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

      3. *-commutative 63.5%

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

      4. *-commutative 63.5%

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

      5. *-commutative 63.5%

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

   5. Simplified 63.5%

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

   6. Taylor expanded in dY.u around inf 58.7%

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

      1. unpow2 58.7%

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

   8. Simplified 58.7%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 180:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloorw\right\rfloor\right)}^{2}, dX.u \cdot dX.u, {\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}\right), {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right)}\right)\\ \end{array} \]

Alternative 3: 56.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloord\right\rfloor \cdot dX.w\\ \mathbf{if}\;dX.v \leq 180:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({t_0}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(t_0, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor d) dX.w)))
   (if (<= dX.v 180.0)
     (log2
      (sqrt
       (fmax
        (pow t_0 2.0)
        (pow
         (hypot
          (* (floor d) dY.w)
          (hypot (* (floor w) dY.u) (* (floor h) dY.v)))
         2.0))))
     (log2
      (sqrt
       (fmax
        (pow (hypot t_0 (hypot (* (floor w) dX.u) (* (floor h) dX.v))) 2.0)
        (* (* dY.u dY.u) (pow (floor 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(d) * dX_46_w;
	float tmp;
	if (dX_46_v <= 180.0f) {
		tmp = log2f(sqrtf(fmaxf(powf(t_0, 2.0f), powf(hypotf((floorf(d) * dY_46_w), hypotf((floorf(w) * dY_46_u), (floorf(h) * dY_46_v))), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf(t_0, hypotf((floorf(w) * dX_46_u), (floorf(h) * dX_46_v))), 2.0f), ((dY_46_u * dY_46_u) * powf(floorf(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(d) * dX_46_w)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(180.0))
		tmp = log2(sqrt((((t_0 ^ Float32(2.0)) != (t_0 ^ Float32(2.0))) ? (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) : (((hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0))) ? (t_0 ^ Float32(2.0)) : max((t_0 ^ Float32(2.0)), (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt((((hypot(t_0, hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0)) != (hypot(t_0, hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0))) ? Float32(Float32(dY_46_u * dY_46_u) * (floor(w) ^ Float32(2.0))) : ((Float32(Float32(dY_46_u * dY_46_u) * (floor(w) ^ Float32(2.0))) != Float32(Float32(dY_46_u * dY_46_u) * (floor(w) ^ Float32(2.0)))) ? (hypot(t_0, hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0)) : max((hypot(t_0, hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0)), Float32(Float32(dY_46_u * dY_46_u) * (floor(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(d) * dX_46_w;
	tmp = single(0.0);
	if (dX_46_v <= single(180.0))
		tmp = log2(sqrt(max((t_0 ^ single(2.0)), (hypot((floor(d) * dY_46_w), hypot((floor(w) * dY_46_u), (floor(h) * dY_46_v))) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max((hypot(t_0, hypot((floor(w) * dX_46_u), (floor(h) * dX_46_v))) ^ single(2.0)), ((dY_46_u * dY_46_u) * (floor(w) ^ single(2.0))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 180

   1. Initial program 69.8%

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

      1. expm1-log1p-u 69.1%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)\right)\right)} \]

      2. expm1-udef 68.8%

         \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)} - 1\right)} \]

   3. Applied egg-rr 68.8%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   4. Step-by-step derivation

      1. expm1-def 69.1%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 69.8%

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

      3. *-commutative 69.8%

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

      4. *-commutative 69.8%

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

      5. *-commutative 69.8%

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

   5. Simplified 69.8%

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

   6. Taylor expanded in dX.w around inf 58.4%

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

      1. unpow2 58.4%

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

      2. unpow2 58.4%

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

      3. swap-sqr 58.4%

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

      4. unpow2 58.4%

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

   8. Simplified 58.4%

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

   if 180 < dX.v

   1. Initial program 63.5%

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

      1. expm1-log1p-u 63.1%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)\right)\right)} \]

      2. expm1-udef 63.1%

         \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)} - 1\right)} \]

   3. Applied egg-rr 63.1%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   4. Step-by-step derivation

      1. expm1-def 63.1%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 63.5%

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

      3. *-commutative 63.5%

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

      4. *-commutative 63.5%

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

      5. *-commutative 63.5%

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

   5. Simplified 63.5%

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

   6. Taylor expanded in dY.u around inf 58.7%

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

      1. unpow2 58.7%

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

   8. Simplified 58.7%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 180:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right)}\right)\\ \end{array} \]

Alternative 4: 55.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;dX.v \leq 20000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (if (<= dX.v 20000.0)
   (log2
    (sqrt
     (fmax
      (pow (* (floor d) dX.w) 2.0)
      (pow
       (hypot (* (floor d) dY.w) (hypot (* (floor w) dY.u) (* (floor h) dY.v)))
       2.0))))
   (log2
    (sqrt
     (fmax
      (pow (hypot (* (floor w) dX.u) (* (floor h) dX.v)) 2.0)
      (* (* dY.u dY.u) (pow (floor 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 tmp;
	if (dX_46_v <= 20000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(d) * dX_46_w), 2.0f), powf(hypotf((floorf(d) * dY_46_w), hypotf((floorf(w) * dY_46_u), (floorf(h) * dY_46_v))), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf((floorf(w) * dX_46_u), (floorf(h) * dX_46_v)), 2.0f), ((dY_46_u * dY_46_u) * powf(floorf(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)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(20000.0))
		tmp = log2(sqrt((((Float32(floor(d) * dX_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dX_46_w) ^ Float32(2.0))) ? (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) : (((hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0))) ? (Float32(floor(d) * dX_46_w) ^ Float32(2.0)) : max((Float32(floor(d) * dX_46_w) ^ Float32(2.0)), (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt((((hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v)) ^ Float32(2.0)) != (hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v)) ^ Float32(2.0))) ? Float32(Float32(dY_46_u * dY_46_u) * (floor(w) ^ Float32(2.0))) : ((Float32(Float32(dY_46_u * dY_46_u) * (floor(w) ^ Float32(2.0))) != Float32(Float32(dY_46_u * dY_46_u) * (floor(w) ^ Float32(2.0)))) ? (hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v)) ^ Float32(2.0)) : max((hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v)) ^ Float32(2.0)), Float32(Float32(dY_46_u * dY_46_u) * (floor(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)
	tmp = single(0.0);
	if (dX_46_v <= single(20000.0))
		tmp = log2(sqrt(max(((floor(d) * dX_46_w) ^ single(2.0)), (hypot((floor(d) * dY_46_w), hypot((floor(w) * dY_46_u), (floor(h) * dY_46_v))) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max((hypot((floor(w) * dX_46_u), (floor(h) * dX_46_v)) ^ single(2.0)), ((dY_46_u * dY_46_u) * (floor(w) ^ single(2.0))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 2e4

   1. Initial program 70.5%

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

      1. expm1-log1p-u 69.9%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)\right)\right)} \]

      2. expm1-udef 69.6%

         \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)} - 1\right)} \]

   3. Applied egg-rr 69.6%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   4. Step-by-step derivation

      1. expm1-def 69.9%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 70.5%

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

      3. *-commutative 70.5%

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

      4. *-commutative 70.5%

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

      5. *-commutative 70.5%

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

   5. Simplified 70.5%

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

   6. Taylor expanded in dX.w around inf 58.5%

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

      1. unpow2 58.5%

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

      2. unpow2 58.5%

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

      3. swap-sqr 58.5%

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

      4. unpow2 58.5%

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

   8. Simplified 58.5%

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

   if 2e4 < dX.v

   1. Initial program 59.9%

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

      1. expm1-log1p-u 59.5%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)\right)\right)} \]

      2. expm1-udef 59.5%

         \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)} - 1\right)} \]

   3. Applied egg-rr 59.5%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   4. Step-by-step derivation

      1. expm1-def 59.5%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 59.9%

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

      3. *-commutative 59.9%

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

      4. *-commutative 59.9%

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

      5. *-commutative 59.9%

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

   5. Simplified 59.9%

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

   6. Taylor expanded in dY.u around inf 55.8%

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

      1. unpow2 55.8%

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

   8. Simplified 55.8%

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

   9. Taylor expanded in dX.w around 0 53.5%

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

       1. *-commutative 53.5%

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

       2. *-commutative 53.5%

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

       3. unpow2 53.5%

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

       4. unpow2 53.5%

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

       5. swap-sqr 53.5%

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

       6. unpow2 53.5%

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

       7. unpow2 53.5%

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

       8. swap-sqr 53.5%

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

       9. hypot-def 53.5%

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

       10. *-commutative 53.5%

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

       11. *-commutative 53.5%

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

   11. Simplified 53.5%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 20000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right)}\right)\\ \end{array} \]

Alternative 5: 47.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;dX.v \leq 20000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (if (<= dX.v 20000.0)
   (log2
    (sqrt
     (fmax
      (pow (* (floor d) dX.w) 2.0)
      (pow (hypot (* (floor w) dY.u) (* (floor h) dY.v)) 2.0))))
   (log2
    (sqrt
     (fmax
      (pow (hypot (* (floor w) dX.u) (* (floor h) dX.v)) 2.0)
      (* (* dY.u dY.u) (pow (floor 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 tmp;
	if (dX_46_v <= 20000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(d) * dX_46_w), 2.0f), powf(hypotf((floorf(w) * dY_46_u), (floorf(h) * dY_46_v)), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf((floorf(w) * dX_46_u), (floorf(h) * dX_46_v)), 2.0f), ((dY_46_u * dY_46_u) * powf(floorf(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)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(20000.0))
		tmp = log2(sqrt((((Float32(floor(d) * dX_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dX_46_w) ^ Float32(2.0))) ? (hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v)) ^ Float32(2.0)) : (((hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v)) ^ Float32(2.0)) != (hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v)) ^ Float32(2.0))) ? (Float32(floor(d) * dX_46_w) ^ Float32(2.0)) : max((Float32(floor(d) * dX_46_w) ^ Float32(2.0)), (hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v)) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt((((hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v)) ^ Float32(2.0)) != (hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v)) ^ Float32(2.0))) ? Float32(Float32(dY_46_u * dY_46_u) * (floor(w) ^ Float32(2.0))) : ((Float32(Float32(dY_46_u * dY_46_u) * (floor(w) ^ Float32(2.0))) != Float32(Float32(dY_46_u * dY_46_u) * (floor(w) ^ Float32(2.0)))) ? (hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v)) ^ Float32(2.0)) : max((hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v)) ^ Float32(2.0)), Float32(Float32(dY_46_u * dY_46_u) * (floor(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)
	tmp = single(0.0);
	if (dX_46_v <= single(20000.0))
		tmp = log2(sqrt(max(((floor(d) * dX_46_w) ^ single(2.0)), (hypot((floor(w) * dY_46_u), (floor(h) * dY_46_v)) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max((hypot((floor(w) * dX_46_u), (floor(h) * dX_46_v)) ^ single(2.0)), ((dY_46_u * dY_46_u) * (floor(w) ^ single(2.0))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 2e4

   1. Initial program 70.5%

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

      1. expm1-log1p-u 69.9%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)\right)\right)} \]

      2. expm1-udef 69.6%

         \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)} - 1\right)} \]

   3. Applied egg-rr 69.6%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   4. Step-by-step derivation

      1. expm1-def 69.9%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 70.5%

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

      3. *-commutative 70.5%

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

      4. *-commutative 70.5%

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

      5. *-commutative 70.5%

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

   5. Simplified 70.5%

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

   6. Taylor expanded in dX.w around inf 58.5%

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

      1. unpow2 58.5%

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

      2. unpow2 58.5%

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

      3. swap-sqr 58.5%

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

      4. unpow2 58.5%

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

   8. Simplified 58.5%

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

   9. Taylor expanded in dY.w around 0 51.5%

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

       1. unpow1 51.5%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, \color{blue}{{\left({dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}\right)}^{1}}\right)}\right) \]

       2. sqr-pow 51.5%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, \color{blue}{{\left({dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left({dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}\right)}^{\left(\frac{1}{2}\right)}}\right)}\right) \]

   11. Simplified 51.5%

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

   if 2e4 < dX.v

   1. Initial program 59.9%

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

      1. expm1-log1p-u 59.5%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)\right)\right)} \]

      2. expm1-udef 59.5%

         \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)} - 1\right)} \]

   3. Applied egg-rr 59.5%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   4. Step-by-step derivation

      1. expm1-def 59.5%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 59.9%

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

      3. *-commutative 59.9%

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

      4. *-commutative 59.9%

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

      5. *-commutative 59.9%

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

   5. Simplified 59.9%

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

   6. Taylor expanded in dY.u around inf 55.8%

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

      1. unpow2 55.8%

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

   8. Simplified 55.8%

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

   9. Taylor expanded in dX.w around 0 53.5%

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

       1. *-commutative 53.5%

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

       2. *-commutative 53.5%

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

       3. unpow2 53.5%

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

       4. unpow2 53.5%

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

       5. swap-sqr 53.5%

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

       6. unpow2 53.5%

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

       7. unpow2 53.5%

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

       8. swap-sqr 53.5%

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

       9. hypot-def 53.5%

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

       10. *-commutative 53.5%

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

       11. *-commutative 53.5%

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

   11. Simplified 53.5%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 20000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right)}\right)\\ \end{array} \]

Alternative 6: 47.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;dY.u \leq 32:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (if (<= dY.u 32.0)
   (log2
    (sqrt
     (fmax
      (pow (* (floor d) dX.w) 2.0)
      (pow (hypot (* (floor d) dY.w) (* (floor h) dY.v)) 2.0))))
   (log2
    (sqrt
     (fmax
      (pow (* (floor h) dX.v) 2.0)
      (* (* dY.u dY.u) (pow (floor 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 tmp;
	if (dY_46_u <= 32.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(d) * dX_46_w), 2.0f), powf(hypotf((floorf(d) * dY_46_w), (floorf(h) * dY_46_v)), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(h) * dX_46_v), 2.0f), ((dY_46_u * dY_46_u) * powf(floorf(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)
	tmp = Float32(0.0)
	if (dY_46_u <= Float32(32.0))
		tmp = log2(sqrt((((Float32(floor(d) * dX_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dX_46_w) ^ Float32(2.0))) ? (hypot(Float32(floor(d) * dY_46_w), Float32(floor(h) * dY_46_v)) ^ Float32(2.0)) : (((hypot(Float32(floor(d) * dY_46_w), Float32(floor(h) * dY_46_v)) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dY_46_w), Float32(floor(h) * dY_46_v)) ^ Float32(2.0))) ? (Float32(floor(d) * dX_46_w) ^ Float32(2.0)) : max((Float32(floor(d) * dX_46_w) ^ Float32(2.0)), (hypot(Float32(floor(d) * dY_46_w), Float32(floor(h) * dY_46_v)) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt((((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) != (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) ? Float32(Float32(dY_46_u * dY_46_u) * (floor(w) ^ Float32(2.0))) : ((Float32(Float32(dY_46_u * dY_46_u) * (floor(w) ^ Float32(2.0))) != Float32(Float32(dY_46_u * dY_46_u) * (floor(w) ^ Float32(2.0)))) ? (Float32(floor(h) * dX_46_v) ^ Float32(2.0)) : max((Float32(floor(h) * dX_46_v) ^ Float32(2.0)), Float32(Float32(dY_46_u * dY_46_u) * (floor(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)
	tmp = single(0.0);
	if (dY_46_u <= single(32.0))
		tmp = log2(sqrt(max(((floor(d) * dX_46_w) ^ single(2.0)), (hypot((floor(d) * dY_46_w), (floor(h) * dY_46_v)) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max(((floor(h) * dX_46_v) ^ single(2.0)), ((dY_46_u * dY_46_u) * (floor(w) ^ single(2.0))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 32

   1. Initial program 70.6%

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

      1. expm1-log1p-u 70.0%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)\right)\right)} \]

      2. expm1-udef 69.7%

         \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)} - 1\right)} \]

   3. Applied egg-rr 69.7%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   4. Step-by-step derivation

      1. expm1-def 70.0%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 70.6%

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

      3. *-commutative 70.6%

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

      4. *-commutative 70.6%

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

      5. *-commutative 70.6%

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

   5. Simplified 70.6%

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

   6. Taylor expanded in dX.w around inf 54.6%

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

      1. unpow2 54.6%

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

      2. unpow2 54.6%

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

      3. swap-sqr 54.6%

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

      4. unpow2 54.6%

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

   8. Simplified 54.6%

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

   9. Taylor expanded in dY.u around 0 44.1%

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

       1. *-commutative 44.1%

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

   11. Simplified 44.1%

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

   if 32 < dY.u

   1. Initial program 61.4%

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

      1. expm1-log1p-u 60.9%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)\right)\right)} \]

      2. expm1-udef 60.9%

         \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)} - 1\right)} \]

   3. Applied egg-rr 60.9%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   4. Step-by-step derivation

      1. expm1-def 60.9%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 61.4%

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

      3. *-commutative 61.4%

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

      4. *-commutative 61.4%

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

      5. *-commutative 61.4%

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

   5. Simplified 61.4%

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

   6. Taylor expanded in dY.u around inf 53.5%

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

      1. unpow2 53.5%

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

   8. Simplified 53.5%

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

   9. Taylor expanded in dX.v around inf 45.1%

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

       1. *-commutative 45.1%

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

   11. Simplified 45.1%

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

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.u \leq 32:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right)}\right)\\ \end{array} \]

Alternative 7: 47.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;dX.v \leq 60000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (if (<= dX.v 60000.0)
   (log2
    (sqrt
     (fmax
      (pow (* (floor d) dX.w) 2.0)
      (pow (hypot (* (floor d) dY.w) (* (floor w) dY.u)) 2.0))))
   (log2
    (sqrt
     (fmax
      (pow (* (floor h) dX.v) 2.0)
      (* (* dY.u dY.u) (pow (floor 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 tmp;
	if (dX_46_v <= 60000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(d) * dX_46_w), 2.0f), powf(hypotf((floorf(d) * dY_46_w), (floorf(w) * dY_46_u)), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(h) * dX_46_v), 2.0f), ((dY_46_u * dY_46_u) * powf(floorf(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)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(60000.0))
		tmp = log2(sqrt((((Float32(floor(d) * dX_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dX_46_w) ^ Float32(2.0))) ? (hypot(Float32(floor(d) * dY_46_w), Float32(floor(w) * dY_46_u)) ^ Float32(2.0)) : (((hypot(Float32(floor(d) * dY_46_w), Float32(floor(w) * dY_46_u)) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dY_46_w), Float32(floor(w) * dY_46_u)) ^ Float32(2.0))) ? (Float32(floor(d) * dX_46_w) ^ Float32(2.0)) : max((Float32(floor(d) * dX_46_w) ^ Float32(2.0)), (hypot(Float32(floor(d) * dY_46_w), Float32(floor(w) * dY_46_u)) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt((((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) != (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) ? Float32(Float32(dY_46_u * dY_46_u) * (floor(w) ^ Float32(2.0))) : ((Float32(Float32(dY_46_u * dY_46_u) * (floor(w) ^ Float32(2.0))) != Float32(Float32(dY_46_u * dY_46_u) * (floor(w) ^ Float32(2.0)))) ? (Float32(floor(h) * dX_46_v) ^ Float32(2.0)) : max((Float32(floor(h) * dX_46_v) ^ Float32(2.0)), Float32(Float32(dY_46_u * dY_46_u) * (floor(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)
	tmp = single(0.0);
	if (dX_46_v <= single(60000.0))
		tmp = log2(sqrt(max(((floor(d) * dX_46_w) ^ single(2.0)), (hypot((floor(d) * dY_46_w), (floor(w) * dY_46_u)) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max(((floor(h) * dX_46_v) ^ single(2.0)), ((dY_46_u * dY_46_u) * (floor(w) ^ single(2.0))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 6e4

   1. Initial program 70.7%

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

      1. expm1-log1p-u 70.0%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)\right)\right)} \]

      2. expm1-udef 69.7%

         \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)} - 1\right)} \]

   3. Applied egg-rr 69.7%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   4. Step-by-step derivation

      1. expm1-def 70.0%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 70.7%

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

      3. *-commutative 70.7%

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

      4. *-commutative 70.7%

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

      5. *-commutative 70.7%

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

   5. Simplified 70.7%

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

   6. Taylor expanded in dX.w around inf 58.3%

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

      1. unpow2 58.3%

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

      2. unpow2 58.3%

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

      3. swap-sqr 58.3%

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

      4. unpow2 58.3%

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

   8. Simplified 58.3%

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

   9. Taylor expanded in dY.u around inf 49.5%

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

       1. *-commutative 49.5%

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

   11. Simplified 49.5%

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

   if 6e4 < dX.v

   1. Initial program 59.2%

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

      1. expm1-log1p-u 58.7%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)\right)\right)} \]

      2. expm1-udef 58.7%

         \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)} - 1\right)} \]

   3. Applied egg-rr 58.7%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   4. Step-by-step derivation

      1. expm1-def 58.7%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 59.2%

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

      3. *-commutative 59.2%

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

      4. *-commutative 59.2%

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

      5. *-commutative 59.2%

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

   5. Simplified 59.2%

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

   6. Taylor expanded in dY.u around inf 55.0%

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

      1. unpow2 55.0%

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

   8. Simplified 55.0%

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

   9. Taylor expanded in dX.v around inf 46.2%

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

       1. *-commutative 46.2%

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

   11. Simplified 46.2%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 60000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right)}\right)\\ \end{array} \]

Alternative 8: 46.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;dX.v \leq 20000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (if (<= dX.v 20000.0)
   (log2
    (sqrt
     (fmax
      (pow (* (floor d) dX.w) 2.0)
      (pow (hypot (* (floor w) dY.u) (* (floor h) dY.v)) 2.0))))
   (log2
    (sqrt
     (fmax
      (pow (* (floor h) dX.v) 2.0)
      (* (* dY.u dY.u) (pow (floor 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 tmp;
	if (dX_46_v <= 20000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(d) * dX_46_w), 2.0f), powf(hypotf((floorf(w) * dY_46_u), (floorf(h) * dY_46_v)), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(h) * dX_46_v), 2.0f), ((dY_46_u * dY_46_u) * powf(floorf(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)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(20000.0))
		tmp = log2(sqrt((((Float32(floor(d) * dX_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dX_46_w) ^ Float32(2.0))) ? (hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v)) ^ Float32(2.0)) : (((hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v)) ^ Float32(2.0)) != (hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v)) ^ Float32(2.0))) ? (Float32(floor(d) * dX_46_w) ^ Float32(2.0)) : max((Float32(floor(d) * dX_46_w) ^ Float32(2.0)), (hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v)) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt((((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) != (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) ? Float32(Float32(dY_46_u * dY_46_u) * (floor(w) ^ Float32(2.0))) : ((Float32(Float32(dY_46_u * dY_46_u) * (floor(w) ^ Float32(2.0))) != Float32(Float32(dY_46_u * dY_46_u) * (floor(w) ^ Float32(2.0)))) ? (Float32(floor(h) * dX_46_v) ^ Float32(2.0)) : max((Float32(floor(h) * dX_46_v) ^ Float32(2.0)), Float32(Float32(dY_46_u * dY_46_u) * (floor(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)
	tmp = single(0.0);
	if (dX_46_v <= single(20000.0))
		tmp = log2(sqrt(max(((floor(d) * dX_46_w) ^ single(2.0)), (hypot((floor(w) * dY_46_u), (floor(h) * dY_46_v)) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max(((floor(h) * dX_46_v) ^ single(2.0)), ((dY_46_u * dY_46_u) * (floor(w) ^ single(2.0))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 2e4

   1. Initial program 70.5%

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

      1. expm1-log1p-u 69.9%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)\right)\right)} \]

      2. expm1-udef 69.6%

         \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)} - 1\right)} \]

   3. Applied egg-rr 69.6%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   4. Step-by-step derivation

      1. expm1-def 69.9%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 70.5%

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

      3. *-commutative 70.5%

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

      4. *-commutative 70.5%

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

      5. *-commutative 70.5%

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

   5. Simplified 70.5%

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

   6. Taylor expanded in dX.w around inf 58.5%

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

      1. unpow2 58.5%

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

      2. unpow2 58.5%

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

      3. swap-sqr 58.5%

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

      4. unpow2 58.5%

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

   8. Simplified 58.5%

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

   9. Taylor expanded in dY.w around 0 51.5%

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

       1. unpow1 51.5%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, \color{blue}{{\left({dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}\right)}^{1}}\right)}\right) \]

       2. sqr-pow 51.5%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, \color{blue}{{\left({dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left({dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}\right)}^{\left(\frac{1}{2}\right)}}\right)}\right) \]

   11. Simplified 51.5%

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

   if 2e4 < dX.v

   1. Initial program 59.9%

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

      1. expm1-log1p-u 59.5%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)\right)\right)} \]

      2. expm1-udef 59.5%

         \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)} - 1\right)} \]

   3. Applied egg-rr 59.5%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   4. Step-by-step derivation

      1. expm1-def 59.5%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 59.9%

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

      3. *-commutative 59.9%

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

      4. *-commutative 59.9%

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

      5. *-commutative 59.9%

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

   5. Simplified 59.9%

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

   6. Taylor expanded in dY.u around inf 55.8%

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

      1. unpow2 55.8%

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

   8. Simplified 55.8%

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

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

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

       1. *-commutative 45.9%

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

   11. Simplified 45.9%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 20000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right)}\right)\\ \end{array} \]

Alternative 9: 47.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloord\right\rfloor \cdot dX.w\\ t_1 := \left\lfloorw\right\rfloor \cdot dY.u\\ \mathbf{if}\;dX.v \leq 160:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({t_0}^{2}, {\left(\mathsf{hypot}\left(t_1, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(t_0, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {t_1}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor d) dX.w)) (t_1 (* (floor w) dY.u)))
   (if (<= dX.v 160.0)
     (log2
      (sqrt (fmax (pow t_0 2.0) (pow (hypot t_1 (* (floor h) dY.v)) 2.0))))
     (log2
      (sqrt (fmax (pow (hypot t_0 (* (floor h) dX.v)) 2.0) (pow t_1 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) * dX_46_w;
	float t_1 = floorf(w) * dY_46_u;
	float tmp;
	if (dX_46_v <= 160.0f) {
		tmp = log2f(sqrtf(fmaxf(powf(t_0, 2.0f), powf(hypotf(t_1, (floorf(h) * dY_46_v)), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf(t_0, (floorf(h) * dX_46_v)), 2.0f), powf(t_1, 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) * dX_46_w)
	t_1 = Float32(floor(w) * dY_46_u)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(160.0))
		tmp = log2(sqrt((((t_0 ^ Float32(2.0)) != (t_0 ^ Float32(2.0))) ? (hypot(t_1, Float32(floor(h) * dY_46_v)) ^ Float32(2.0)) : (((hypot(t_1, Float32(floor(h) * dY_46_v)) ^ Float32(2.0)) != (hypot(t_1, Float32(floor(h) * dY_46_v)) ^ Float32(2.0))) ? (t_0 ^ Float32(2.0)) : max((t_0 ^ Float32(2.0)), (hypot(t_1, Float32(floor(h) * dY_46_v)) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt((((hypot(t_0, Float32(floor(h) * dX_46_v)) ^ Float32(2.0)) != (hypot(t_0, Float32(floor(h) * dX_46_v)) ^ Float32(2.0))) ? (t_1 ^ Float32(2.0)) : (((t_1 ^ Float32(2.0)) != (t_1 ^ Float32(2.0))) ? (hypot(t_0, Float32(floor(h) * dX_46_v)) ^ Float32(2.0)) : max((hypot(t_0, Float32(floor(h) * dX_46_v)) ^ Float32(2.0)), (t_1 ^ 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) * dX_46_w;
	t_1 = floor(w) * dY_46_u;
	tmp = single(0.0);
	if (dX_46_v <= single(160.0))
		tmp = log2(sqrt(max((t_0 ^ single(2.0)), (hypot(t_1, (floor(h) * dY_46_v)) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max((hypot(t_0, (floor(h) * dX_46_v)) ^ single(2.0)), (t_1 ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 160

   1. Initial program 69.8%

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

      1. expm1-log1p-u 69.1%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)\right)\right)} \]

      2. expm1-udef 68.8%

         \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)} - 1\right)} \]

   3. Applied egg-rr 68.8%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   4. Step-by-step derivation

      1. expm1-def 69.1%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 69.8%

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

      3. *-commutative 69.8%

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

      4. *-commutative 69.8%

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

      5. *-commutative 69.8%

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

   5. Simplified 69.8%

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

   6. Taylor expanded in dX.w around inf 58.4%

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

      1. unpow2 58.4%

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

      2. unpow2 58.4%

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

      3. swap-sqr 58.4%

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

      4. unpow2 58.4%

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

   8. Simplified 58.4%

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

   9. Taylor expanded in dY.w around 0 51.2%

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

       1. unpow1 51.2%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, \color{blue}{{\left({dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}\right)}^{1}}\right)}\right) \]

       2. sqr-pow 51.2%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.w \cdot \left\lfloord\right\rfloor\right)}^{2}, \color{blue}{{\left({dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left({dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}\right)}^{\left(\frac{1}{2}\right)}}\right)}\right) \]

   11. Simplified 51.2%

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

   if 160 < dX.v

   1. Initial program 63.5%

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

      1. expm1-log1p-u 63.1%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)\right)\right)} \]

      2. expm1-udef 63.1%

         \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)} - 1\right)} \]

   3. Applied egg-rr 63.1%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   4. Step-by-step derivation

      1. expm1-def 63.1%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 63.5%

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

      3. *-commutative 63.5%

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

      4. *-commutative 63.5%

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

      5. *-commutative 63.5%

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

   5. Simplified 63.5%

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

   6. Taylor expanded in dY.u around inf 58.7%

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

      1. unpow2 58.7%

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

   8. Simplified 58.7%

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

   9. Taylor expanded in dX.u around 0 48.2%

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

       1. unpow2 48.2%

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

       2. unpow2 48.2%

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

       3. swap-sqr 48.2%

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

       4. unpow2 48.2%

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

       5. +-commutative 48.2%

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

       6. unpow2 48.2%

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

       7. unpow2 48.2%

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

       8. unpow2 48.2%

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

       9. swap-sqr 48.2%

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

       10. hypot-def 48.2%

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

       11. *-commutative 48.2%

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

   11. Simplified 48.2%

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

   12. Taylor expanded in dX.w around 0 48.2%

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

       1. *-commutative 48.2%

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

       2. unpow2 48.2%

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

       3. *-commutative 48.2%

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

       4. unpow2 48.2%

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

       5. unswap-sqr 48.2%

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

       6. unpow2 48.2%

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

   14. Simplified 48.2%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 160:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}\right)\\ \end{array} \]

Alternative 10: 39.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\\ \mathbf{if}\;dX.w \leq 20000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot \left(dX.v \cdot dX.v\right), t_0\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, t_0\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (* (floor d) dY.w) 2.0)))
   (if (<= dX.w 20000.0)
     (log2 (sqrt (fmax (* (pow (floor h) 2.0) (* dX.v dX.v)) t_0)))
     (log2 (sqrt (fmax (pow (* (floor d) dX.w) 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((floorf(d) * dY_46_w), 2.0f);
	float tmp;
	if (dX_46_w <= 20000.0f) {
		tmp = log2f(sqrtf(fmaxf((powf(floorf(h), 2.0f) * (dX_46_v * dX_46_v)), t_0)));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(d) * dX_46_w), 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 = Float32(floor(d) * dY_46_w) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dX_46_w <= Float32(20000.0))
		tmp = log2(sqrt(((Float32((floor(h) ^ Float32(2.0)) * Float32(dX_46_v * dX_46_v)) != Float32((floor(h) ^ Float32(2.0)) * Float32(dX_46_v * dX_46_v))) ? t_0 : ((t_0 != t_0) ? Float32((floor(h) ^ Float32(2.0)) * Float32(dX_46_v * dX_46_v)) : max(Float32((floor(h) ^ Float32(2.0)) * Float32(dX_46_v * dX_46_v)), t_0)))));
	else
		tmp = log2(sqrt((((Float32(floor(d) * dX_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dX_46_w) ^ Float32(2.0))) ? t_0 : ((t_0 != t_0) ? (Float32(floor(d) * dX_46_w) ^ Float32(2.0)) : max((Float32(floor(d) * dX_46_w) ^ 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 = (floor(d) * dY_46_w) ^ single(2.0);
	tmp = single(0.0);
	if (dX_46_w <= single(20000.0))
		tmp = log2(sqrt(max(((floor(h) ^ single(2.0)) * (dX_46_v * dX_46_v)), t_0)));
	else
		tmp = log2(sqrt(max(((floor(d) * dX_46_w) ^ single(2.0)), t_0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.w < 2e4

   1. Initial program 69.2%

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

      1. expm1-log1p-u 68.7%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)\right)\right)} \]

      2. expm1-udef 68.4%

         \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)} - 1\right)} \]

   3. Applied egg-rr 68.4%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   4. Step-by-step derivation

      1. expm1-def 68.7%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 69.2%

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

      3. *-commutative 69.2%

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

      4. *-commutative 69.2%

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

      5. *-commutative 69.2%

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

   5. Simplified 69.2%

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

   6. Taylor expanded in dX.v around inf 58.0%

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

      1. *-commutative 58.0%

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

      2. unpow2 58.0%

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

   8. Simplified 58.0%

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

   9. Taylor expanded in dY.w around inf 35.7%

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

       1. *-commutative 29.1%

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

       2. unpow2 29.1%

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

       3. unpow2 29.1%

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

       4. swap-sqr 29.1%

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

       5. unpow2 29.1%

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

   11. Simplified 35.7%

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

   if 2e4 < dX.w

   1. Initial program 63.0%

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

      1. expm1-log1p-u 62.3%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)\right)\right)} \]

      2. expm1-udef 62.3%

         \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)} - 1\right)} \]

   3. Applied egg-rr 62.3%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   4. Step-by-step derivation

      1. expm1-def 62.3%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 63.0%

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

      3. *-commutative 63.0%

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

      4. *-commutative 63.0%

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

      5. *-commutative 63.0%

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

   5. Simplified 63.0%

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

   6. Taylor expanded in dX.w around inf 53.2%

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

      1. unpow2 53.2%

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

      2. unpow2 53.2%

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

      3. swap-sqr 53.2%

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

      4. unpow2 53.2%

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

   8. Simplified 53.2%

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

   9. Taylor expanded in dY.w around inf 47.9%

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

       1. *-commutative 47.9%

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

       2. unpow2 47.9%

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

       3. unpow2 47.9%

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

       4. swap-sqr 47.9%

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

       5. unpow2 47.9%

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

   11. Simplified 47.9%

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

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.w \leq 20000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot \left(dX.v \cdot dX.v\right), {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \end{array} \]

Alternative 11: 38.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;dX.v \leq 500000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot \left(dX.v \cdot dX.v\right), {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (if (<= dX.v 500000.0)
   (log2
    (sqrt
     (fmax
      (pow (* (floor d) dX.w) 2.0)
      (* (* dY.u dY.u) (pow (floor w) 2.0)))))
   (log2
    (sqrt
     (fmax
      (* (pow (floor h) 2.0) (* dX.v dX.v))
      (pow (* (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 tmp;
	if (dX_46_v <= 500000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(d) * dX_46_w), 2.0f), ((dY_46_u * dY_46_u) * powf(floorf(w), 2.0f)))));
	} else {
		tmp = log2f(sqrtf(fmaxf((powf(floorf(h), 2.0f) * (dX_46_v * dX_46_v)), powf((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)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(500000.0))
		tmp = log2(sqrt((((Float32(floor(d) * dX_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dX_46_w) ^ Float32(2.0))) ? Float32(Float32(dY_46_u * dY_46_u) * (floor(w) ^ Float32(2.0))) : ((Float32(Float32(dY_46_u * dY_46_u) * (floor(w) ^ Float32(2.0))) != Float32(Float32(dY_46_u * dY_46_u) * (floor(w) ^ Float32(2.0)))) ? (Float32(floor(d) * dX_46_w) ^ Float32(2.0)) : max((Float32(floor(d) * dX_46_w) ^ Float32(2.0)), Float32(Float32(dY_46_u * dY_46_u) * (floor(w) ^ Float32(2.0))))))));
	else
		tmp = log2(sqrt(((Float32((floor(h) ^ Float32(2.0)) * Float32(dX_46_v * dX_46_v)) != Float32((floor(h) ^ Float32(2.0)) * Float32(dX_46_v * dX_46_v))) ? (Float32(floor(d) * dY_46_w) ^ Float32(2.0)) : (((Float32(floor(d) * dY_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dY_46_w) ^ Float32(2.0))) ? Float32((floor(h) ^ Float32(2.0)) * Float32(dX_46_v * dX_46_v)) : max(Float32((floor(h) ^ Float32(2.0)) * Float32(dX_46_v * dX_46_v)), (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)
	tmp = single(0.0);
	if (dX_46_v <= single(500000.0))
		tmp = log2(sqrt(max(((floor(d) * dX_46_w) ^ single(2.0)), ((dY_46_u * dY_46_u) * (floor(w) ^ single(2.0))))));
	else
		tmp = log2(sqrt(max(((floor(h) ^ single(2.0)) * (dX_46_v * dX_46_v)), ((floor(d) * dY_46_w) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 5e5

   1. Initial program 70.8%

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

      1. expm1-log1p-u 70.1%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)\right)\right)} \]

      2. expm1-udef 69.8%

         \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)} - 1\right)} \]

   3. Applied egg-rr 69.8%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   4. Step-by-step derivation

      1. expm1-def 70.1%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 70.8%

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

      3. *-commutative 70.8%

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

      4. *-commutative 70.8%

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

      5. *-commutative 70.8%

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

   5. Simplified 70.8%

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

   6. Taylor expanded in dY.u around inf 57.2%

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

      1. unpow2 57.2%

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

   8. Simplified 57.2%

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

   9. Taylor expanded in dX.w around inf 41.8%

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

   if 5e5 < dX.v

   1. Initial program 57.8%

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

      1. expm1-log1p-u 57.4%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)\right)\right)} \]

      2. expm1-udef 57.4%

         \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)} - 1\right)} \]

   3. Applied egg-rr 57.4%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   4. Step-by-step derivation

      1. expm1-def 57.4%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 57.8%

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

      3. *-commutative 57.8%

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

      4. *-commutative 57.8%

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

      5. *-commutative 57.8%

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

   5. Simplified 57.8%

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

   6. Taylor expanded in dX.v around inf 49.6%

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

      1. *-commutative 49.6%

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

      2. unpow2 49.6%

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

   8. Simplified 49.6%

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

   9. Taylor expanded in dY.w around inf 45.9%

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

       1. *-commutative 25.3%

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

       2. unpow2 25.3%

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

       3. unpow2 25.3%

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

       4. swap-sqr 25.3%

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

       5. unpow2 25.3%

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

   11. Simplified 45.9%

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

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 500000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot \left(dX.v \cdot dX.v\right), {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \end{array} \]

Alternative 12: 38.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;dX.w \leq 8500:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, \left(dY.v \cdot dY.v\right) \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (if (<= dX.w 8500.0)
   (log2
    (sqrt
     (fmax
      (pow (* (floor h) dX.v) 2.0)
      (* (* dY.u dY.u) (pow (floor w) 2.0)))))
   (log2
    (sqrt
     (fmax
      (pow (* (floor d) dX.w) 2.0)
      (* (* dY.v dY.v) (pow (floor h) 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 tmp;
	if (dX_46_w <= 8500.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(h) * dX_46_v), 2.0f), ((dY_46_u * dY_46_u) * powf(floorf(w), 2.0f)))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(d) * dX_46_w), 2.0f), ((dY_46_v * dY_46_v) * powf(floorf(h), 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)
	tmp = Float32(0.0)
	if (dX_46_w <= Float32(8500.0))
		tmp = log2(sqrt((((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) != (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) ? Float32(Float32(dY_46_u * dY_46_u) * (floor(w) ^ Float32(2.0))) : ((Float32(Float32(dY_46_u * dY_46_u) * (floor(w) ^ Float32(2.0))) != Float32(Float32(dY_46_u * dY_46_u) * (floor(w) ^ Float32(2.0)))) ? (Float32(floor(h) * dX_46_v) ^ Float32(2.0)) : max((Float32(floor(h) * dX_46_v) ^ Float32(2.0)), Float32(Float32(dY_46_u * dY_46_u) * (floor(w) ^ Float32(2.0))))))));
	else
		tmp = log2(sqrt((((Float32(floor(d) * dX_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dX_46_w) ^ Float32(2.0))) ? Float32(Float32(dY_46_v * dY_46_v) * (floor(h) ^ Float32(2.0))) : ((Float32(Float32(dY_46_v * dY_46_v) * (floor(h) ^ Float32(2.0))) != Float32(Float32(dY_46_v * dY_46_v) * (floor(h) ^ Float32(2.0)))) ? (Float32(floor(d) * dX_46_w) ^ Float32(2.0)) : max((Float32(floor(d) * dX_46_w) ^ Float32(2.0)), Float32(Float32(dY_46_v * dY_46_v) * (floor(h) ^ 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)
	tmp = single(0.0);
	if (dX_46_w <= single(8500.0))
		tmp = log2(sqrt(max(((floor(h) * dX_46_v) ^ single(2.0)), ((dY_46_u * dY_46_u) * (floor(w) ^ single(2.0))))));
	else
		tmp = log2(sqrt(max(((floor(d) * dX_46_w) ^ single(2.0)), ((dY_46_v * dY_46_v) * (floor(h) ^ single(2.0))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.w < 8500

   1. Initial program 68.8%

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

      1. expm1-log1p-u 68.2%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)\right)\right)} \]

      2. expm1-udef 67.9%

         \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)} - 1\right)} \]

   3. Applied egg-rr 67.9%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   4. Step-by-step derivation

      1. expm1-def 68.2%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 68.8%

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

      3. *-commutative 68.8%

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

      4. *-commutative 68.8%

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

      5. *-commutative 68.8%

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

   5. Simplified 68.8%

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

   6. Taylor expanded in dY.u around inf 55.8%

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

      1. unpow2 55.8%

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

   8. Simplified 55.8%

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

   9. Taylor expanded in dX.v around inf 42.1%

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

       1. *-commutative 42.1%

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

   11. Simplified 42.1%

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

   if 8500 < dX.w

   1. Initial program 65.4%

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

      1. expm1-log1p-u 64.7%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)\right)\right)} \]

      2. expm1-udef 64.7%

         \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)} - 1\right)} \]

   3. Applied egg-rr 64.7%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
   4. Step-by-step derivation

      1. expm1-def 64.7%

         \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

      2. expm1-log1p 65.4%

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

      3. *-commutative 65.4%

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

      4. *-commutative 65.4%

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

      5. *-commutative 65.4%

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

   5. Simplified 65.4%

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

   6. Taylor expanded in dX.w around inf 54.7%

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

      1. unpow2 54.7%

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

      2. unpow2 54.7%

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

      3. swap-sqr 54.7%

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

      4. unpow2 54.7%

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

   8. Simplified 54.7%

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

   9. Taylor expanded in dY.v around inf 48.0%

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

       1. unpow2 48.0%

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

   11. Simplified 48.0%

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

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.w \leq 8500:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, \left(dY.v \cdot dY.v\right) \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}\right)}\right)\\ \end{array} \]

Alternative 13: 35.8% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (log2
  (sqrt (fmax (pow (* (floor d) dX.w) 2.0) (pow (* (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(d) * dX_46_w), 2.0f), powf((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(d) * dX_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dX_46_w) ^ Float32(2.0))) ? (Float32(floor(d) * dY_46_w) ^ Float32(2.0)) : (((Float32(floor(d) * dY_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dY_46_w) ^ Float32(2.0))) ? (Float32(floor(d) * dX_46_w) ^ Float32(2.0)) : max((Float32(floor(d) * dX_46_w) ^ Float32(2.0)), (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(d) * dX_46_w) ^ single(2.0)), ((floor(d) * dY_46_w) ^ single(2.0)))));
end

Derivation

1. Initial program 68.2%

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

   1. expm1-log1p-u 67.6%

      \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)\right)\right)} \]

   2. expm1-udef 67.4%

      \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right)} - 1\right)} \]

3. Applied egg-rr 67.4%

   \[\leadsto \log_{2} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)} - 1\right)} \]
4. Step-by-step derivation

   1. expm1-def 67.6%

      \[\leadsto \log_{2} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\right)\right)} \]

   2. expm1-log1p 68.2%

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

   3. *-commutative 68.2%

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

   4. *-commutative 68.2%

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

   5. *-commutative 68.2%

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

5. Simplified 68.2%

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

6. Taylor expanded in dX.w around inf 54.2%

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

   1. unpow2 54.2%

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

   2. unpow2 54.2%

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

   3. swap-sqr 54.2%

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

   4. unpow2 54.2%

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

8. Simplified 54.2%

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

9. Taylor expanded in dY.w around inf 32.3%

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

    1. *-commutative 32.3%

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

    2. unpow2 32.3%

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

    3. unpow2 32.3%

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

    4. swap-sqr 32.3%

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

    5. unpow2 32.3%

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

11. Simplified 32.3%

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

12. Final simplification 32.3%

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

Reproduce

herbie shell --seed 2023293 
(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)))))))