Isotropic LOD (LOD)

Percentage Accurate: 67.7% → 67.7%

Time: 20.5s

Alternatives: 10

Speedup: N/A×

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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_1 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_2 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_3 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_4 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_5 := \left\lfloor w\right\rfloor \cdot dX.u\\ \log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_5 \cdot t\_5 + t\_2 \cdot t\_2\right) + t\_4 \cdot t\_4, \left(t\_0 \cdot t\_0 + t\_1 \cdot t\_1\right) + t\_3 \cdot t\_3\right)}\right) \end{array} \end{array} \]

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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_1 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_2 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_3 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_4 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_5 := \left\lfloor w\right\rfloor \cdot dX.u\\ \log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_5 \cdot t\_5 + t\_2 \cdot t\_2\right) + t\_4 \cdot t\_4, \left(t\_0 \cdot t\_0 + t\_1 \cdot t\_1\right) + t\_3 \cdot t\_3\right)}\right) \end{array} \end{array} \]

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

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

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 (* t_4 t_4))
        (t_6 (* (floor w) dX.u))
        (t_7 (+ (+ (* t_6 t_6) (* t_2 t_2)) t_5)))
   (if (<= (fmax t_7 (+ (+ (* t_0 t_0) (* t_1 t_1)) (* t_3 t_3))) INFINITY)
     (log2
      (sqrt
       (fmax
        t_7
        (/ 1.0 (/ 1.0 (+ (pow t_1 2.0) (+ (pow t_0 2.0) (pow t_3 2.0))))))))
     (log2
      (sqrt
       (fmax
        (+ t_5 (* dX.v (* dX.v (pow (floor h) 2.0))))
        (* (pow (floor w) 2.0) (* dY.u dY.u))))))))

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 = t_4 * t_4;
	float t_6 = floorf(w) * dX_46_u;
	float t_7 = ((t_6 * t_6) + (t_2 * t_2)) + t_5;
	float tmp;
	if (fmaxf(t_7, (((t_0 * t_0) + (t_1 * t_1)) + (t_3 * t_3))) <= ((float) INFINITY)) {
		tmp = log2f(sqrtf(fmaxf(t_7, (1.0f / (1.0f / (powf(t_1, 2.0f) + (powf(t_0, 2.0f) + powf(t_3, 2.0f))))))));
	} else {
		tmp = log2f(sqrtf(fmaxf((t_5 + (dX_46_v * (dX_46_v * powf(floorf(h), 2.0f)))), (powf(floorf(w), 2.0f) * (dY_46_u * dY_46_u)))));
	}
	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(t_4 * t_4)
	t_6 = Float32(floor(w) * dX_46_u)
	t_7 = Float32(Float32(Float32(t_6 * t_6) + Float32(t_2 * t_2)) + t_5)
	tmp = Float32(0.0)
	if (((t_7 != t_7) ? 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))) ? t_7 : max(t_7, Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3))))) <= Float32(Inf))
		tmp = log2(sqrt(((t_7 != t_7) ? Float32(Float32(1.0) / Float32(Float32(1.0) / Float32((t_1 ^ Float32(2.0)) + Float32((t_0 ^ Float32(2.0)) + (t_3 ^ Float32(2.0)))))) : ((Float32(Float32(1.0) / Float32(Float32(1.0) / Float32((t_1 ^ Float32(2.0)) + Float32((t_0 ^ Float32(2.0)) + (t_3 ^ Float32(2.0)))))) != Float32(Float32(1.0) / Float32(Float32(1.0) / Float32((t_1 ^ Float32(2.0)) + Float32((t_0 ^ Float32(2.0)) + (t_3 ^ Float32(2.0))))))) ? t_7 : max(t_7, Float32(Float32(1.0) / Float32(Float32(1.0) / Float32((t_1 ^ Float32(2.0)) + Float32((t_0 ^ Float32(2.0)) + (t_3 ^ Float32(2.0)))))))))));
	else
		tmp = log2(sqrt(((Float32(t_5 + Float32(dX_46_v * Float32(dX_46_v * (floor(h) ^ Float32(2.0))))) != Float32(t_5 + Float32(dX_46_v * Float32(dX_46_v * (floor(h) ^ Float32(2.0)))))) ? Float32((floor(w) ^ Float32(2.0)) * Float32(dY_46_u * dY_46_u)) : ((Float32((floor(w) ^ Float32(2.0)) * Float32(dY_46_u * dY_46_u)) != Float32((floor(w) ^ Float32(2.0)) * Float32(dY_46_u * dY_46_u))) ? Float32(t_5 + Float32(dX_46_v * Float32(dX_46_v * (floor(h) ^ Float32(2.0))))) : max(Float32(t_5 + Float32(dX_46_v * Float32(dX_46_v * (floor(h) ^ Float32(2.0))))), Float32((floor(w) ^ Float32(2.0)) * Float32(dY_46_u * dY_46_u)))))));
	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 = t_4 * t_4;
	t_6 = floor(w) * dX_46_u;
	t_7 = ((t_6 * t_6) + (t_2 * t_2)) + t_5;
	tmp = single(0.0);
	if (max(t_7, (((t_0 * t_0) + (t_1 * t_1)) + (t_3 * t_3))) <= single(Inf))
		tmp = log2(sqrt(max(t_7, (single(1.0) / (single(1.0) / ((t_1 ^ single(2.0)) + ((t_0 ^ single(2.0)) + (t_3 ^ single(2.0)))))))));
	else
		tmp = log2(sqrt(max((t_5 + (dX_46_v * (dX_46_v * (floor(h) ^ single(2.0))))), ((floor(w) ^ single(2.0)) * (dY_46_u * dY_46_u)))));
	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 71.5%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)}\right)}\right) \]

      2. flip3-+ N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\frac{{\left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{3} + {\left(\left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}^{3}}{\left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) \cdot \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left(\left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right) \cdot \left(\left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right) - \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) \cdot \left(\left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)\right)}}\right)}\right) \]

      3. clear-num N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\frac{1}{\frac{\left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) \cdot \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left(\left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right) \cdot \left(\left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right) - \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) \cdot \left(\left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)\right)}{{\left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{3} + {\left(\left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}^{3}}}}\right)}\right) \]

   4. Applied rewrites 71.5%

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

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

   1. Initial program 71.5%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)}\right)}\right) \]

      2. flip3-+ N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\frac{{\left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{3} + {\left(\left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}^{3}}{\left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) \cdot \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left(\left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right) \cdot \left(\left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right) - \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) \cdot \left(\left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)\right)}}\right)}\right) \]

      3. clear-num N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\frac{1}{\frac{\left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) \cdot \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left(\left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right) \cdot \left(\left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right) - \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) \cdot \left(\left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)\right)}{{\left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{3} + {\left(\left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}^{3}}}}\right)}\right) \]

   4. Applied rewrites 71.5%

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

   5. Taylor expanded in dY.w around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-pow.f32 N/A

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

      6. lower-floor.f32 56.2

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

   7. Applied rewrites 56.2%

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

   8. Taylor expanded in dX.u around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}} + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 46.9

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

   10. Applied rewrites 46.9%

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

   11. Taylor expanded in dY.u around inf

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

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

       3. lower-pow.f32 N/A

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

       4. lower-floor.f32 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f32 47.3

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

   13. Applied rewrites 47.3%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right) \leq \infty:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \frac{1}{\frac{1}{{\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2} + \left({\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2}\right)}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right) + dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 62.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_1 := {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\\ t_2 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_3 := \left\lfloor d\right\rfloor \cdot dX.w\\ \mathbf{if}\;dY.w \leq 0.5:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_0 \cdot t\_0 + t\_2 \cdot t\_2\right) + t\_3 \cdot t\_3, \mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}, dY.u \cdot dY.u, t\_1\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({t\_0}^{2} + {t\_3}^{2}, t\_1 + \left({\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2}\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor w) dX.u))
        (t_1 (pow (* (floor h) dY.v) 2.0))
        (t_2 (* (floor h) dX.v))
        (t_3 (* (floor d) dX.w)))
   (if (<= dY.w 0.5)
     (log2
      (sqrt
       (fmax
        (+ (+ (* t_0 t_0) (* t_2 t_2)) (* t_3 t_3))
        (fma (pow (floor w) 2.0) (* dY.u dY.u) t_1))))
     (log2
      (sqrt
       (fmax
        (+ (pow t_0 2.0) (pow t_3 2.0))
        (+
         t_1
         (+ (pow (* (floor w) dY.u) 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) {
	float t_0 = floorf(w) * dX_46_u;
	float t_1 = powf((floorf(h) * dY_46_v), 2.0f);
	float t_2 = floorf(h) * dX_46_v;
	float t_3 = floorf(d) * dX_46_w;
	float tmp;
	if (dY_46_w <= 0.5f) {
		tmp = log2f(sqrtf(fmaxf((((t_0 * t_0) + (t_2 * t_2)) + (t_3 * t_3)), fmaf(powf(floorf(w), 2.0f), (dY_46_u * dY_46_u), t_1))));
	} else {
		tmp = log2f(sqrtf(fmaxf((powf(t_0, 2.0f) + powf(t_3, 2.0f)), (t_1 + (powf((floorf(w) * dY_46_u), 2.0f) + 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)
	t_0 = Float32(floor(w) * dX_46_u)
	t_1 = Float32(floor(h) * dY_46_v) ^ Float32(2.0)
	t_2 = Float32(floor(h) * dX_46_v)
	t_3 = Float32(floor(d) * dX_46_w)
	tmp = Float32(0.0)
	if (dY_46_w <= Float32(0.5))
		tmp = log2(sqrt(((Float32(Float32(Float32(t_0 * t_0) + Float32(t_2 * t_2)) + Float32(t_3 * t_3)) != Float32(Float32(Float32(t_0 * t_0) + Float32(t_2 * t_2)) + Float32(t_3 * t_3))) ? fma((floor(w) ^ Float32(2.0)), Float32(dY_46_u * dY_46_u), t_1) : ((fma((floor(w) ^ Float32(2.0)), Float32(dY_46_u * dY_46_u), t_1) != fma((floor(w) ^ Float32(2.0)), Float32(dY_46_u * dY_46_u), t_1)) ? Float32(Float32(Float32(t_0 * t_0) + Float32(t_2 * t_2)) + Float32(t_3 * t_3)) : max(Float32(Float32(Float32(t_0 * t_0) + Float32(t_2 * t_2)) + Float32(t_3 * t_3)), fma((floor(w) ^ Float32(2.0)), Float32(dY_46_u * dY_46_u), t_1))))));
	else
		tmp = log2(sqrt(((Float32((t_0 ^ Float32(2.0)) + (t_3 ^ Float32(2.0))) != Float32((t_0 ^ Float32(2.0)) + (t_3 ^ Float32(2.0)))) ? Float32(t_1 + Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + (Float32(floor(d) * dY_46_w) ^ Float32(2.0)))) : ((Float32(t_1 + Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + (Float32(floor(d) * dY_46_w) ^ Float32(2.0)))) != Float32(t_1 + Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + (Float32(floor(d) * dY_46_w) ^ Float32(2.0))))) ? Float32((t_0 ^ Float32(2.0)) + (t_3 ^ Float32(2.0))) : max(Float32((t_0 ^ Float32(2.0)) + (t_3 ^ Float32(2.0))), Float32(t_1 + Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + (Float32(floor(d) * dY_46_w) ^ Float32(2.0)))))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.w < 0.5

   1. Initial program 69.1%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)}\right)}\right) \]

      2. flip3-+ N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\frac{{\left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{3} + {\left(\left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}^{3}}{\left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) \cdot \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left(\left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right) \cdot \left(\left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right) - \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) \cdot \left(\left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)\right)}}\right)}\right) \]

      3. clear-num N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\frac{1}{\frac{\left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) \cdot \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left(\left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right) \cdot \left(\left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right) - \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) \cdot \left(\left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)\right)}{{\left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{3} + {\left(\left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}^{3}}}}\right)}\right) \]

   4. Applied rewrites 69.1%

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

   5. Taylor expanded in dY.w around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-pow.f32 N/A

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

      6. lower-floor.f32 53.7

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

   7. Applied rewrites 53.7%

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

   8. Taylor expanded in dY.w around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. unswap-sqr N/A

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

      10. unpow2 N/A

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

      11. lower-pow.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. lower-floor.f32 64.0

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

   10. Applied rewrites 64.0%

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

   if 0.5 < dY.w

   1. Initial program 63.3%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dX.v around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. lower-pow.f32 N/A

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

      16. lower-floor.f32 58.5

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

   5. Applied rewrites 58.5%

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

   7. Applied rewrites 58.5%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.w \leq 0.5:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({\left(\left\lfloor w\right\rfloor \right)}^{2}, dY.u \cdot dY.u, {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2} + {\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2} + \left({\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2}\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024223 
(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)))))))