Isotropic LOD (LOD)

Percentage Accurate: 67.7% → 67.7%

Time: 16.1s

Alternatives: 7

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(fmax(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(fmax(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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 67.7%

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

3. Applied rewrites 67.7%

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

Alternative 2: 67.7% accurate, 1.0× speedup

Math

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

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 (* (floor d) (floor d)))
        (t_2 (* (floor h) (floor h)))
        (t_3 (* (floor w) dY.u)))
   (log2
    (sqrt
     (fmax
      (fma (* t_1 dX.w) dX.w (fma (* t_2 dX.v) dX.v (* t_0 t_0)))
      (fma (* t_1 dY.w) dY.w (fma (* t_2 dY.v) dY.v (* 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) * dX_46_u;
	float t_1 = floorf(d) * floorf(d);
	float t_2 = floorf(h) * floorf(h);
	float t_3 = floorf(w) * dY_46_u;
	return log2f(sqrtf(fmaxf(fmaf((t_1 * dX_46_w), dX_46_w, fmaf((t_2 * dX_46_v), dX_46_v, (t_0 * t_0))), fmaf((t_1 * dY_46_w), dY_46_w, fmaf((t_2 * dY_46_v), dY_46_v, (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) * dX_46_u)
	t_1 = Float32(floor(d) * floor(d))
	t_2 = Float32(floor(h) * floor(h))
	t_3 = Float32(floor(w) * dY_46_u)
	return log2(sqrt(fmax(fma(Float32(t_1 * dX_46_w), dX_46_w, fma(Float32(t_2 * dX_46_v), dX_46_v, Float32(t_0 * t_0))), fma(Float32(t_1 * dY_46_w), dY_46_w, fma(Float32(t_2 * dY_46_v), dY_46_v, Float32(t_3 * t_3))))))
end

Derivation

1. Initial program 67.7%

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

3. Applied rewrites 67.7%

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

Alternative 3: 63.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_1 := \left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor \\ t_2 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_3 := t\_2 \cdot t\_2\\ t_4 := \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \\ t_5 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_6 := \left\lfloor d\right\rfloor \cdot dX.w\\ \mathbf{if}\;dY.w \leq 62000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_1 \cdot dX.w, dX.w, \mathsf{fma}\left(t\_4 \cdot dX.v, dX.v, t\_5 \cdot t\_5\right)\right), \mathsf{fma}\left(dY.u \cdot dY.u, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor , t\_3\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v \cdot dX.v, t\_4, t\_6 \cdot t\_6\right), \mathsf{fma}\left(t\_0, t\_0, \mathsf{fma}\left(t\_1 \cdot dY.w, dY.w, t\_3\right)\right)\right)}\right)\\ \end{array} \end{array} \]

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 d) (floor d)))
        (t_2 (* (floor h) dY.v))
        (t_3 (* t_2 t_2))
        (t_4 (* (floor h) (floor h)))
        (t_5 (* (floor w) dX.u))
        (t_6 (* (floor d) dX.w)))
   (if (<= dY.w 62000.0)
     (log2
      (sqrt
       (fmax
        (fma (* t_1 dX.w) dX.w (fma (* t_4 dX.v) dX.v (* t_5 t_5)))
        (fma (* dY.u dY.u) (* (floor w) (floor w)) t_3))))
     (log2
      (sqrt
       (fmax
        (fma (* dX.v dX.v) t_4 (* t_6 t_6))
        (fma t_0 t_0 (fma (* t_1 dY.w) dY.w 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(d) * floorf(d);
	float t_2 = floorf(h) * dY_46_v;
	float t_3 = t_2 * t_2;
	float t_4 = floorf(h) * floorf(h);
	float t_5 = floorf(w) * dX_46_u;
	float t_6 = floorf(d) * dX_46_w;
	float tmp;
	if (dY_46_w <= 62000.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf((t_1 * dX_46_w), dX_46_w, fmaf((t_4 * dX_46_v), dX_46_v, (t_5 * t_5))), fmaf((dY_46_u * dY_46_u), (floorf(w) * floorf(w)), t_3))));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf((dX_46_v * dX_46_v), t_4, (t_6 * t_6)), fmaf(t_0, t_0, fmaf((t_1 * dY_46_w), dY_46_w, t_3)))));
	}
	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(d) * floor(d))
	t_2 = Float32(floor(h) * dY_46_v)
	t_3 = Float32(t_2 * t_2)
	t_4 = Float32(floor(h) * floor(h))
	t_5 = Float32(floor(w) * dX_46_u)
	t_6 = Float32(floor(d) * dX_46_w)
	tmp = Float32(0.0)
	if (dY_46_w <= Float32(62000.0))
		tmp = log2(sqrt(fmax(fma(Float32(t_1 * dX_46_w), dX_46_w, fma(Float32(t_4 * dX_46_v), dX_46_v, Float32(t_5 * t_5))), fma(Float32(dY_46_u * dY_46_u), Float32(floor(w) * floor(w)), t_3))));
	else
		tmp = log2(sqrt(fmax(fma(Float32(dX_46_v * dX_46_v), t_4, Float32(t_6 * t_6)), fma(t_0, t_0, fma(Float32(t_1 * dY_46_w), dY_46_w, t_3)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.w < 62000

   1. Initial program 67.7%

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

   2. Taylor expanded in dY.u around 0

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

   4. Applied rewrites 60.5%

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

   5. Taylor expanded in dY.w around 0

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

   7. Applied rewrites 60.6%

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

   9. Applied rewrites 60.6%

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

   if 62000 < dY.w

   1. Initial program 67.7%

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

   2. Taylor expanded in dX.u around 0

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

   4. Applied rewrites 60.9%

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

   6. Applied rewrites 60.9%

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

Alternative 4: 63.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_1 := \left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor \\ t_2 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_3 := \mathsf{fma}\left(t\_1 \cdot dY.w, dY.w, t\_2 \cdot t\_2\right)\\ t_4 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_5 := \left\lfloor d\right\rfloor \cdot dX.w\\ \mathbf{if}\;dY.u \leq 0.20000000298023224:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_1 \cdot dX.w, dX.w, \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, dX.v, t\_4 \cdot t\_4\right)\right), t\_3\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u \cdot dX.u, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor , t\_5 \cdot t\_5\right), \mathsf{fma}\left(t\_0, t\_0, t\_3\right)\right)}\right)\\ \end{array} \end{array} \]

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 d) (floor d)))
        (t_2 (* (floor h) dY.v))
        (t_3 (fma (* t_1 dY.w) dY.w (* t_2 t_2)))
        (t_4 (* (floor w) dX.u))
        (t_5 (* (floor d) dX.w)))
   (if (<= dY.u 0.20000000298023224)
     (log2
      (sqrt
       (fmax
        (fma
         (* t_1 dX.w)
         dX.w
         (fma (* (* (floor h) (floor h)) dX.v) dX.v (* t_4 t_4)))
        t_3)))
     (log2
      (sqrt
       (fmax
        (fma (* dX.u dX.u) (* (floor w) (floor w)) (* t_5 t_5))
        (fma t_0 t_0 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(d) * floorf(d);
	float t_2 = floorf(h) * dY_46_v;
	float t_3 = fmaf((t_1 * dY_46_w), dY_46_w, (t_2 * t_2));
	float t_4 = floorf(w) * dX_46_u;
	float t_5 = floorf(d) * dX_46_w;
	float tmp;
	if (dY_46_u <= 0.20000000298023224f) {
		tmp = log2f(sqrtf(fmaxf(fmaf((t_1 * dX_46_w), dX_46_w, fmaf(((floorf(h) * floorf(h)) * dX_46_v), dX_46_v, (t_4 * t_4))), t_3)));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf((dX_46_u * dX_46_u), (floorf(w) * floorf(w)), (t_5 * t_5)), fmaf(t_0, t_0, t_3))));
	}
	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(d) * floor(d))
	t_2 = Float32(floor(h) * dY_46_v)
	t_3 = fma(Float32(t_1 * dY_46_w), dY_46_w, Float32(t_2 * t_2))
	t_4 = Float32(floor(w) * dX_46_u)
	t_5 = Float32(floor(d) * dX_46_w)
	tmp = Float32(0.0)
	if (dY_46_u <= Float32(0.20000000298023224))
		tmp = log2(sqrt(fmax(fma(Float32(t_1 * dX_46_w), dX_46_w, fma(Float32(Float32(floor(h) * floor(h)) * dX_46_v), dX_46_v, Float32(t_4 * t_4))), t_3)));
	else
		tmp = log2(sqrt(fmax(fma(Float32(dX_46_u * dX_46_u), Float32(floor(w) * floor(w)), Float32(t_5 * t_5)), fma(t_0, t_0, t_3))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 0.200000003

   1. Initial program 67.7%

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

   2. Taylor expanded in dY.u around 0

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

   4. Applied rewrites 60.5%

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

   6. Applied rewrites 60.5%

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

   if 0.200000003 < dY.u

   1. Initial program 67.7%

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

   2. 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. Applied rewrites 60.4%

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

   6. Applied rewrites 60.4%

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

Alternative 5: 62.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_1 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_2 := t\_1 \cdot t\_1\\ t_3 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_4 := \mathsf{fma}\left(t\_3, t\_3, \mathsf{fma}\left(\left(\left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor \right) \cdot dY.w, dY.w, t\_0 \cdot t\_0\right)\right)\\ \mathbf{if}\;dX.u \leq 4:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v \cdot dX.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , t\_2\right), t\_4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u \cdot dX.u, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor , t\_2\right), t\_4\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor h) dY.v))
        (t_1 (* (floor d) dX.w))
        (t_2 (* t_1 t_1))
        (t_3 (* (floor w) dY.u))
        (t_4
         (fma
          t_3
          t_3
          (fma (* (* (floor d) (floor d)) dY.w) dY.w (* t_0 t_0)))))
   (if (<= dX.u 4.0)
     (log2 (sqrt (fmax (fma (* dX.v dX.v) (* (floor h) (floor h)) t_2) t_4)))
     (log2
      (sqrt (fmax (fma (* dX.u dX.u) (* (floor w) (floor w)) t_2) t_4))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(h) * dY_46_v;
	float t_1 = floorf(d) * dX_46_w;
	float t_2 = t_1 * t_1;
	float t_3 = floorf(w) * dY_46_u;
	float t_4 = fmaf(t_3, t_3, fmaf(((floorf(d) * floorf(d)) * dY_46_w), dY_46_w, (t_0 * t_0)));
	float tmp;
	if (dX_46_u <= 4.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf((dX_46_v * dX_46_v), (floorf(h) * floorf(h)), t_2), t_4)));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf((dX_46_u * dX_46_u), (floorf(w) * floorf(w)), t_2), t_4)));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(h) * dY_46_v)
	t_1 = Float32(floor(d) * dX_46_w)
	t_2 = Float32(t_1 * t_1)
	t_3 = Float32(floor(w) * dY_46_u)
	t_4 = fma(t_3, t_3, fma(Float32(Float32(floor(d) * floor(d)) * dY_46_w), dY_46_w, Float32(t_0 * t_0)))
	tmp = Float32(0.0)
	if (dX_46_u <= Float32(4.0))
		tmp = log2(sqrt(fmax(fma(Float32(dX_46_v * dX_46_v), Float32(floor(h) * floor(h)), t_2), t_4)));
	else
		tmp = log2(sqrt(fmax(fma(Float32(dX_46_u * dX_46_u), Float32(floor(w) * floor(w)), t_2), t_4)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 4

   1. Initial program 67.7%

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

   2. Taylor expanded in dX.u around 0

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

   4. Applied rewrites 60.9%

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

   6. Applied rewrites 60.9%

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

   if 4 < dX.u

   1. Initial program 67.7%

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

   2. 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. Applied rewrites 60.4%

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

   6. Applied rewrites 60.4%

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

Alternative 6: 62.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_1 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_2 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_3 := \left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor \\ t_4 := \left\lfloor w\right\rfloor \cdot dX.u\\ \mathbf{if}\;dX.v \leq 700000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u \cdot dX.u, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor , t\_1 \cdot t\_1\right), \mathsf{fma}\left(t\_2, t\_2, \mathsf{fma}\left(t\_3 \cdot dY.w, dY.w, t\_0 \cdot t\_0\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_3 \cdot dX.w, dX.w, \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, dX.v, t\_4 \cdot t\_4\right)\right), \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor h) dY.v))
        (t_1 (* (floor d) dX.w))
        (t_2 (* (floor w) dY.u))
        (t_3 (* (floor d) (floor d)))
        (t_4 (* (floor w) dX.u)))
   (if (<= dX.v 700000.0)
     (log2
      (sqrt
       (fmax
        (fma (* dX.u dX.u) (* (floor w) (floor w)) (* t_1 t_1))
        (fma t_2 t_2 (fma (* t_3 dY.w) dY.w (* t_0 t_0))))))
     (log2
      (sqrt
       (fmax
        (fma
         (* t_3 dX.w)
         dX.w
         (fma (* (* (floor h) (floor h)) dX.v) dX.v (* t_4 t_4)))
        (* (* (* dY.u dY.u) (floor w)) (floor w))))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(h) * dY_46_v;
	float t_1 = floorf(d) * dX_46_w;
	float t_2 = floorf(w) * dY_46_u;
	float t_3 = floorf(d) * floorf(d);
	float t_4 = floorf(w) * dX_46_u;
	float tmp;
	if (dX_46_v <= 700000.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf((dX_46_u * dX_46_u), (floorf(w) * floorf(w)), (t_1 * t_1)), fmaf(t_2, t_2, fmaf((t_3 * dY_46_w), dY_46_w, (t_0 * t_0))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf((t_3 * dX_46_w), dX_46_w, fmaf(((floorf(h) * floorf(h)) * dX_46_v), dX_46_v, (t_4 * t_4))), (((dY_46_u * dY_46_u) * floorf(w)) * floorf(w)))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(h) * dY_46_v)
	t_1 = Float32(floor(d) * dX_46_w)
	t_2 = Float32(floor(w) * dY_46_u)
	t_3 = Float32(floor(d) * floor(d))
	t_4 = Float32(floor(w) * dX_46_u)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(700000.0))
		tmp = log2(sqrt(fmax(fma(Float32(dX_46_u * dX_46_u), Float32(floor(w) * floor(w)), Float32(t_1 * t_1)), fma(t_2, t_2, fma(Float32(t_3 * dY_46_w), dY_46_w, Float32(t_0 * t_0))))));
	else
		tmp = log2(sqrt(fmax(fma(Float32(t_3 * dX_46_w), dX_46_w, fma(Float32(Float32(floor(h) * floor(h)) * dX_46_v), dX_46_v, Float32(t_4 * t_4))), Float32(Float32(Float32(dY_46_u * dY_46_u) * floor(w)) * floor(w)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 7e5

   1. Initial program 67.7%

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

   2. 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. Applied rewrites 60.4%

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

   6. Applied rewrites 60.4%

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

   if 7e5 < dX.v

   1. Initial program 67.7%

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

   2. Taylor expanded in dY.u around 0

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

   4. Applied rewrites 60.5%

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

   5. Taylor expanded in dY.u around inf

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

   7. Applied rewrites 53.4%

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

   9. Applied rewrites 53.4%

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

Alternative 7: 53.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dX.u\\ \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(\left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w, dX.w, \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, dX.v, t\_0 \cdot t\_0\right)\right), \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)}\right) \end{array} \end{array} \]

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)))
   (log2
    (sqrt
     (fmax
      (fma
       (* (* (floor d) (floor d)) dX.w)
       dX.w
       (fma (* (* (floor h) (floor h)) dX.v) dX.v (* t_0 t_0)))
      (* (* (* dY.u dY.u) (floor w)) (floor w)))))))

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;
	return log2f(sqrtf(fmaxf(fmaf(((floorf(d) * floorf(d)) * dX_46_w), dX_46_w, fmaf(((floorf(h) * floorf(h)) * dX_46_v), dX_46_v, (t_0 * t_0))), (((dY_46_u * dY_46_u) * floorf(w)) * floorf(w)))));
}

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)
	return log2(sqrt(fmax(fma(Float32(Float32(floor(d) * floor(d)) * dX_46_w), dX_46_w, fma(Float32(Float32(floor(h) * floor(h)) * dX_46_v), dX_46_v, Float32(t_0 * t_0))), Float32(Float32(Float32(dY_46_u * dY_46_u) * floor(w)) * floor(w)))))
end

Derivation

1. Initial program 67.7%

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

2. Taylor expanded in dY.u around 0

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

4. Applied rewrites 60.5%

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

5. Taylor expanded in dY.u around inf

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

7. Applied rewrites 53.4%

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

9. Applied rewrites 53.4%

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

Reproduce

herbie shell --seed 2025153 
(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)))))))