Isotropic LOD (LOD)

Percentage Accurate: 68.2% → 71.7%

Time: 11.2s

Alternatives: 9

Speedup: 1.0×

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

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

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: 68.2% accurate, 1.0× speedup

Math

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

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: 71.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_1 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_2 := t\_0 \cdot t\_0 + t\_1 \cdot t\_1\\ t_3 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_4 := \left\lfloor d\right\rfloor \cdot \left|dY.w\right|\\ t_5 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_6 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_7 := \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \\ \mathbf{if}\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_6 \cdot t\_6 + t\_3 \cdot t\_3\right) + t\_5 \cdot t\_5, t\_2 + t\_4 \cdot t\_4\right)}\right) \leq 100:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_7 \cdot dX.v, dX.v, \mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot \left\lfloor d\right\rfloor , dX.w, \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor , dY.u \cdot dY.u, \mathsf{fma}\left(t\_7 \cdot dY.v, dY.v, \left(\left(\left|dY.w\right| \cdot \left\lfloor d\right\rfloor \right) \cdot \left|dY.w\right|\right) \cdot \left\lfloor d\right\rfloor \right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({dX.u}^{2}, {\left(\left\lfloor w\right\rfloor \right)}^{2}, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), t\_2 + e^{\log \left(\left(-\left|dY.w\right|\right) \cdot \left\lfloor d\right\rfloor \right) \cdot 2}\right)}\right)\\ \end{array} \]

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 (+ (* t_0 t_0) (* t_1 t_1)))
        (t_3 (* (floor h) dX.v))
        (t_4 (* (floor d) (fabs dY.w)))
        (t_5 (* (floor d) dX.w))
        (t_6 (* (floor w) dX.u))
        (t_7 (* (floor h) (floor h))))
   (if (<=
        (log2
         (sqrt
          (fmax
           (+ (+ (* t_6 t_6) (* t_3 t_3)) (* t_5 t_5))
           (+ t_2 (* t_4 t_4)))))
        100.0)
     (log2
      (sqrt
       (fmax
        (fma
         (* t_7 dX.v)
         dX.v
         (fma
          (* (* dX.w (floor d)) (floor d))
          dX.w
          (* (* (* dX.u (floor w)) dX.u) (floor w))))
        (fma
         (* (floor w) (floor w))
         (* dY.u dY.u)
         (fma
          (* t_7 dY.v)
          dY.v
          (* (* (* (fabs dY.w) (floor d)) (fabs dY.w)) (floor d)))))))
     (log2
      (sqrt
       (fmax
        (fma
         (pow dX.u 2.0)
         (pow (floor w) 2.0)
         (* (pow dX.w 2.0) (pow (floor d) 2.0)))
        (+ t_2 (exp (* (log (* (- (fabs dY.w)) (floor d))) 2.0)))))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(w) * dY_46_u;
	float t_1 = floorf(h) * dY_46_v;
	float t_2 = (t_0 * t_0) + (t_1 * t_1);
	float t_3 = floorf(h) * dX_46_v;
	float t_4 = floorf(d) * fabsf(dY_46_w);
	float t_5 = floorf(d) * dX_46_w;
	float t_6 = floorf(w) * dX_46_u;
	float t_7 = floorf(h) * floorf(h);
	float tmp;
	if (log2f(sqrtf(fmaxf((((t_6 * t_6) + (t_3 * t_3)) + (t_5 * t_5)), (t_2 + (t_4 * t_4))))) <= 100.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf((t_7 * dX_46_v), dX_46_v, fmaf(((dX_46_w * floorf(d)) * floorf(d)), dX_46_w, (((dX_46_u * floorf(w)) * dX_46_u) * floorf(w)))), fmaf((floorf(w) * floorf(w)), (dY_46_u * dY_46_u), fmaf((t_7 * dY_46_v), dY_46_v, (((fabsf(dY_46_w) * floorf(d)) * fabsf(dY_46_w)) * floorf(d)))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf(powf(dX_46_u, 2.0f), powf(floorf(w), 2.0f), (powf(dX_46_w, 2.0f) * powf(floorf(d), 2.0f))), (t_2 + expf((logf((-fabsf(dY_46_w) * floorf(d))) * 2.0f))))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * dY_46_u)
	t_1 = Float32(floor(h) * dY_46_v)
	t_2 = Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1))
	t_3 = Float32(floor(h) * dX_46_v)
	t_4 = Float32(floor(d) * abs(dY_46_w))
	t_5 = Float32(floor(d) * dX_46_w)
	t_6 = Float32(floor(w) * dX_46_u)
	t_7 = Float32(floor(h) * floor(h))
	tmp = Float32(0.0)
	if (log2(sqrt(fmax(Float32(Float32(Float32(t_6 * t_6) + Float32(t_3 * t_3)) + Float32(t_5 * t_5)), Float32(t_2 + Float32(t_4 * t_4))))) <= Float32(100.0))
		tmp = log2(sqrt(fmax(fma(Float32(t_7 * dX_46_v), dX_46_v, fma(Float32(Float32(dX_46_w * floor(d)) * floor(d)), dX_46_w, Float32(Float32(Float32(dX_46_u * floor(w)) * dX_46_u) * floor(w)))), fma(Float32(floor(w) * floor(w)), Float32(dY_46_u * dY_46_u), fma(Float32(t_7 * dY_46_v), dY_46_v, Float32(Float32(Float32(abs(dY_46_w) * floor(d)) * abs(dY_46_w)) * floor(d)))))));
	else
		tmp = log2(sqrt(fmax(fma((dX_46_u ^ Float32(2.0)), (floor(w) ^ Float32(2.0)), Float32((dX_46_w ^ Float32(2.0)) * (floor(d) ^ Float32(2.0)))), Float32(t_2 + exp(Float32(log(Float32(Float32(-abs(dY_46_w)) * floor(d))) * Float32(2.0)))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log2.f32 (sqrt.f32 (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)))))) < 100

   1. Initial program 68.2%

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

   3. Applied rewrites 68.2%

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

   5. Applied rewrites 68.2%

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

   if 100 < (log2.f32 (sqrt.f32 (fmax.f32 (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (*.f32 (*.f32 (floor.f32 d) dX.w) (*.f32 (floor.f32 d) dX.w))) (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))) (*.f32 (*.f32 (floor.f32 d) dY.w) (*.f32 (floor.f32 d) dY.w))))))

   1. Initial program 68.2%

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

      1. lift-*.f32 N/A

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

      2. sqr-neg-rev N/A

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

      3. pow2 N/A

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

      4. pow-to-exp N/A

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

      5. lower-unsound-exp.f32 N/A

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

      6. lower-unsound-*.f32 N/A

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

      7. lower-unsound-log.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-*.f32 N/A

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

      12. lower-neg.f32 57.9

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

   3. Applied rewrites 57.9%

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

   4. Taylor expanded in dX.v around 0

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

      1. lower-fma.f32 N/A

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

      2. lower-pow.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 49.3

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

   6. Applied rewrites 49.3%

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

Alternative 2: 71.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \\ t_1 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_2 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_3 := \left|dX.u\right| \cdot \left\lfloor w\right\rfloor \\ t_4 := \left\lfloor d\right\rfloor \cdot \left|dY.w\right|\\ t_5 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_6 := t\_5 \cdot t\_5\\ t_7 := \left\lfloor w\right\rfloor \cdot \left|dX.u\right|\\ t_8 := t\_2 \cdot t\_2\\ t_9 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_10 := t\_1 \cdot t\_1 + t\_9 \cdot t\_9\\ \mathbf{if}\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_7 \cdot t\_7 + t\_8\right) + t\_6, t\_10 + t\_4 \cdot t\_4\right)}\right) \leq 100:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_0 \cdot dX.v, dX.v, \mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot \left\lfloor d\right\rfloor , dX.w, \left(t\_3 \cdot \left|dX.u\right|\right) \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor , dY.u \cdot dY.u, \mathsf{fma}\left(t\_0 \cdot dY.v, dY.v, \left(\left(\left|dY.w\right| \cdot \left\lfloor d\right\rfloor \right) \cdot \left|dY.w\right|\right) \cdot \left\lfloor d\right\rfloor \right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(e^{\log t\_3 \cdot 2} + t\_8\right) + t\_6, t\_10 + e^{\log \left(\left(-\left|dY.w\right|\right) \cdot \left\lfloor d\right\rfloor \right) \cdot 2}\right)}\right)\\ \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor h) (floor h)))
        (t_1 (* (floor w) dY.u))
        (t_2 (* (floor h) dX.v))
        (t_3 (* (fabs dX.u) (floor w)))
        (t_4 (* (floor d) (fabs dY.w)))
        (t_5 (* (floor d) dX.w))
        (t_6 (* t_5 t_5))
        (t_7 (* (floor w) (fabs dX.u)))
        (t_8 (* t_2 t_2))
        (t_9 (* (floor h) dY.v))
        (t_10 (+ (* t_1 t_1) (* t_9 t_9))))
   (if (<=
        (log2 (sqrt (fmax (+ (+ (* t_7 t_7) t_8) t_6) (+ t_10 (* t_4 t_4)))))
        100.0)
     (log2
      (sqrt
       (fmax
        (fma
         (* t_0 dX.v)
         dX.v
         (fma
          (* (* dX.w (floor d)) (floor d))
          dX.w
          (* (* t_3 (fabs dX.u)) (floor w))))
        (fma
         (* (floor w) (floor w))
         (* dY.u dY.u)
         (fma
          (* t_0 dY.v)
          dY.v
          (* (* (* (fabs dY.w) (floor d)) (fabs dY.w)) (floor d)))))))
     (log2
      (sqrt
       (fmax
        (+ (+ (exp (* (log t_3) 2.0)) t_8) t_6)
        (+ t_10 (exp (* (log (* (- (fabs dY.w)) (floor d))) 2.0)))))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(h) * floorf(h);
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = floorf(h) * dX_46_v;
	float t_3 = fabsf(dX_46_u) * floorf(w);
	float t_4 = floorf(d) * fabsf(dY_46_w);
	float t_5 = floorf(d) * dX_46_w;
	float t_6 = t_5 * t_5;
	float t_7 = floorf(w) * fabsf(dX_46_u);
	float t_8 = t_2 * t_2;
	float t_9 = floorf(h) * dY_46_v;
	float t_10 = (t_1 * t_1) + (t_9 * t_9);
	float tmp;
	if (log2f(sqrtf(fmaxf((((t_7 * t_7) + t_8) + t_6), (t_10 + (t_4 * t_4))))) <= 100.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf((t_0 * dX_46_v), dX_46_v, fmaf(((dX_46_w * floorf(d)) * floorf(d)), dX_46_w, ((t_3 * fabsf(dX_46_u)) * floorf(w)))), fmaf((floorf(w) * floorf(w)), (dY_46_u * dY_46_u), fmaf((t_0 * dY_46_v), dY_46_v, (((fabsf(dY_46_w) * floorf(d)) * fabsf(dY_46_w)) * floorf(d)))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(((expf((logf(t_3) * 2.0f)) + t_8) + t_6), (t_10 + expf((logf((-fabsf(dY_46_w) * floorf(d))) * 2.0f))))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(h) * floor(h))
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(h) * dX_46_v)
	t_3 = Float32(abs(dX_46_u) * floor(w))
	t_4 = Float32(floor(d) * abs(dY_46_w))
	t_5 = Float32(floor(d) * dX_46_w)
	t_6 = Float32(t_5 * t_5)
	t_7 = Float32(floor(w) * abs(dX_46_u))
	t_8 = Float32(t_2 * t_2)
	t_9 = Float32(floor(h) * dY_46_v)
	t_10 = Float32(Float32(t_1 * t_1) + Float32(t_9 * t_9))
	tmp = Float32(0.0)
	if (log2(sqrt(fmax(Float32(Float32(Float32(t_7 * t_7) + t_8) + t_6), Float32(t_10 + Float32(t_4 * t_4))))) <= Float32(100.0))
		tmp = log2(sqrt(fmax(fma(Float32(t_0 * dX_46_v), dX_46_v, fma(Float32(Float32(dX_46_w * floor(d)) * floor(d)), dX_46_w, Float32(Float32(t_3 * abs(dX_46_u)) * floor(w)))), fma(Float32(floor(w) * floor(w)), Float32(dY_46_u * dY_46_u), fma(Float32(t_0 * dY_46_v), dY_46_v, Float32(Float32(Float32(abs(dY_46_w) * floor(d)) * abs(dY_46_w)) * floor(d)))))));
	else
		tmp = log2(sqrt(fmax(Float32(Float32(exp(Float32(log(t_3) * Float32(2.0))) + t_8) + t_6), Float32(t_10 + exp(Float32(log(Float32(Float32(-abs(dY_46_w)) * floor(d))) * Float32(2.0)))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log2.f32 (sqrt.f32 (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)))))) < 100

   1. Initial program 68.2%

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

   3. Applied rewrites 68.2%

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

   5. Applied rewrites 68.2%

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

   if 100 < (log2.f32 (sqrt.f32 (fmax.f32 (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (*.f32 (*.f32 (floor.f32 d) dX.w) (*.f32 (floor.f32 d) dX.w))) (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))) (*.f32 (*.f32 (floor.f32 d) dY.w) (*.f32 (floor.f32 d) dY.w))))))

   1. Initial program 68.2%

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

      1. lift-*.f32 N/A

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

      2. sqr-neg-rev N/A

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

      3. pow2 N/A

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

      4. pow-to-exp N/A

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

      5. lower-unsound-exp.f32 N/A

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

      6. lower-unsound-*.f32 N/A

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

      7. lower-unsound-log.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-*.f32 N/A

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

      12. lower-neg.f32 57.9

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

   3. Applied rewrites 57.9%

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

      1. lift-*.f32 N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. lower-unsound-exp.f32 N/A

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

      5. lower-unsound-*.f32 N/A

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

      6. lower-unsound-log.f32 41.8

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

      7. lift-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f32 41.8

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

   5. Applied rewrites 41.8%

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

Alternative 3: 71.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_1 := dX.w \cdot \left\lfloor d\right\rfloor \\ t_2 := \left\lfloor h\right\rfloor \cdot \left|dY.v\right|\\ t_3 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_4 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_5 := \left(dY.w \cdot \left\lfloor d\right\rfloor \right) \cdot dY.w\\ t_6 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_7 := \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \\ t_8 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_9 := \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\\ \mathbf{if}\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_6 \cdot t\_6 + t\_8 \cdot t\_8\right) + t\_4 \cdot t\_4, \left(t\_0 \cdot t\_0 + t\_2 \cdot t\_2\right) + t\_3 \cdot t\_3\right)}\right) \leq 100:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_7 \cdot dX.v, dX.v, \mathsf{fma}\left(t\_1 \cdot \left\lfloor d\right\rfloor , dX.w, t\_9 \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor , dY.u \cdot dY.u, \mathsf{fma}\left(t\_7 \cdot \left|dY.v\right|, \left|dY.v\right|, t\_5 \cdot \left\lfloor d\right\rfloor \right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_5, \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , e^{\log \left(\left(-\left|dY.v\right|\right) \cdot \left\lfloor h\right\rfloor \right) \cdot 2}\right)\right), \mathsf{fma}\left(t\_1 \cdot dX.w, \left\lfloor d\right\rfloor , \mathsf{fma}\left(t\_9, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)\right)}\right)\\ \end{array} \]

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 (* dX.w (floor d)))
        (t_2 (* (floor h) (fabs dY.v)))
        (t_3 (* (floor d) dY.w))
        (t_4 (* (floor d) dX.w))
        (t_5 (* (* dY.w (floor d)) dY.w))
        (t_6 (* (floor w) dX.u))
        (t_7 (* (floor h) (floor h)))
        (t_8 (* (floor h) dX.v))
        (t_9 (* (* dX.u (floor w)) dX.u)))
   (if (<=
        (log2
         (sqrt
          (fmax
           (+ (+ (* t_6 t_6) (* t_8 t_8)) (* t_4 t_4))
           (+ (+ (* t_0 t_0) (* t_2 t_2)) (* t_3 t_3)))))
        100.0)
     (log2
      (sqrt
       (fmax
        (fma (* t_7 dX.v) dX.v (fma (* t_1 (floor d)) dX.w (* t_9 (floor w))))
        (fma
         (* (floor w) (floor w))
         (* dY.u dY.u)
         (fma (* t_7 (fabs dY.v)) (fabs dY.v) (* t_5 (floor d)))))))
     (log2
      (sqrt
       (fmax
        (fma
         t_5
         (floor d)
         (fma
          (* (* dY.u (floor w)) dY.u)
          (floor w)
          (exp (* (log (* (- (fabs dY.v)) (floor h))) 2.0))))
        (fma
         (* t_1 dX.w)
         (floor d)
         (fma t_9 (floor w) (* (* (* dX.v (floor h)) dX.v) (floor h))))))))))

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 = dX_46_w * floorf(d);
	float t_2 = floorf(h) * fabsf(dY_46_v);
	float t_3 = floorf(d) * dY_46_w;
	float t_4 = floorf(d) * dX_46_w;
	float t_5 = (dY_46_w * floorf(d)) * dY_46_w;
	float t_6 = floorf(w) * dX_46_u;
	float t_7 = floorf(h) * floorf(h);
	float t_8 = floorf(h) * dX_46_v;
	float t_9 = (dX_46_u * floorf(w)) * dX_46_u;
	float tmp;
	if (log2f(sqrtf(fmaxf((((t_6 * t_6) + (t_8 * t_8)) + (t_4 * t_4)), (((t_0 * t_0) + (t_2 * t_2)) + (t_3 * t_3))))) <= 100.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf((t_7 * dX_46_v), dX_46_v, fmaf((t_1 * floorf(d)), dX_46_w, (t_9 * floorf(w)))), fmaf((floorf(w) * floorf(w)), (dY_46_u * dY_46_u), fmaf((t_7 * fabsf(dY_46_v)), fabsf(dY_46_v), (t_5 * floorf(d)))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf(t_5, floorf(d), fmaf(((dY_46_u * floorf(w)) * dY_46_u), floorf(w), expf((logf((-fabsf(dY_46_v) * floorf(h))) * 2.0f)))), fmaf((t_1 * dX_46_w), floorf(d), fmaf(t_9, floorf(w), (((dX_46_v * floorf(h)) * dX_46_v) * floorf(h)))))));
	}
	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(dX_46_w * floor(d))
	t_2 = Float32(floor(h) * abs(dY_46_v))
	t_3 = Float32(floor(d) * dY_46_w)
	t_4 = Float32(floor(d) * dX_46_w)
	t_5 = Float32(Float32(dY_46_w * floor(d)) * dY_46_w)
	t_6 = Float32(floor(w) * dX_46_u)
	t_7 = Float32(floor(h) * floor(h))
	t_8 = Float32(floor(h) * dX_46_v)
	t_9 = Float32(Float32(dX_46_u * floor(w)) * dX_46_u)
	tmp = Float32(0.0)
	if (log2(sqrt(fmax(Float32(Float32(Float32(t_6 * t_6) + Float32(t_8 * t_8)) + Float32(t_4 * t_4)), Float32(Float32(Float32(t_0 * t_0) + Float32(t_2 * t_2)) + Float32(t_3 * t_3))))) <= Float32(100.0))
		tmp = log2(sqrt(fmax(fma(Float32(t_7 * dX_46_v), dX_46_v, fma(Float32(t_1 * floor(d)), dX_46_w, Float32(t_9 * floor(w)))), fma(Float32(floor(w) * floor(w)), Float32(dY_46_u * dY_46_u), fma(Float32(t_7 * abs(dY_46_v)), abs(dY_46_v), Float32(t_5 * floor(d)))))));
	else
		tmp = log2(sqrt(fmax(fma(t_5, floor(d), fma(Float32(Float32(dY_46_u * floor(w)) * dY_46_u), floor(w), exp(Float32(log(Float32(Float32(-abs(dY_46_v)) * floor(h))) * Float32(2.0))))), fma(Float32(t_1 * dX_46_w), floor(d), fma(t_9, floor(w), Float32(Float32(Float32(dX_46_v * floor(h)) * dX_46_v) * floor(h)))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log2.f32 (sqrt.f32 (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)))))) < 100

   1. Initial program 68.2%

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

   3. Applied rewrites 68.2%

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

   5. Applied rewrites 68.2%

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

   if 100 < (log2.f32 (sqrt.f32 (fmax.f32 (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (*.f32 (*.f32 (floor.f32 d) dX.w) (*.f32 (floor.f32 d) dX.w))) (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))) (*.f32 (*.f32 (floor.f32 d) dY.w) (*.f32 (floor.f32 d) dY.w))))))

   1. Initial program 68.2%

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

   3. Applied rewrites 68.2%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f32 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f32 N/A

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

      9. sqr-neg-rev N/A

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

      10. pow2 N/A

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

      11. pow-to-exp N/A

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

      12. lower-unsound-exp.f32 N/A

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

      13. lower-unsound-*.f32 N/A

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

      14. lower-unsound-log.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. distribute-lft-neg-in N/A

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

      18. lower-*.f32 N/A

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

      19. lower-neg.f32 56.9

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

   5. Applied rewrites 56.9%

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

Alternative 4: 68.2% accurate, 1.0× speedup

Math

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

FPCore

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

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) * floorf(h);
	return log2f(sqrtf(fmaxf(fmaf((t_0 * dX_46_v), dX_46_v, fmaf(((dX_46_w * floorf(d)) * floorf(d)), dX_46_w, (((dX_46_u * floorf(w)) * dX_46_u) * floorf(w)))), fmaf((floorf(w) * floorf(w)), (dY_46_u * dY_46_u), fmaf((t_0 * dY_46_v), dY_46_v, (((dY_46_w * floorf(d)) * dY_46_w) * floorf(d)))))));
}

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) * floor(h))
	return log2(sqrt(fmax(fma(Float32(t_0 * dX_46_v), dX_46_v, fma(Float32(Float32(dX_46_w * floor(d)) * floor(d)), dX_46_w, Float32(Float32(Float32(dX_46_u * floor(w)) * dX_46_u) * floor(w)))), fma(Float32(floor(w) * floor(w)), Float32(dY_46_u * dY_46_u), fma(Float32(t_0 * dY_46_v), dY_46_v, Float32(Float32(Float32(dY_46_w * floor(d)) * dY_46_w) * floor(d)))))))
end

Derivation

1. Initial program 68.2%

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

3. Applied rewrites 68.2%

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

5. Applied rewrites 68.2%

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

Alternative 5: 68.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	return log2(sqrt(fmax(fma(Float32(Float32(dY_46_w * floor(d)) * dY_46_w), floor(d), fma(Float32(Float32(dY_46_u * floor(w)) * dY_46_u), floor(w), Float32(Float32(Float32(dY_46_v * floor(h)) * dY_46_v) * floor(h)))), fma(Float32(Float32(dX_46_w * floor(d)) * dX_46_w), floor(d), fma(Float32(Float32(dX_46_u * floor(w)) * dX_46_u), floor(w), Float32(Float32(Float32(dX_46_v * floor(h)) * dX_46_v) * floor(h)))))))
end

Derivation

1. Initial program 68.2%

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

3. Applied rewrites 68.2%

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

Alternative 6: 65.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_1 := \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \\ \mathbf{if}\;\left|dX.v\right| \leq 0.0035000001080334187:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \mathsf{fma}\left(dY.v \cdot dY.v, t\_1, \left(\left(dY.w \cdot \left\lfloor d\right\rfloor \right) \cdot dY.w\right) \cdot \left\lfloor d\right\rfloor \right)\right), \mathsf{fma}\left(\left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot dX.w, \left\lfloor d\right\rfloor , t\_0 \cdot t\_0\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_1 \cdot \left|dX.v\right|, \left|dX.v\right|, \mathsf{fma}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot \left\lfloor d\right\rfloor , dX.w, \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, dY.u, \left(\left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot dY.w\right) \cdot \left\lfloor d\right\rfloor \right)\right)}\right)\\ \end{array} \]

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 h) (floor h))))
   (if (<= (fabs dX.v) 0.0035000001080334187)
     (log2
      (sqrt
       (fmax
        (fma
         (* (* dY.u (floor w)) dY.u)
         (floor w)
         (fma (* dY.v dY.v) t_1 (* (* (* dY.w (floor d)) dY.w) (floor d))))
        (fma (* (* (floor d) dX.w) dX.w) (floor d) (* t_0 t_0)))))
     (log2
      (sqrt
       (fmax
        (fma
         (* t_1 (fabs dX.v))
         (fabs dX.v)
         (fma
          (* (* dX.w (floor d)) (floor d))
          dX.w
          (* (* (* dX.u (floor w)) dX.u) (floor w))))
        (fma
         (* (* (floor w) (floor w)) dY.u)
         dY.u
         (* (* (* (floor d) dY.w) dY.w) (floor d)))))))))

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(h) * floorf(h);
	float tmp;
	if (fabsf(dX_46_v) <= 0.0035000001080334187f) {
		tmp = log2f(sqrtf(fmaxf(fmaf(((dY_46_u * floorf(w)) * dY_46_u), floorf(w), fmaf((dY_46_v * dY_46_v), t_1, (((dY_46_w * floorf(d)) * dY_46_w) * floorf(d)))), fmaf(((floorf(d) * dX_46_w) * dX_46_w), floorf(d), (t_0 * t_0)))));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf((t_1 * fabsf(dX_46_v)), fabsf(dX_46_v), fmaf(((dX_46_w * floorf(d)) * floorf(d)), dX_46_w, (((dX_46_u * floorf(w)) * dX_46_u) * floorf(w)))), fmaf(((floorf(w) * floorf(w)) * dY_46_u), dY_46_u, (((floorf(d) * dY_46_w) * dY_46_w) * floorf(d))))));
	}
	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) * floor(h))
	tmp = Float32(0.0)
	if (abs(dX_46_v) <= Float32(0.0035000001080334187))
		tmp = log2(sqrt(fmax(fma(Float32(Float32(dY_46_u * floor(w)) * dY_46_u), floor(w), fma(Float32(dY_46_v * dY_46_v), t_1, Float32(Float32(Float32(dY_46_w * floor(d)) * dY_46_w) * floor(d)))), fma(Float32(Float32(floor(d) * dX_46_w) * dX_46_w), floor(d), Float32(t_0 * t_0)))));
	else
		tmp = log2(sqrt(fmax(fma(Float32(t_1 * abs(dX_46_v)), abs(dX_46_v), fma(Float32(Float32(dX_46_w * floor(d)) * floor(d)), dX_46_w, Float32(Float32(Float32(dX_46_u * floor(w)) * dX_46_u) * floor(w)))), fma(Float32(Float32(floor(w) * floor(w)) * dY_46_u), dY_46_u, Float32(Float32(Float32(floor(d) * dY_46_w) * dY_46_w) * floor(d))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 0.00350000011

   1. Initial program 68.2%

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

   3. Applied rewrites 68.2%

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

   4. Taylor expanded in dX.v around 0

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

      1. lower-fma.f32 N/A

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

      2. lower-pow.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 61.2

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

   6. Applied rewrites 61.2%

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

   8. Applied rewrites 61.2%

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

   if 0.00350000011 < dX.v

   1. Initial program 68.2%

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

   3. Applied rewrites 68.2%

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in dY.v around 0

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

      1. lower-fma.f32 N/A

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

      2. lower-pow.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 60.9

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

   8. Applied rewrites 60.9%

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

      1. lift-fma.f32 N/A

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

      2. lift-pow.f32 N/A

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

      3. pow2 N/A

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

      4. lift-pow.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. lift-*.f32 N/A

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

      11. lift-pow.f32 N/A

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

      12. lift-pow.f32 N/A

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

      13. pow-prod-down N/A

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

      14. lift-*.f32 N/A

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

      15. pow2 N/A

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

      16. lift-*.f32 N/A

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

      17. associate-*l* N/A

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

      18. lift-*.f32 N/A

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

      19. lift-*.f32 N/A

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

      20. lower-fma.f32 N/A

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

   10. Applied rewrites 60.9%

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

Alternative 7: 60.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot \left|dX.u\right|\\ t_1 := \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \\ \mathbf{if}\;\left|dX.u\right| \leq 0.09399999678134918:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dY.u \cdot dY.u, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(dY.v \cdot dY.v, t\_1, \left(\left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot dY.w\right) \cdot \left\lfloor d\right\rfloor \right)\right), \left(dX.v \cdot dX.v\right) \cdot t\_1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, \left\lfloor w\right\rfloor , \mathsf{fma}\left(dY.v \cdot dY.v, t\_1, \left(\left(dY.w \cdot \left\lfloor d\right\rfloor \right) \cdot dY.w\right) \cdot \left\lfloor d\right\rfloor \right)\right), \mathsf{fma}\left(\left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot dX.w, \left\lfloor d\right\rfloor , t\_0 \cdot t\_0\right)\right)}\right)\\ \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor w) (fabs dX.u))) (t_1 (* (floor h) (floor h))))
   (if (<= (fabs dX.u) 0.09399999678134918)
     (log2
      (sqrt
       (fmax
        (fma
         (* dY.u dY.u)
         (* (floor w) (floor w))
         (fma (* dY.v dY.v) t_1 (* (* (* (floor d) dY.w) dY.w) (floor d))))
        (* (* dX.v dX.v) t_1))))
     (log2
      (sqrt
       (fmax
        (fma
         (* (* dY.u (floor w)) dY.u)
         (floor w)
         (fma (* dY.v dY.v) t_1 (* (* (* dY.w (floor d)) dY.w) (floor d))))
        (fma (* (* (floor d) dX.w) dX.w) (floor d) (* t_0 t_0))))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(w) * fabsf(dX_46_u);
	float t_1 = floorf(h) * floorf(h);
	float tmp;
	if (fabsf(dX_46_u) <= 0.09399999678134918f) {
		tmp = log2f(sqrtf(fmaxf(fmaf((dY_46_u * dY_46_u), (floorf(w) * floorf(w)), fmaf((dY_46_v * dY_46_v), t_1, (((floorf(d) * dY_46_w) * dY_46_w) * floorf(d)))), ((dX_46_v * dX_46_v) * t_1))));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf(((dY_46_u * floorf(w)) * dY_46_u), floorf(w), fmaf((dY_46_v * dY_46_v), t_1, (((dY_46_w * floorf(d)) * dY_46_w) * floorf(d)))), fmaf(((floorf(d) * dX_46_w) * dX_46_w), floorf(d), (t_0 * t_0)))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * abs(dX_46_u))
	t_1 = Float32(floor(h) * floor(h))
	tmp = Float32(0.0)
	if (abs(dX_46_u) <= Float32(0.09399999678134918))
		tmp = log2(sqrt(fmax(fma(Float32(dY_46_u * dY_46_u), Float32(floor(w) * floor(w)), fma(Float32(dY_46_v * dY_46_v), t_1, Float32(Float32(Float32(floor(d) * dY_46_w) * dY_46_w) * floor(d)))), Float32(Float32(dX_46_v * dX_46_v) * t_1))));
	else
		tmp = log2(sqrt(fmax(fma(Float32(Float32(dY_46_u * floor(w)) * dY_46_u), floor(w), fma(Float32(dY_46_v * dY_46_v), t_1, Float32(Float32(Float32(dY_46_w * floor(d)) * dY_46_w) * floor(d)))), fma(Float32(Float32(floor(d) * dX_46_w) * dX_46_w), floor(d), Float32(t_0 * t_0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 0.0939999968

   1. Initial program 68.2%

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

   3. Applied rewrites 68.2%

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

   5. Applied rewrites 64.4%

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

   6. Taylor expanded in dX.v around inf

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-fabs.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-fabs.f32 N/A

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

      6. lower-floor.f32 N/A

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

      7. lower-floor.f32 50.7

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

   8. Applied rewrites 50.7%

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

   10. Applied rewrites 54.2%

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

   if 0.0939999968 < dX.u

   1. Initial program 68.2%

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

   3. Applied rewrites 68.2%

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

   4. Taylor expanded in dX.v around 0

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

      1. lower-fma.f32 N/A

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

      2. lower-pow.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-floor.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 61.2

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

   6. Applied rewrites 61.2%

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

   8. Applied rewrites 61.2%

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

Alternative 8: 54.2% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (log2
  (sqrt
   (fmax
    (* (fabs dX.v) (* (fabs (fabs dX.v)) (* (fabs (floor h)) (floor h))))
    (fma
     (* (floor w) (floor w))
     (* dY.u dY.u)
     (fma
      (floor h)
      (* (floor h) (* dY.v dY.v))
      (* (* (* (floor d) dY.w) dY.w) (floor d))))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	return log2f(sqrtf(fmaxf((fabsf(dX_46_v) * (fabsf(fabsf(dX_46_v)) * (fabsf(floorf(h)) * floorf(h)))), fmaf((floorf(w) * floorf(w)), (dY_46_u * dY_46_u), fmaf(floorf(h), (floorf(h) * (dY_46_v * dY_46_v)), (((floorf(d) * dY_46_w) * dY_46_w) * floorf(d)))))));
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	return log2(sqrt(fmax(Float32(abs(dX_46_v) * Float32(abs(abs(dX_46_v)) * Float32(abs(floor(h)) * floor(h)))), fma(Float32(floor(w) * floor(w)), Float32(dY_46_u * dY_46_u), fma(floor(h), Float32(floor(h) * Float32(dY_46_v * dY_46_v)), Float32(Float32(Float32(floor(d) * dY_46_w) * dY_46_w) * floor(d)))))))
end

Derivation

1. Initial program 68.2%

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

3. Applied rewrites 68.2%

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

5. Applied rewrites 64.4%

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

6. Taylor expanded in dX.v around inf

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-fabs.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-fabs.f32 N/A

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

   6. lower-floor.f32 N/A

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

   7. lower-floor.f32 50.7

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

8. Applied rewrites 50.7%

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

   1. lift-fma.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

   9. lower-fma.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 50.7

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 50.7

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

10. Applied rewrites 50.7%

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

Alternative 9: 54.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

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) * floor(h))
	return log2(sqrt(fmax(fma(Float32(dY_46_u * dY_46_u), Float32(floor(w) * floor(w)), fma(Float32(dY_46_v * dY_46_v), t_0, Float32(Float32(Float32(floor(d) * dY_46_w) * dY_46_w) * floor(d)))), Float32(Float32(dX_46_v * dX_46_v) * t_0))))
end

Derivation

1. Initial program 68.2%

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

3. Applied rewrites 68.2%

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

5. Applied rewrites 64.4%

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

6. Taylor expanded in dX.v around inf

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-fabs.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-fabs.f32 N/A

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

   6. lower-floor.f32 N/A

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

   7. lower-floor.f32 50.7

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

8. Applied rewrites 50.7%

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

10. Applied rewrites 54.2%

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

Reproduce

herbie shell --seed 2025167 
(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)))))))