Isotropic LOD (LOD)

Percentage Accurate: 67.6% → 67.6%

Time: 7.1s

Alternatives: 8

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

C

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

Julia

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

MATLAB

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

Initial Program: 67.6% accurate, 1.0× speedup

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

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

C

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

Julia

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

MATLAB

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

Alternative 1: 67.6% accurate, 1.0× speedup

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

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 67.6%

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

3. Applied rewrites 67.6%

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

Alternative 2: 67.6% accurate, 1.0× speedup

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

Math

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

FPCore

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

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

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(dY_46_u * floor(w))
	t_1 = Float32(dX_46_w * floor(d))
	t_2 = Float32(dY_46_w * floor(d))
	t_3 = Float32(dX_46_v * floor(h))
	t_4 = Float32(dY_46_v * floor(h))
	t_5 = Float32(dX_46_u * floor(w))
	return log2(sqrt(fmax(fma(t_2, t_2, fma(t_4, t_4, Float32(t_0 * t_0))), fma(t_1, t_1, fma(t_3, t_3, Float32(t_5 * t_5))))))
end

Derivation

1. Initial program 67.6%

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

3. Applied rewrites 67.6%

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

Alternative 3: 65.6% accurate, 1.1× speedup

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

Math

\[\begin{array}{l} t_0 := \left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor \\ t_1 := \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \\ t_2 := \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \\ t_3 := \mathsf{fma}\left(t\_2, dX.u \cdot dX.u, t\_0 \cdot \left(dX.w \cdot dX.w\right)\right)\\ \mathbf{if}\;\left|dY.v\right| \leq 1.3647814989089966:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_2 \cdot dY.u, dY.u, \left(dY.w \cdot dY.w\right) \cdot t\_0\right), \mathsf{fma}\left(t\_1 \cdot dX.v, dX.v, t\_3\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_3, \mathsf{fma}\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor , \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(dY.w \cdot dY.w\right) \cdot \left\lfloor d\right\rfloor , \left\lfloor d\right\rfloor , \left(\left|dY.v\right| \cdot \left|dY.v\right|\right) \cdot t\_1\right)\right)\right)}\right)\\ \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
  :precision binary32
  :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)))
  (let* ((t_0 (* (floor d) (floor d)))
       (t_1 (* (floor h) (floor h)))
       (t_2 (* (floor w) (floor w)))
       (t_3 (fma t_2 (* dX.u dX.u) (* t_0 (* dX.w dX.w)))))
  (if (<= (fabs dY.v) 1.3647814989089966)
    (log2
     (sqrt
      (fmax
       (fma (* t_2 dY.u) dY.u (* (* dY.w dY.w) t_0))
       (fma (* t_1 dX.v) dX.v t_3))))
    (log2
     (sqrt
      (fmax
       t_3
       (fma
        (* (* dY.u dY.u) (floor w))
        (floor w)
        (fma
         (* (* dY.w dY.w) (floor d))
         (floor d)
         (* (* (fabs dY.v) (fabs dY.v)) t_1)))))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(d) * floorf(d);
	float t_1 = floorf(h) * floorf(h);
	float t_2 = floorf(w) * floorf(w);
	float t_3 = fmaf(t_2, (dX_46_u * dX_46_u), (t_0 * (dX_46_w * dX_46_w)));
	float tmp;
	if (fabsf(dY_46_v) <= 1.3647814989089966f) {
		tmp = log2f(sqrtf(fmaxf(fmaf((t_2 * dY_46_u), dY_46_u, ((dY_46_w * dY_46_w) * t_0)), fmaf((t_1 * dX_46_v), dX_46_v, t_3))));
	} else {
		tmp = log2f(sqrtf(fmaxf(t_3, fmaf(((dY_46_u * dY_46_u) * floorf(w)), floorf(w), fmaf(((dY_46_w * dY_46_w) * floorf(d)), floorf(d), ((fabsf(dY_46_v) * fabsf(dY_46_v)) * t_1))))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(d) * floor(d))
	t_1 = Float32(floor(h) * floor(h))
	t_2 = Float32(floor(w) * floor(w))
	t_3 = fma(t_2, Float32(dX_46_u * dX_46_u), Float32(t_0 * Float32(dX_46_w * dX_46_w)))
	tmp = Float32(0.0)
	if (abs(dY_46_v) <= Float32(1.3647814989089966))
		tmp = log2(sqrt(fmax(fma(Float32(t_2 * dY_46_u), dY_46_u, Float32(Float32(dY_46_w * dY_46_w) * t_0)), fma(Float32(t_1 * dX_46_v), dX_46_v, t_3))));
	else
		tmp = log2(sqrt(fmax(t_3, fma(Float32(Float32(dY_46_u * dY_46_u) * floor(w)), floor(w), fma(Float32(Float32(dY_46_w * dY_46_w) * floor(d)), floor(d), Float32(Float32(abs(dY_46_v) * abs(dY_46_v)) * t_1))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.v < 1.3647815

   1. Initial program 67.6%

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

   2. Taylor expanded in dY.v around 0

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

   4. Applied rewrites 60.5%

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

   6. Applied rewrites 60.5%

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

   if 1.3647815 < dY.v

   1. Initial program 67.6%

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

   2. Taylor expanded in dX.v around 0

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

   4. Applied rewrites 60.5%

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

   6. Applied rewrites 60.5%

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

Alternative 4: 65.2% accurate, 1.1× speedup

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

Math

\[\begin{array}{l} t_0 := \left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor \\ t_1 := \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \\ t_2 := \mathsf{fma}\left(\left(dY.w \cdot dY.w\right) \cdot \left\lfloor d\right\rfloor , \left\lfloor d\right\rfloor , \left(dY.v \cdot dY.v\right) \cdot t\_1\right)\\ \mathbf{if}\;\left|dY.u\right| \leq 39545.984375:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_2, \mathsf{fma}\left(t\_1 \cdot dX.v, dX.v, \mathsf{fma}\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor , dX.u \cdot dX.u, t\_0 \cdot \left(dX.w \cdot dX.w\right)\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_0, dX.w \cdot dX.w, t\_1 \cdot \left(dX.v \cdot dX.v\right)\right), \mathsf{fma}\left(\left(\left|dY.u\right| \cdot \left|dY.u\right|\right) \cdot \left\lfloor w\right\rfloor , \left\lfloor w\right\rfloor , t\_2\right)\right)}\right)\\ \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
  :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)))
  (let* ((t_0 (* (floor d) (floor d)))
       (t_1 (* (floor h) (floor h)))
       (t_2
        (fma
         (* (* dY.w dY.w) (floor d))
         (floor d)
         (* (* dY.v dY.v) t_1))))
  (if (<= (fabs dY.u) 39545.984375)
    (log2
     (sqrt
      (fmax
       t_2
       (fma
        (* t_1 dX.v)
        dX.v
        (fma
         (* (floor w) (floor w))
         (* dX.u dX.u)
         (* t_0 (* dX.w dX.w)))))))
    (log2
     (sqrt
      (fmax
       (fma t_0 (* dX.w dX.w) (* t_1 (* dX.v dX.v)))
       (fma
        (* (* (fabs dY.u) (fabs dY.u)) (floor w))
        (floor w)
        t_2)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(d) * floorf(d);
	float t_1 = floorf(h) * floorf(h);
	float t_2 = fmaf(((dY_46_w * dY_46_w) * floorf(d)), floorf(d), ((dY_46_v * dY_46_v) * t_1));
	float tmp;
	if (fabsf(dY_46_u) <= 39545.984375f) {
		tmp = log2f(sqrtf(fmaxf(t_2, fmaf((t_1 * dX_46_v), dX_46_v, fmaf((floorf(w) * floorf(w)), (dX_46_u * dX_46_u), (t_0 * (dX_46_w * dX_46_w)))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf(t_0, (dX_46_w * dX_46_w), (t_1 * (dX_46_v * dX_46_v))), fmaf(((fabsf(dY_46_u) * fabsf(dY_46_u)) * floorf(w)), floorf(w), t_2))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(d) * floor(d))
	t_1 = Float32(floor(h) * floor(h))
	t_2 = fma(Float32(Float32(dY_46_w * dY_46_w) * floor(d)), floor(d), Float32(Float32(dY_46_v * dY_46_v) * t_1))
	tmp = Float32(0.0)
	if (abs(dY_46_u) <= Float32(39545.984375))
		tmp = log2(sqrt(fmax(t_2, fma(Float32(t_1 * dX_46_v), dX_46_v, fma(Float32(floor(w) * floor(w)), Float32(dX_46_u * dX_46_u), Float32(t_0 * Float32(dX_46_w * dX_46_w)))))));
	else
		tmp = log2(sqrt(fmax(fma(t_0, Float32(dX_46_w * dX_46_w), Float32(t_1 * Float32(dX_46_v * dX_46_v))), fma(Float32(Float32(abs(dY_46_u) * abs(dY_46_u)) * floor(w)), floor(w), t_2))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 39545.9844

   1. Initial program 67.6%

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

   2. Taylor expanded in dY.u around 0

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

   4. Applied rewrites 61.0%

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

   6. Applied rewrites 61.0%

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

   if 39545.9844 < dY.u

   1. Initial program 67.6%

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

   2. Taylor expanded in dX.u around 0

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

   4. Applied rewrites 60.6%

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

   6. Applied rewrites 60.6%

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

Alternative 5: 65.1% accurate, 1.1× speedup

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

Math

\[\begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \\ t_1 := \left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor \\ t_2 := \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \\ t_3 := \mathsf{fma}\left(t\_2 \cdot dX.v, dX.v, \mathsf{fma}\left(t\_0, dX.u \cdot dX.u, t\_1 \cdot \left(dX.w \cdot dX.w\right)\right)\right)\\ \mathbf{if}\;\left|dY.v\right| \leq 24.979936599731445:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_0 \cdot dY.u, dY.u, \left(dY.w \cdot dY.w\right) \cdot t\_1\right), t\_3\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dY.w \cdot dY.w\right) \cdot \left\lfloor d\right\rfloor , \left\lfloor d\right\rfloor , \left(\left|dY.v\right| \cdot \left|dY.v\right|\right) \cdot t\_2\right), t\_3\right)}\right)\\ \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
  :precision binary32
  :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)))
  (let* ((t_0 (* (floor w) (floor w)))
       (t_1 (* (floor d) (floor d)))
       (t_2 (* (floor h) (floor h)))
       (t_3
        (fma
         (* t_2 dX.v)
         dX.v
         (fma t_0 (* dX.u dX.u) (* t_1 (* dX.w dX.w))))))
  (if (<= (fabs dY.v) 24.979936599731445)
    (log2
     (sqrt (fmax (fma (* t_0 dY.u) dY.u (* (* dY.w dY.w) t_1)) t_3)))
    (log2
     (sqrt
      (fmax
       (fma
        (* (* dY.w dY.w) (floor d))
        (floor d)
        (* (* (fabs dY.v) (fabs dY.v)) t_2))
       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) * floorf(w);
	float t_1 = floorf(d) * floorf(d);
	float t_2 = floorf(h) * floorf(h);
	float t_3 = fmaf((t_2 * dX_46_v), dX_46_v, fmaf(t_0, (dX_46_u * dX_46_u), (t_1 * (dX_46_w * dX_46_w))));
	float tmp;
	if (fabsf(dY_46_v) <= 24.979936599731445f) {
		tmp = log2f(sqrtf(fmaxf(fmaf((t_0 * dY_46_u), dY_46_u, ((dY_46_w * dY_46_w) * t_1)), t_3)));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf(((dY_46_w * dY_46_w) * floorf(d)), floorf(d), ((fabsf(dY_46_v) * fabsf(dY_46_v)) * t_2)), t_3)));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * floor(w))
	t_1 = Float32(floor(d) * floor(d))
	t_2 = Float32(floor(h) * floor(h))
	t_3 = fma(Float32(t_2 * dX_46_v), dX_46_v, fma(t_0, Float32(dX_46_u * dX_46_u), Float32(t_1 * Float32(dX_46_w * dX_46_w))))
	tmp = Float32(0.0)
	if (abs(dY_46_v) <= Float32(24.979936599731445))
		tmp = log2(sqrt(fmax(fma(Float32(t_0 * dY_46_u), dY_46_u, Float32(Float32(dY_46_w * dY_46_w) * t_1)), t_3)));
	else
		tmp = log2(sqrt(fmax(fma(Float32(Float32(dY_46_w * dY_46_w) * floor(d)), floor(d), Float32(Float32(abs(dY_46_v) * abs(dY_46_v)) * t_2)), t_3)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.v < 24.9799366

   1. Initial program 67.6%

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

   2. Taylor expanded in dY.v around 0

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

   4. Applied rewrites 60.5%

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

   6. Applied rewrites 60.5%

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

   if 24.9799366 < dY.v

   1. Initial program 67.6%

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

   2. Taylor expanded in dY.u around 0

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

   4. Applied rewrites 61.0%

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

   6. Applied rewrites 61.0%

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

Alternative 6: 61.0% accurate, 1.2× speedup

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

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 67.6%

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

2. Taylor expanded in dY.u around 0

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

4. Applied rewrites 61.0%

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

6. Applied rewrites 61.0%

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

Alternative 7: 57.6% accurate, 1.1× speedup

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

Math

\[\begin{array}{l} t_0 := dX.w \cdot \left\lfloor d\right\rfloor \\ t_1 := \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \\ t_2 := \left\lfloor 0\right\rfloor \cdot \left\lfloor 0\right\rfloor \\ t_3 := \left\lfloor 0\right\rfloor \cdot dY.u\\ t_4 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_5 := dY.w \cdot \left\lfloor d\right\rfloor \\ \mathbf{if}\;\left|dY.v\right| \leq 205517184:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_2 \cdot dY.u, dY.u, t\_5 \cdot t\_5\right), \mathsf{fma}\left(\left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor , dX.w \cdot dX.w, \mathsf{fma}\left(t\_1 \cdot dX.v, dX.v, t\_4 \cdot t\_4\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_2, dX.u \cdot dX.u, t\_0 \cdot t\_0\right), \mathsf{fma}\left(\left(dY.w \cdot dY.w\right) \cdot \left\lfloor d\right\rfloor , \left\lfloor d\right\rfloor , \mathsf{fma}\left(t\_3, t\_3, \left(\left|dY.v\right| \cdot \left|dY.v\right|\right) \cdot t\_1\right)\right)\right)}\right)\\ \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
  :precision binary32
  :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)))
  (let* ((t_0 (* dX.w (floor d)))
       (t_1 (* (floor h) (floor h)))
       (t_2 (* (floor 0.0) (floor 0.0)))
       (t_3 (* (floor 0.0) dY.u))
       (t_4 (* dX.u (floor w)))
       (t_5 (* dY.w (floor d))))
  (if (<= (fabs dY.v) 205517184.0)
    (log2
     (sqrt
      (fmax
       (fma (* t_2 dY.u) dY.u (* t_5 t_5))
       (fma
        (* (floor d) (floor d))
        (* dX.w dX.w)
        (fma (* t_1 dX.v) dX.v (* t_4 t_4))))))
    (log2
     (sqrt
      (fmax
       (fma t_2 (* dX.u dX.u) (* t_0 t_0))
       (fma
        (* (* dY.w dY.w) (floor d))
        (floor d)
        (fma t_3 t_3 (* (* (fabs dY.v) (fabs dY.v)) t_1)))))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = dX_46_w * floorf(d);
	float t_1 = floorf(h) * floorf(h);
	float t_2 = floorf(0.0f) * floorf(0.0f);
	float t_3 = floorf(0.0f) * dY_46_u;
	float t_4 = dX_46_u * floorf(w);
	float t_5 = dY_46_w * floorf(d);
	float tmp;
	if (fabsf(dY_46_v) <= 205517184.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf((t_2 * dY_46_u), dY_46_u, (t_5 * t_5)), fmaf((floorf(d) * floorf(d)), (dX_46_w * dX_46_w), fmaf((t_1 * dX_46_v), dX_46_v, (t_4 * t_4))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf(t_2, (dX_46_u * dX_46_u), (t_0 * t_0)), fmaf(((dY_46_w * dY_46_w) * floorf(d)), floorf(d), fmaf(t_3, t_3, ((fabsf(dY_46_v) * fabsf(dY_46_v)) * t_1))))));
	}
	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(dX_46_w * floor(d))
	t_1 = Float32(floor(h) * floor(h))
	t_2 = Float32(floor(Float32(0.0)) * floor(Float32(0.0)))
	t_3 = Float32(floor(Float32(0.0)) * dY_46_u)
	t_4 = Float32(dX_46_u * floor(w))
	t_5 = Float32(dY_46_w * floor(d))
	tmp = Float32(0.0)
	if (abs(dY_46_v) <= Float32(205517184.0))
		tmp = log2(sqrt(fmax(fma(Float32(t_2 * dY_46_u), dY_46_u, Float32(t_5 * t_5)), fma(Float32(floor(d) * floor(d)), Float32(dX_46_w * dX_46_w), fma(Float32(t_1 * dX_46_v), dX_46_v, Float32(t_4 * t_4))))));
	else
		tmp = log2(sqrt(fmax(fma(t_2, Float32(dX_46_u * dX_46_u), Float32(t_0 * t_0)), fma(Float32(Float32(dY_46_w * dY_46_w) * floor(d)), floor(d), fma(t_3, t_3, Float32(Float32(abs(dY_46_v) * abs(dY_46_v)) * t_1))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.v < 205517184

   1. Initial program 67.6%

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

   2. Taylor expanded in dY.v around 0

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

   4. Applied rewrites 60.5%

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

   5. Taylor expanded in undef-var around zero

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor 0\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor 0\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({dY.u}^{2}, {\left(\left\lfloor 0\right\rfloor \right)}^{2}, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 45.7%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor 0\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor 0\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({dY.u}^{2}, {\left(\left\lfloor 0\right\rfloor \right)}^{2}, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

   8. Taylor expanded in w around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({dY.u}^{2}, {\left(\left\lfloor 0\right\rfloor \right)}^{2}, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 54.1%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left({dY.u}^{2}, {\left(\left\lfloor 0\right\rfloor \right)}^{2}, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

   12. Applied rewrites 54.1%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(\left\lfloor 0\right\rfloor \cdot \left\lfloor 0\right\rfloor \right) \cdot dY.u, dY.u, \left(dY.w \cdot \left\lfloor d\right\rfloor \right) \cdot \left(dY.w \cdot \left\lfloor d\right\rfloor \right)\right), \mathsf{fma}\left(\left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor , dX.w \cdot dX.w, \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, dX.v, \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right)\right)\right)}\right) \]

   if 205517184 < dY.v

   1. Initial program 67.6%

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

   2. Taylor expanded in dX.v around 0

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

   4. Applied rewrites 60.5%

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

   5. Taylor expanded in undef-var around zero

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({dX.u}^{2}, {\left(\left\lfloor 0\right\rfloor \right)}^{2}, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \left(\left(\left\lfloor 0\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor 0\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 45.5%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({dX.u}^{2}, {\left(\left\lfloor 0\right\rfloor \right)}^{2}, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \left(\left(\left\lfloor 0\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor 0\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   9. Applied rewrites 45.5%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloor 0\right\rfloor \cdot \left\lfloor 0\right\rfloor , dX.u \cdot dX.u, \left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot \left(dX.w \cdot \left\lfloor d\right\rfloor \right)\right), \mathsf{fma}\left(\left(dY.w \cdot dY.w\right) \cdot \left\lfloor d\right\rfloor , \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left\lfloor 0\right\rfloor \cdot dY.u, \left\lfloor 0\right\rfloor \cdot dY.u, \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)\right)\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 45.5% accurate, 1.2× speedup

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

Math

\[\begin{array}{l} t_0 := dX.w \cdot \left\lfloor d\right\rfloor \\ t_1 := \left\lfloor 0\right\rfloor \cdot dY.u\\ \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, \left(\left\lfloor 0\right\rfloor \cdot \left\lfloor 0\right\rfloor \right) \cdot dX.u, t\_0 \cdot t\_0\right), \mathsf{fma}\left(\left(dY.w \cdot dY.w\right) \cdot \left\lfloor d\right\rfloor , \left\lfloor d\right\rfloor , \mathsf{fma}\left(t\_1, t\_1, \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)\right)\right)}\right) \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
  :precision binary32
  :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)))
  (let* ((t_0 (* dX.w (floor d))) (t_1 (* (floor 0.0) dY.u)))
  (log2
   (sqrt
    (fmax
     (fma dX.u (* (* (floor 0.0) (floor 0.0)) dX.u) (* t_0 t_0))
     (fma
      (* (* dY.w dY.w) (floor d))
      (floor d)
      (fma t_1 t_1 (* (* dY.v dY.v) (* (floor h) (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 = dX_46_w * floorf(d);
	float t_1 = floorf(0.0f) * dY_46_u;
	return log2f(sqrtf(fmaxf(fmaf(dX_46_u, ((floorf(0.0f) * floorf(0.0f)) * dX_46_u), (t_0 * t_0)), fmaf(((dY_46_w * dY_46_w) * floorf(d)), floorf(d), fmaf(t_1, t_1, ((dY_46_v * dY_46_v) * (floorf(h) * 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)
	t_0 = Float32(dX_46_w * floor(d))
	t_1 = Float32(floor(Float32(0.0)) * dY_46_u)
	return log2(sqrt(fmax(fma(dX_46_u, Float32(Float32(floor(Float32(0.0)) * floor(Float32(0.0))) * dX_46_u), Float32(t_0 * t_0)), fma(Float32(Float32(dY_46_w * dY_46_w) * floor(d)), floor(d), fma(t_1, t_1, Float32(Float32(dY_46_v * dY_46_v) * Float32(floor(h) * floor(h))))))))
end

Derivation

1. Initial program 67.6%

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

2. Taylor expanded in dX.v around 0

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

4. Applied rewrites 60.5%

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

5. Taylor expanded in undef-var around zero

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({dX.u}^{2}, {\left(\left\lfloor 0\right\rfloor \right)}^{2}, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \left(\left(\left\lfloor 0\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor 0\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
6. Step-by-step derivation

7. Applied rewrites 45.5%

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({dX.u}^{2}, {\left(\left\lfloor 0\right\rfloor \right)}^{2}, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \left(\left(\left\lfloor 0\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor 0\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

9. Applied rewrites 45.5%

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloor 0\right\rfloor \cdot \left\lfloor 0\right\rfloor , dX.u \cdot dX.u, \left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot \left(dX.w \cdot \left\lfloor d\right\rfloor \right)\right), \mathsf{fma}\left(\left(dY.w \cdot dY.w\right) \cdot \left\lfloor d\right\rfloor , \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left\lfloor 0\right\rfloor \cdot dY.u, \left\lfloor 0\right\rfloor \cdot dY.u, \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)\right)\right)}\right) \]
10. Step-by-step derivation

11. Applied rewrites 45.5%

    \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, \left(\left\lfloor 0\right\rfloor \cdot \left\lfloor 0\right\rfloor \right) \cdot dX.u, \left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot \left(dX.w \cdot \left\lfloor d\right\rfloor \right)\right), \mathsf{fma}\left(\left(dY.w \cdot dY.w\right) \cdot \left\lfloor d\right\rfloor , \left\lfloor d\right\rfloor , \mathsf{fma}\left(\left\lfloor 0\right\rfloor \cdot dY.u, \left\lfloor 0\right\rfloor \cdot dY.u, \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)\right)\right)}\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2026086 
(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)))))))