UniformSampleCone 2

Percentage Accurate: 98.9% → 99.0%

Time: 22.2s

Alternatives: 21

Speedup: 1.0×

Specification

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\\ t_1 := \sqrt{1 - t\_0 \cdot t\_0}\\ t_2 := \left(uy \cdot 2\right) \cdot \pi\\ \left(\left(\cos t\_2 \cdot t\_1\right) \cdot xi + \left(\sin t\_2 \cdot t\_1\right) \cdot yi\right) + t\_0 \cdot zi \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux))
        (t_1 (sqrt (- 1.0 (* t_0 t_0))))
        (t_2 (* (* uy 2.0) PI)))
   (+ (+ (* (* (cos t_2) t_1) xi) (* (* (sin t_2) t_1) yi)) (* t_0 zi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((1.0f - ux) * maxCos) * ux;
	float t_1 = sqrtf((1.0f - (t_0 * t_0)));
	float t_2 = (uy * 2.0f) * ((float) M_PI);
	return (((cosf(t_2) * t_1) * xi) + ((sinf(t_2) * t_1) * yi)) + (t_0 * zi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux)
	t_1 = sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)))
	t_2 = Float32(Float32(uy * Float32(2.0)) * Float32(pi))
	return Float32(Float32(Float32(Float32(cos(t_2) * t_1) * xi) + Float32(Float32(sin(t_2) * t_1) * yi)) + Float32(t_0 * zi))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = ((single(1.0) - ux) * maxCos) * ux;
	t_1 = sqrt((single(1.0) - (t_0 * t_0)));
	t_2 = (uy * single(2.0)) * single(pi);
	tmp = (((cos(t_2) * t_1) * xi) + ((sin(t_2) * t_1) * yi)) + (t_0 * zi);
end

Initial Program: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\\ t_1 := \sqrt{1 - t\_0 \cdot t\_0}\\ t_2 := \left(uy \cdot 2\right) \cdot \pi\\ \left(\left(\cos t\_2 \cdot t\_1\right) \cdot xi + \left(\sin t\_2 \cdot t\_1\right) \cdot yi\right) + t\_0 \cdot zi \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux))
        (t_1 (sqrt (- 1.0 (* t_0 t_0))))
        (t_2 (* (* uy 2.0) PI)))
   (+ (+ (* (* (cos t_2) t_1) xi) (* (* (sin t_2) t_1) yi)) (* t_0 zi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((1.0f - ux) * maxCos) * ux;
	float t_1 = sqrtf((1.0f - (t_0 * t_0)));
	float t_2 = (uy * 2.0f) * ((float) M_PI);
	return (((cosf(t_2) * t_1) * xi) + ((sinf(t_2) * t_1) * yi)) + (t_0 * zi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux)
	t_1 = sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)))
	t_2 = Float32(Float32(uy * Float32(2.0)) * Float32(pi))
	return Float32(Float32(Float32(Float32(cos(t_2) * t_1) * xi) + Float32(Float32(sin(t_2) * t_1) * yi)) + Float32(t_0 * zi))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = ((single(1.0) - ux) * maxCos) * ux;
	t_1 = sqrt((single(1.0) - (t_0 * t_0)));
	t_2 = (uy * single(2.0)) * single(pi);
	tmp = (((cos(t_2) * t_1) * xi) + ((sin(t_2) * t_1) * yi)) + (t_0 * zi);
end

Alternative 1: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ t_1 := \sqrt{\mathsf{fma}\left(ux, \left(1 - ux\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(ux + -1\right) \cdot maxCos\right)\right), 1\right)}\\ \mathsf{fma}\left(t\_1 \cdot \sin t\_0, yi, t\_1 \cdot \left(\cos t\_0 \cdot xi\right)\right) + \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy PI)))
        (t_1
         (sqrt
          (fma
           ux
           (* (- 1.0 ux) (* (* ux maxCos) (* (+ ux -1.0) maxCos)))
           1.0))))
   (+
    (fma (* t_1 (sin t_0)) yi (* t_1 (* (cos t_0) xi)))
    (* (* ux (* (- 1.0 ux) maxCos)) zi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = 2.0f * (uy * ((float) M_PI));
	float t_1 = sqrtf(fmaf(ux, ((1.0f - ux) * ((ux * maxCos) * ((ux + -1.0f) * maxCos))), 1.0f));
	return fmaf((t_1 * sinf(t_0)), yi, (t_1 * (cosf(t_0) * xi))) + ((ux * ((1.0f - ux) * maxCos)) * zi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	t_1 = sqrt(fma(ux, Float32(Float32(Float32(1.0) - ux) * Float32(Float32(ux * maxCos) * Float32(Float32(ux + Float32(-1.0)) * maxCos))), Float32(1.0)))
	return Float32(fma(Float32(t_1 * sin(t_0)), yi, Float32(t_1 * Float32(cos(t_0) * xi))) + Float32(Float32(ux * Float32(Float32(Float32(1.0) - ux) * maxCos)) * zi))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. Add Preprocessing
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi + \left(\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi\right)} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   2. accelerator-lowering-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}, yi, \left(\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi\right)} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

4. Applied egg-rr 98.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(ux, \left(0 - \left(1 - ux\right)\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right), 1\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right), yi, \sqrt{\mathsf{fma}\left(ux, \left(0 - \left(1 - ux\right)\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right), 1\right)} \cdot \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot xi\right)\right)} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

5. Final simplification 98.9%

   \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(ux, \left(1 - ux\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(ux + -1\right) \cdot maxCos\right)\right), 1\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right), yi, \sqrt{\mathsf{fma}\left(ux, \left(1 - ux\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(ux + -1\right) \cdot maxCos\right)\right), 1\right)} \cdot \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot xi\right)\right) + \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi \]
6. Add Preprocessing

Alternative 2: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ t_1 := \sqrt{\mathsf{fma}\left(ux, \left(1 - ux\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(ux + -1\right) \cdot maxCos\right)\right), 1\right)}\\ \mathsf{fma}\left(t\_1 \cdot \sin t\_0, yi, \mathsf{fma}\left(t\_1, \cos t\_0 \cdot xi, \left(1 - ux\right) \cdot \left(maxCos \cdot \left(ux \cdot zi\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy PI)))
        (t_1
         (sqrt
          (fma
           ux
           (* (- 1.0 ux) (* (* ux maxCos) (* (+ ux -1.0) maxCos)))
           1.0))))
   (fma
    (* t_1 (sin t_0))
    yi
    (fma t_1 (* (cos t_0) xi) (* (- 1.0 ux) (* maxCos (* ux zi)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = 2.0f * (uy * ((float) M_PI));
	float t_1 = sqrtf(fmaf(ux, ((1.0f - ux) * ((ux * maxCos) * ((ux + -1.0f) * maxCos))), 1.0f));
	return fmaf((t_1 * sinf(t_0)), yi, fmaf(t_1, (cosf(t_0) * xi), ((1.0f - ux) * (maxCos * (ux * zi)))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	t_1 = sqrt(fma(ux, Float32(Float32(Float32(1.0) - ux) * Float32(Float32(ux * maxCos) * Float32(Float32(ux + Float32(-1.0)) * maxCos))), Float32(1.0)))
	return fma(Float32(t_1 * sin(t_0)), yi, fma(t_1, Float32(cos(t_0) * xi), Float32(Float32(Float32(1.0) - ux) * Float32(maxCos * Float32(ux * zi)))))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. Add Preprocessing

4. Applied egg-rr 98.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(ux, \left(0 - \left(1 - ux\right)\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right), 1\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right), yi, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(ux, \left(0 - \left(1 - ux\right)\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right), 1\right)}, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot xi, \left(1 - ux\right) \cdot \left(maxCos \cdot \left(ux \cdot zi\right)\right)\right)\right)} \]

5. Final simplification 98.9%

   \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(ux, \left(1 - ux\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(ux + -1\right) \cdot maxCos\right)\right), 1\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right), yi, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(ux, \left(1 - ux\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(ux + -1\right) \cdot maxCos\right)\right), 1\right)}, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot xi, \left(1 - ux\right) \cdot \left(maxCos \cdot \left(ux \cdot zi\right)\right)\right)\right) \]
6. Add Preprocessing

Alternative 3: 98.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := uy \cdot \left(2 \cdot \pi\right)\\ \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi + xi \cdot \left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(\sin t\_0, \frac{yi}{xi}, \cos t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* uy (* 2.0 PI))))
   (+
    (* (* ux (* (- 1.0 ux) maxCos)) zi)
    (*
     xi
     (*
      (sqrt
       (fma (* maxCos maxCos) (* (* ux ux) (* (- 1.0 ux) (+ ux -1.0))) 1.0))
      (fma (sin t_0) (/ yi xi) (cos t_0)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = uy * (2.0f * ((float) M_PI));
	return ((ux * ((1.0f - ux) * maxCos)) * zi) + (xi * (sqrtf(fmaf((maxCos * maxCos), ((ux * ux) * ((1.0f - ux) * (ux + -1.0f))), 1.0f)) * fmaf(sinf(t_0), (yi / xi), cosf(t_0))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(uy * Float32(Float32(2.0) * Float32(pi)))
	return Float32(Float32(Float32(ux * Float32(Float32(Float32(1.0) - ux) * maxCos)) * zi) + Float32(xi * Float32(sqrt(fma(Float32(maxCos * maxCos), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))), Float32(1.0))) * fma(sin(t_0), Float32(yi / xi), cos(t_0)))))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. Add Preprocessing

3. Taylor expanded in xi around inf

   \[\leadsto \color{blue}{xi \cdot \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi} \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

5. Simplified 98.7%

   \[\leadsto \color{blue}{xi \cdot \left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right), \frac{yi}{xi}, \cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right)} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

6. Final simplification 98.7%

   \[\leadsto \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi + xi \cdot \left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right), \frac{yi}{xi}, \cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right) \]
7. Add Preprocessing

Alternative 4: 98.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ \mathsf{fma}\left(maxCos, \left(1 - ux\right) \cdot \left(ux \cdot zi\right), \mathsf{fma}\left(xi, \cos t\_0, \sin t\_0 \cdot yi\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy PI))))
   (fma maxCos (* (- 1.0 ux) (* ux zi)) (fma xi (cos t_0) (* (sin t_0) yi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = 2.0f * (uy * ((float) M_PI));
	return fmaf(maxCos, ((1.0f - ux) * (ux * zi)), fmaf(xi, cosf(t_0), (sinf(t_0) * yi)));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	return fma(maxCos, Float32(Float32(Float32(1.0) - ux) * Float32(ux * zi)), fma(xi, cos(t_0), Float32(sin(t_0) * yi)))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. Add Preprocessing
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi + \left(\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi\right)} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   2. accelerator-lowering-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}, yi, \left(\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi\right)} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

4. Applied egg-rr 98.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(ux, \left(0 - \left(1 - ux\right)\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right), 1\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right), yi, \sqrt{\mathsf{fma}\left(ux, \left(0 - \left(1 - ux\right)\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right), 1\right)} \cdot \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot xi\right)\right)} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

5. Taylor expanded in maxCos around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. accelerator-lowering-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{\left(ux \cdot zi\right) \cdot \left(1 - ux\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{\left(ux \cdot zi\right) \cdot \left(1 - ux\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{\left(ux \cdot zi\right)} \cdot \left(1 - ux\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   5. --lowering--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \left(ux \cdot zi\right) \cdot \color{blue}{\left(1 - ux\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   6. accelerator-lowering-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \left(ux \cdot zi\right) \cdot \left(1 - ux\right), \color{blue}{\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \]

   7. cos-lowering-cos.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \left(ux \cdot zi\right) \cdot \left(1 - ux\right), \mathsf{fma}\left(xi, \color{blue}{\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \left(ux \cdot zi\right) \cdot \left(1 - ux\right), \mathsf{fma}\left(xi, \cos \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \left(ux \cdot zi\right) \cdot \left(1 - ux\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   10. PI-lowering-PI.f32 N/A

       \[\leadsto \mathsf{fma}\left(maxCos, \left(ux \cdot zi\right) \cdot \left(1 - ux\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   11. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(maxCos, \left(ux \cdot zi\right) \cdot \left(1 - ux\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

   12. sin-lowering-sin.f32 N/A

       \[\leadsto \mathsf{fma}\left(maxCos, \left(ux \cdot zi\right) \cdot \left(1 - ux\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \color{blue}{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

   13. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(maxCos, \left(ux \cdot zi\right) \cdot \left(1 - ux\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(maxCos, \left(ux \cdot zi\right) \cdot \left(1 - ux\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \]

   15. PI-lowering-PI.f32 98.6

       \[\leadsto \mathsf{fma}\left(maxCos, \left(ux \cdot zi\right) \cdot \left(1 - ux\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \color{blue}{\pi}\right)\right)\right)\right) \]

7. Simplified 98.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, \left(ux \cdot zi\right) \cdot \left(1 - ux\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]

8. Final simplification 98.6%

   \[\leadsto \mathsf{fma}\left(maxCos, \left(1 - ux\right) \cdot \left(ux \cdot zi\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right)\right) \]
9. Add Preprocessing

Alternative 5: 98.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := uy \cdot \left(2 \cdot \pi\right)\\ \mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \mathsf{fma}\left(xi, \cos t\_0, yi \cdot \sin t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* uy (* 2.0 PI))))
   (fma maxCos (* ux (* (- 1.0 ux) zi)) (fma xi (cos t_0) (* yi (sin t_0))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = uy * (2.0f * ((float) M_PI));
	return fmaf(maxCos, (ux * ((1.0f - ux) * zi)), fmaf(xi, cosf(t_0), (yi * sinf(t_0))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(uy * Float32(Float32(2.0) * Float32(pi)))
	return fma(maxCos, Float32(ux * Float32(Float32(Float32(1.0) - ux) * zi)), fma(xi, cos(t_0), Float32(yi * sin(t_0))))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. Add Preprocessing

3. Taylor expanded in maxCos around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
4. Step-by-step derivation

   1. accelerator-lowering-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

   2. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(\left(1 - ux\right) \cdot zi\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(\left(1 - ux\right) \cdot zi\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   5. --lowering--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(\color{blue}{\left(1 - ux\right)} \cdot zi\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   6. accelerator-lowering-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \color{blue}{\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)}\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \mathsf{fma}\left(xi, \cos \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   9. cos-lowering-cos.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \mathsf{fma}\left(xi, \color{blue}{\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \mathsf{fma}\left(xi, \cos \color{blue}{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot uy\right)\right)}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \mathsf{fma}\left(xi, \cos \color{blue}{\left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   13. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \mathsf{fma}\left(xi, \cos \color{blue}{\left(uy \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \mathsf{fma}\left(xi, \cos \left(uy \cdot \color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)}\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \mathsf{fma}\left(xi, \cos \color{blue}{\left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   16. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \mathsf{fma}\left(xi, \cos \left(uy \cdot \color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)}\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   17. PI-lowering-PI.f32 N/A

       \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   18. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

5. Simplified 98.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \pi\right)\right), yi \cdot \sin \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right)} \]
6. Add Preprocessing

Alternative 6: 97.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := uy \cdot \left(2 \cdot \pi\right)\\ t_1 := \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}\\ \mathbf{if}\;2 \cdot uy \leq 0.029999999329447746:\\ \;\;\;\;\mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \mathsf{fma}\left(t\_1, \mathsf{fma}\left(2, \pi \cdot \left(uy \cdot yi\right), xi\right), \left(t\_1 \cdot \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), -1.3333333333333333 \cdot \left(uy \cdot \left(\pi \cdot \left(yi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\right) \cdot \left(uy \cdot uy\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, \cos t\_0, yi \cdot \sin t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* uy (* 2.0 PI)))
        (t_1
         (sqrt
          (fma
           (* maxCos maxCos)
           (* (* ux ux) (* (- 1.0 ux) (+ ux -1.0)))
           1.0))))
   (if (<= (* 2.0 uy) 0.029999999329447746)
     (fma
      maxCos
      (* ux (* (- 1.0 ux) zi))
      (fma
       t_1
       (fma 2.0 (* PI (* uy yi)) xi)
       (*
        (*
         t_1
         (fma
          -2.0
          (* xi (* PI PI))
          (* -1.3333333333333333 (* uy (* PI (* yi (* PI PI)))))))
        (* uy uy))))
     (fma xi (cos t_0) (* yi (sin t_0))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = uy * (2.0f * ((float) M_PI));
	float t_1 = sqrtf(fmaf((maxCos * maxCos), ((ux * ux) * ((1.0f - ux) * (ux + -1.0f))), 1.0f));
	float tmp;
	if ((2.0f * uy) <= 0.029999999329447746f) {
		tmp = fmaf(maxCos, (ux * ((1.0f - ux) * zi)), fmaf(t_1, fmaf(2.0f, (((float) M_PI) * (uy * yi)), xi), ((t_1 * fmaf(-2.0f, (xi * (((float) M_PI) * ((float) M_PI))), (-1.3333333333333333f * (uy * (((float) M_PI) * (yi * (((float) M_PI) * ((float) M_PI)))))))) * (uy * uy))));
	} else {
		tmp = fmaf(xi, cosf(t_0), (yi * sinf(t_0)));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(uy * Float32(Float32(2.0) * Float32(pi)))
	t_1 = sqrt(fma(Float32(maxCos * maxCos), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))), Float32(1.0)))
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.029999999329447746))
		tmp = fma(maxCos, Float32(ux * Float32(Float32(Float32(1.0) - ux) * zi)), fma(t_1, fma(Float32(2.0), Float32(Float32(pi) * Float32(uy * yi)), xi), Float32(Float32(t_1 * fma(Float32(-2.0), Float32(xi * Float32(Float32(pi) * Float32(pi))), Float32(Float32(-1.3333333333333333) * Float32(uy * Float32(Float32(pi) * Float32(yi * Float32(Float32(pi) * Float32(pi)))))))) * Float32(uy * uy))));
	else
		tmp = fma(xi, cos(t_0), Float32(yi * sin(t_0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.0299999993

   1. Initial program 99.3%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \left(uy \cdot \left(2 \cdot \left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) + uy \cdot \left(-2 \cdot \left(\left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) + \frac{-4}{3} \cdot \left(\left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

   5. Simplified 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)}, \mathsf{fma}\left(2, \left(uy \cdot yi\right) \cdot \pi, xi\right), \left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), -1.3333333333333333 \cdot \left(uy \cdot \left(\pi \cdot \left(\left(\pi \cdot \pi\right) \cdot yi\right)\right)\right)\right)\right) \cdot \left(uy \cdot uy\right)\right)\right)} \]

   if 0.0299999993 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 97.1%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   2. Add Preprocessing

   3. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   4. Step-by-step derivation

      1. accelerator-lowering-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)}\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. cos-lowering-cos.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \color{blue}{\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \color{blue}{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot uy\right)\right)}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \color{blue}{\left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \color{blue}{\left(uy \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(uy \cdot \color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)}\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      10. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(xi, \cos \color{blue}{\left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      11. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(xi, \cos \left(uy \cdot \color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)}\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      12. PI-lowering-PI.f32 N/A

          \[\leadsto \mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      13. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)}\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)}\right) \]

      16. sin-lowering-sin.f32 N/A

          \[\leadsto \mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \color{blue}{\sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)}\right) \]

      17. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \color{blue}{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot uy\right)\right)}\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \color{blue}{\left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}\right) \]

      20. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \color{blue}{\left(uy \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)}\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(uy \cdot \color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

   5. Simplified 91.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \pi\right)\right), yi \cdot \sin \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.029999999329447746:\\ \;\;\;\;\mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}, \mathsf{fma}\left(2, \pi \cdot \left(uy \cdot yi\right), xi\right), \left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), -1.3333333333333333 \cdot \left(uy \cdot \left(\pi \cdot \left(yi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\right) \cdot \left(uy \cdot uy\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \pi\right)\right), yi \cdot \sin \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := uy \cdot \left(2 \cdot \pi\right)\\ \mathsf{fma}\left(xi, \cos t\_0, \mathsf{fma}\left(yi, \sin t\_0, maxCos \cdot \left(ux \cdot zi\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* uy (* 2.0 PI))))
   (fma xi (cos t_0) (fma yi (sin t_0) (* maxCos (* ux zi))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = uy * (2.0f * ((float) M_PI));
	return fmaf(xi, cosf(t_0), fmaf(yi, sinf(t_0), (maxCos * (ux * zi))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(uy * Float32(Float32(2.0) * Float32(pi)))
	return fma(xi, cos(t_0), fma(yi, sin(t_0), Float32(maxCos * Float32(ux * zi))))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. Add Preprocessing

3. Taylor expanded in ux around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) + maxCos \cdot \left(ux \cdot zi\right)} \]

   2. associate-+l+ N/A

      \[\leadsto \color{blue}{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot zi\right)\right)} \]

   3. accelerator-lowering-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot zi\right)\right)} \]

   4. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)}\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot zi\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot zi\right)\right) \]

   6. cos-lowering-cos.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \color{blue}{\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot zi\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \color{blue}{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot uy\right)\right)}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot zi\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot zi\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \color{blue}{\left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot zi\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(xi, \cos \color{blue}{\left(uy \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot zi\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(uy \cdot \color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)}\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot zi\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(xi, \cos \color{blue}{\left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot zi\right)\right) \]

   13. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(uy \cdot \color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)}\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot zi\right)\right) \]

   14. PI-lowering-PI.f32 N/A

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot zi\right)\right) \]

   15. accelerator-lowering-fma.f32 N/A

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\mathsf{fma}\left(yi, \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), maxCos \cdot \left(ux \cdot zi\right)\right)}\right) \]

5. Simplified 95.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \pi\right)\right), \mathsf{fma}\left(yi, \sin \left(uy \cdot \left(2 \cdot \pi\right)\right), maxCos \cdot \left(ux \cdot zi\right)\right)\right)} \]
6. Add Preprocessing

Alternative 8: 91.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}\\ \mathbf{if}\;2 \cdot uy \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \mathsf{fma}\left(t\_0, \mathsf{fma}\left(2, \pi \cdot \left(uy \cdot yi\right), xi\right), \left(t\_0 \cdot \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), -1.3333333333333333 \cdot \left(uy \cdot \left(\pi \cdot \left(yi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\right) \cdot \left(uy \cdot uy\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi, t\_0, maxCos \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot zi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0
         (sqrt
          (fma
           (* maxCos maxCos)
           (* (* ux ux) (* (- 1.0 ux) (+ ux -1.0)))
           1.0))))
   (if (<= (* 2.0 uy) 0.20000000298023224)
     (fma
      maxCos
      (* ux (* (- 1.0 ux) zi))
      (fma
       t_0
       (fma 2.0 (* PI (* uy yi)) xi)
       (*
        (*
         t_0
         (fma
          -2.0
          (* xi (* PI PI))
          (* -1.3333333333333333 (* uy (* PI (* yi (* PI PI)))))))
        (* uy uy))))
     (fma
      (* (sin (* 2.0 (* uy PI))) yi)
      t_0
      (* maxCos (* (- 1.0 ux) (* ux zi)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = sqrtf(fmaf((maxCos * maxCos), ((ux * ux) * ((1.0f - ux) * (ux + -1.0f))), 1.0f));
	float tmp;
	if ((2.0f * uy) <= 0.20000000298023224f) {
		tmp = fmaf(maxCos, (ux * ((1.0f - ux) * zi)), fmaf(t_0, fmaf(2.0f, (((float) M_PI) * (uy * yi)), xi), ((t_0 * fmaf(-2.0f, (xi * (((float) M_PI) * ((float) M_PI))), (-1.3333333333333333f * (uy * (((float) M_PI) * (yi * (((float) M_PI) * ((float) M_PI)))))))) * (uy * uy))));
	} else {
		tmp = fmaf((sinf((2.0f * (uy * ((float) M_PI)))) * yi), t_0, (maxCos * ((1.0f - ux) * (ux * zi))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = sqrt(fma(Float32(maxCos * maxCos), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))), Float32(1.0)))
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.20000000298023224))
		tmp = fma(maxCos, Float32(ux * Float32(Float32(Float32(1.0) - ux) * zi)), fma(t_0, fma(Float32(2.0), Float32(Float32(pi) * Float32(uy * yi)), xi), Float32(Float32(t_0 * fma(Float32(-2.0), Float32(xi * Float32(Float32(pi) * Float32(pi))), Float32(Float32(-1.3333333333333333) * Float32(uy * Float32(Float32(pi) * Float32(yi * Float32(Float32(pi) * Float32(pi)))))))) * Float32(uy * uy))));
	else
		tmp = fma(Float32(sin(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) * yi), t_0, Float32(maxCos * Float32(Float32(Float32(1.0) - ux) * Float32(ux * zi))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.200000003

   1. Initial program 99.3%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \left(uy \cdot \left(2 \cdot \left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) + uy \cdot \left(-2 \cdot \left(\left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) + \frac{-4}{3} \cdot \left(\left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

   5. Simplified 96.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)}, \mathsf{fma}\left(2, \left(uy \cdot yi\right) \cdot \pi, xi\right), \left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), -1.3333333333333333 \cdot \left(uy \cdot \left(\pi \cdot \left(\left(\pi \cdot \pi\right) \cdot yi\right)\right)\right)\right)\right) \cdot \left(uy \cdot uy\right)\right)\right)} \]

   if 0.200000003 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 95.5%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi + \left(\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi\right)} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      2. accelerator-lowering-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}, yi, \left(\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi\right)} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   4. Applied egg-rr 95.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(ux, \left(0 - \left(1 - ux\right)\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right), 1\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right), yi, \sqrt{\mathsf{fma}\left(ux, \left(0 - \left(1 - ux\right)\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right), 1\right)} \cdot \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot xi\right)\right)} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   5. Taylor expanded in xi around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{1 + {maxCos}^{2} \cdot \left({ux}^{2} \cdot \left(\left(1 - ux\right) \cdot \left(ux - 1\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{1 + {maxCos}^{2} \cdot \left({ux}^{2} \cdot \left(\left(1 - ux\right) \cdot \left(ux - 1\right)\right)\right)} + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]

      2. accelerator-lowering-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \sqrt{1 + {maxCos}^{2} \cdot \left({ux}^{2} \cdot \left(\left(1 - ux\right) \cdot \left(ux - 1\right)\right)\right)}, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)} \]

   7. Simplified 63.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right), \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}, maxCos \cdot \left(\left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}, \mathsf{fma}\left(2, \pi \cdot \left(uy \cdot yi\right), xi\right), \left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), -1.3333333333333333 \cdot \left(uy \cdot \left(\pi \cdot \left(yi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\right) \cdot \left(uy \cdot uy\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}, maxCos \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot zi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}\\ t_1 := 2 \cdot \left(\pi \cdot yi\right)\\ \mathbf{if}\;2 \cdot uy \leq 0.0020000000949949026:\\ \;\;\;\;\left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi + \mathsf{fma}\left(xi, t\_0, uy \cdot \left(t\_0 \cdot \mathsf{fma}\left(-2, xi \cdot \left(uy \cdot \left(\pi \cdot \pi\right)\right), t\_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(yi \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0
         (sqrt
          (fma
           (* maxCos maxCos)
           (* (* ux ux) (* (- 1.0 ux) (+ ux -1.0)))
           1.0)))
        (t_1 (* 2.0 (* PI yi))))
   (if (<= (* 2.0 uy) 0.0020000000949949026)
     (+
      (* (* ux (* (- 1.0 ux) maxCos)) zi)
      (fma xi t_0 (* uy (* t_0 (fma -2.0 (* xi (* uy (* PI PI))) t_1)))))
     (fma
      xi
      (cos (* 2.0 (* uy PI)))
      (*
       uy
       (fma -1.3333333333333333 (* (* yi (* uy uy)) (* PI (* PI PI))) t_1))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = sqrtf(fmaf((maxCos * maxCos), ((ux * ux) * ((1.0f - ux) * (ux + -1.0f))), 1.0f));
	float t_1 = 2.0f * (((float) M_PI) * yi);
	float tmp;
	if ((2.0f * uy) <= 0.0020000000949949026f) {
		tmp = ((ux * ((1.0f - ux) * maxCos)) * zi) + fmaf(xi, t_0, (uy * (t_0 * fmaf(-2.0f, (xi * (uy * (((float) M_PI) * ((float) M_PI)))), t_1))));
	} else {
		tmp = fmaf(xi, cosf((2.0f * (uy * ((float) M_PI)))), (uy * fmaf(-1.3333333333333333f, ((yi * (uy * uy)) * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))), t_1)));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = sqrt(fma(Float32(maxCos * maxCos), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))), Float32(1.0)))
	t_1 = Float32(Float32(2.0) * Float32(Float32(pi) * yi))
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.0020000000949949026))
		tmp = Float32(Float32(Float32(ux * Float32(Float32(Float32(1.0) - ux) * maxCos)) * zi) + fma(xi, t_0, Float32(uy * Float32(t_0 * fma(Float32(-2.0), Float32(xi * Float32(uy * Float32(Float32(pi) * Float32(pi)))), t_1)))));
	else
		tmp = fma(xi, cos(Float32(Float32(2.0) * Float32(uy * Float32(pi)))), Float32(uy * fma(Float32(-1.3333333333333333), Float32(Float32(yi * Float32(uy * uy)) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))), t_1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.00200000009

   1. Initial program 99.3%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(uy \cdot \left(-2 \cdot \left(\left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) + 2 \cdot \left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   5. Simplified 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(xi, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)}, uy \cdot \left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(-2, xi \cdot \left(\left(\pi \cdot \pi\right) \cdot uy\right), 2 \cdot \left(yi \cdot \pi\right)\right)\right)\right)} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   if 0.00200000009 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 97.7%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   2. Add Preprocessing

   3. Taylor expanded in zi around 0

      \[\leadsto \color{blue}{\left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)} + \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
   4. Step-by-step derivation

      1. distribute-rgt-out N/A

         \[\leadsto \color{blue}{\sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)} \cdot \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)} \cdot \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

   5. Simplified 91.6%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \pi\right)\right), yi \cdot \sin \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)} \]

   6. Taylor expanded in uy around 0

      \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{uy \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{uy \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), uy \cdot \color{blue}{\left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + \frac{-4}{3} \cdot \left({uy}^{2} \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)}\right) \]

      3. accelerator-lowering-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), uy \cdot \color{blue}{\mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), \frac{-4}{3} \cdot \left({uy}^{2} \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), uy \cdot \mathsf{fma}\left(2, \color{blue}{\mathsf{PI}\left(\right) \cdot yi}, \frac{-4}{3} \cdot \left({uy}^{2} \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), uy \cdot \mathsf{fma}\left(2, \color{blue}{\mathsf{PI}\left(\right) \cdot yi}, \frac{-4}{3} \cdot \left({uy}^{2} \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right) \]

      6. PI-lowering-PI.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), uy \cdot \mathsf{fma}\left(2, \color{blue}{\mathsf{PI}\left(\right)} \cdot yi, \frac{-4}{3} \cdot \left({uy}^{2} \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), uy \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right) \cdot yi, \color{blue}{\left(\frac{-4}{3} \cdot {uy}^{2}\right) \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)}\right)\right) \]

      8. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), uy \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right) \cdot yi, \color{blue}{\left(\frac{-4}{3} \cdot {uy}^{2}\right) \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)}\right)\right) \]

      9. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), uy \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right) \cdot yi, \color{blue}{\left(\frac{-4}{3} \cdot {uy}^{2}\right)} \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), uy \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right) \cdot yi, \left(\frac{-4}{3} \cdot \color{blue}{\left(uy \cdot uy\right)}\right) \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right) \]

      11. *-lowering-*.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), uy \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right) \cdot yi, \left(\frac{-4}{3} \cdot \color{blue}{\left(uy \cdot uy\right)}\right) \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right) \]

      12. *-lowering-*.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), uy \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right) \cdot yi, \left(\frac{-4}{3} \cdot \left(uy \cdot uy\right)\right) \cdot \color{blue}{\left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)}\right)\right) \]

      13. cube-mult N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), uy \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right) \cdot yi, \left(\frac{-4}{3} \cdot \left(uy \cdot uy\right)\right) \cdot \left(yi \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), uy \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right) \cdot yi, \left(\frac{-4}{3} \cdot \left(uy \cdot uy\right)\right) \cdot \left(yi \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)\right)\right)\right) \]

      15. *-lowering-*.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), uy \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right) \cdot yi, \left(\frac{-4}{3} \cdot \left(uy \cdot uy\right)\right) \cdot \left(yi \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right) \]

      16. PI-lowering-PI.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), uy \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right) \cdot yi, \left(\frac{-4}{3} \cdot \left(uy \cdot uy\right)\right) \cdot \left(yi \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), uy \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right) \cdot yi, \left(\frac{-4}{3} \cdot \left(uy \cdot uy\right)\right) \cdot \left(yi \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right)\right) \]

      18. *-lowering-*.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), uy \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right) \cdot yi, \left(\frac{-4}{3} \cdot \left(uy \cdot uy\right)\right) \cdot \left(yi \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right)\right) \]

      19. PI-lowering-PI.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), uy \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right) \cdot yi, \left(\frac{-4}{3} \cdot \left(uy \cdot uy\right)\right) \cdot \left(yi \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \]

      20. PI-lowering-PI.f32 73.6

          \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \pi\right)\right), uy \cdot \mathsf{fma}\left(2, \pi \cdot yi, \left(-1.3333333333333333 \cdot \left(uy \cdot uy\right)\right) \cdot \left(yi \cdot \left(\pi \cdot \left(\pi \cdot \color{blue}{\pi}\right)\right)\right)\right)\right) \]

   8. Simplified 73.6%

      \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \pi\right)\right), \color{blue}{uy \cdot \mathsf{fma}\left(2, \pi \cdot yi, \left(-1.3333333333333333 \cdot \left(uy \cdot uy\right)\right) \cdot \left(yi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)}\right) \]

   9. Taylor expanded in maxCos around 0

      \[\leadsto \color{blue}{uy \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) + xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + uy \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)} \]

       2. accelerator-lowering-fma.f32 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), uy \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

       3. cos-lowering-cos.f32 N/A

          \[\leadsto \mathsf{fma}\left(xi, \color{blue}{\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}, uy \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       4. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(xi, \cos \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}, uy \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       5. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right), uy \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       6. PI-lowering-PI.f32 N/A

          \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right), uy \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       7. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{uy \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

       8. accelerator-lowering-fma.f32 N/A

          \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), uy \cdot \color{blue}{\mathsf{fma}\left(\frac{-4}{3}, {uy}^{2} \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

   11. Simplified 72.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(\left(uy \cdot uy\right) \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \left(yi \cdot \pi\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0020000000949949026:\\ \;\;\;\;\left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi + \mathsf{fma}\left(xi, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}, uy \cdot \left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(-2, xi \cdot \left(uy \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \left(\pi \cdot yi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(yi \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \left(\pi \cdot yi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 88.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}\\ \mathbf{if}\;2 \cdot uy \leq 0.20000000298023224:\\ \;\;\;\;\left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi + \mathsf{fma}\left(xi, t\_0, uy \cdot \left(t\_0 \cdot \mathsf{fma}\left(-2, xi \cdot \left(uy \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \left(\pi \cdot yi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0
         (sqrt
          (fma
           (* maxCos maxCos)
           (* (* ux ux) (* (- 1.0 ux) (+ ux -1.0)))
           1.0))))
   (if (<= (* 2.0 uy) 0.20000000298023224)
     (+
      (* (* ux (* (- 1.0 ux) maxCos)) zi)
      (fma
       xi
       t_0
       (* uy (* t_0 (fma -2.0 (* xi (* uy (* PI PI))) (* 2.0 (* PI yi)))))))
     (* (sin (* 2.0 (* uy PI))) yi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = sqrtf(fmaf((maxCos * maxCos), ((ux * ux) * ((1.0f - ux) * (ux + -1.0f))), 1.0f));
	float tmp;
	if ((2.0f * uy) <= 0.20000000298023224f) {
		tmp = ((ux * ((1.0f - ux) * maxCos)) * zi) + fmaf(xi, t_0, (uy * (t_0 * fmaf(-2.0f, (xi * (uy * (((float) M_PI) * ((float) M_PI)))), (2.0f * (((float) M_PI) * yi))))));
	} else {
		tmp = sinf((2.0f * (uy * ((float) M_PI)))) * yi;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = sqrt(fma(Float32(maxCos * maxCos), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))), Float32(1.0)))
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.20000000298023224))
		tmp = Float32(Float32(Float32(ux * Float32(Float32(Float32(1.0) - ux) * maxCos)) * zi) + fma(xi, t_0, Float32(uy * Float32(t_0 * fma(Float32(-2.0), Float32(xi * Float32(uy * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(2.0) * Float32(Float32(pi) * yi)))))));
	else
		tmp = Float32(sin(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) * yi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.200000003

   1. Initial program 99.3%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(uy \cdot \left(-2 \cdot \left(\left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) + 2 \cdot \left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   5. Simplified 93.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(xi, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)}, uy \cdot \left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(-2, xi \cdot \left(\left(\pi \cdot \pi\right) \cdot uy\right), 2 \cdot \left(yi \cdot \pi\right)\right)\right)\right)} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   if 0.200000003 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 95.5%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   2. Add Preprocessing

   3. Taylor expanded in yi around inf

      \[\leadsto \color{blue}{\left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)} \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)} \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

   5. Simplified 57.2%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \left(yi \cdot \sin \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)} \]

   6. Taylor expanded in maxCos around 0

      \[\leadsto \color{blue}{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. sin-lowering-sin.f32 N/A

         \[\leadsto yi \cdot \color{blue}{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      3. *-lowering-*.f32 N/A

         \[\leadsto yi \cdot \sin \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      4. *-lowering-*.f32 N/A

         \[\leadsto yi \cdot \sin \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      5. PI-lowering-PI.f32 57.2

         \[\leadsto yi \cdot \sin \left(2 \cdot \left(uy \cdot \color{blue}{\pi}\right)\right) \]

   8. Simplified 57.2%

      \[\leadsto \color{blue}{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.20000000298023224:\\ \;\;\;\;\left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi + \mathsf{fma}\left(xi, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}, uy \cdot \left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(-2, xi \cdot \left(uy \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \left(\pi \cdot yi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 88.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}\\ \mathbf{if}\;2 \cdot uy \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{fma}\left(maxCos, \left(1 - ux\right) \cdot \left(ux \cdot zi\right), \mathsf{fma}\left(uy, t\_0 \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \left(\pi \cdot yi\right)\right), xi \cdot t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0
         (sqrt
          (fma
           (* maxCos maxCos)
           (* (* ux ux) (* (- 1.0 ux) (+ ux -1.0)))
           1.0))))
   (if (<= (* 2.0 uy) 0.20000000298023224)
     (fma
      maxCos
      (* (- 1.0 ux) (* ux zi))
      (fma
       uy
       (* t_0 (fma -2.0 (* uy (* xi (* PI PI))) (* 2.0 (* PI yi))))
       (* xi t_0)))
     (* (sin (* 2.0 (* uy PI))) yi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = sqrtf(fmaf((maxCos * maxCos), ((ux * ux) * ((1.0f - ux) * (ux + -1.0f))), 1.0f));
	float tmp;
	if ((2.0f * uy) <= 0.20000000298023224f) {
		tmp = fmaf(maxCos, ((1.0f - ux) * (ux * zi)), fmaf(uy, (t_0 * fmaf(-2.0f, (uy * (xi * (((float) M_PI) * ((float) M_PI)))), (2.0f * (((float) M_PI) * yi)))), (xi * t_0)));
	} else {
		tmp = sinf((2.0f * (uy * ((float) M_PI)))) * yi;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = sqrt(fma(Float32(maxCos * maxCos), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))), Float32(1.0)))
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.20000000298023224))
		tmp = fma(maxCos, Float32(Float32(Float32(1.0) - ux) * Float32(ux * zi)), fma(uy, Float32(t_0 * fma(Float32(-2.0), Float32(uy * Float32(xi * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(2.0) * Float32(Float32(pi) * yi)))), Float32(xi * t_0)));
	else
		tmp = Float32(sin(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) * yi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.200000003

   1. Initial program 99.3%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi + \left(\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi\right)} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      2. accelerator-lowering-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}, yi, \left(\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi\right)} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   4. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(ux, \left(0 - \left(1 - ux\right)\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right), 1\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right), yi, \sqrt{\mathsf{fma}\left(ux, \left(0 - \left(1 - ux\right)\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right), 1\right)} \cdot \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot xi\right)\right)} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   5. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \left(uy \cdot \left(-2 \cdot \left(\left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \sqrt{1 + {maxCos}^{2} \cdot \left({ux}^{2} \cdot \left(\left(1 - ux\right) \cdot \left(ux - 1\right)\right)\right)}\right) + 2 \cdot \left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 + {maxCos}^{2} \cdot \left({ux}^{2} \cdot \left(\left(1 - ux\right) \cdot \left(ux - 1\right)\right)\right)}\right)\right) + xi \cdot \sqrt{1 + {maxCos}^{2} \cdot \left({ux}^{2} \cdot \left(\left(1 - ux\right) \cdot \left(ux - 1\right)\right)\right)}\right)} \]
   6. Step-by-step derivation

      1. accelerator-lowering-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \left(-2 \cdot \left(\left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \sqrt{1 + {maxCos}^{2} \cdot \left({ux}^{2} \cdot \left(\left(1 - ux\right) \cdot \left(ux - 1\right)\right)\right)}\right) + 2 \cdot \left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 + {maxCos}^{2} \cdot \left({ux}^{2} \cdot \left(\left(1 - ux\right) \cdot \left(ux - 1\right)\right)\right)}\right)\right) + xi \cdot \sqrt{1 + {maxCos}^{2} \cdot \left({ux}^{2} \cdot \left(\left(1 - ux\right) \cdot \left(ux - 1\right)\right)\right)}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{\left(ux \cdot zi\right) \cdot \left(1 - ux\right)}, uy \cdot \left(-2 \cdot \left(\left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \sqrt{1 + {maxCos}^{2} \cdot \left({ux}^{2} \cdot \left(\left(1 - ux\right) \cdot \left(ux - 1\right)\right)\right)}\right) + 2 \cdot \left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 + {maxCos}^{2} \cdot \left({ux}^{2} \cdot \left(\left(1 - ux\right) \cdot \left(ux - 1\right)\right)\right)}\right)\right) + xi \cdot \sqrt{1 + {maxCos}^{2} \cdot \left({ux}^{2} \cdot \left(\left(1 - ux\right) \cdot \left(ux - 1\right)\right)\right)}\right) \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{\left(ux \cdot zi\right) \cdot \left(1 - ux\right)}, uy \cdot \left(-2 \cdot \left(\left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \sqrt{1 + {maxCos}^{2} \cdot \left({ux}^{2} \cdot \left(\left(1 - ux\right) \cdot \left(ux - 1\right)\right)\right)}\right) + 2 \cdot \left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 + {maxCos}^{2} \cdot \left({ux}^{2} \cdot \left(\left(1 - ux\right) \cdot \left(ux - 1\right)\right)\right)}\right)\right) + xi \cdot \sqrt{1 + {maxCos}^{2} \cdot \left({ux}^{2} \cdot \left(\left(1 - ux\right) \cdot \left(ux - 1\right)\right)\right)}\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{\left(ux \cdot zi\right)} \cdot \left(1 - ux\right), uy \cdot \left(-2 \cdot \left(\left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \sqrt{1 + {maxCos}^{2} \cdot \left({ux}^{2} \cdot \left(\left(1 - ux\right) \cdot \left(ux - 1\right)\right)\right)}\right) + 2 \cdot \left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 + {maxCos}^{2} \cdot \left({ux}^{2} \cdot \left(\left(1 - ux\right) \cdot \left(ux - 1\right)\right)\right)}\right)\right) + xi \cdot \sqrt{1 + {maxCos}^{2} \cdot \left({ux}^{2} \cdot \left(\left(1 - ux\right) \cdot \left(ux - 1\right)\right)\right)}\right) \]

      5. --lowering--.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, \left(ux \cdot zi\right) \cdot \color{blue}{\left(1 - ux\right)}, uy \cdot \left(-2 \cdot \left(\left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \sqrt{1 + {maxCos}^{2} \cdot \left({ux}^{2} \cdot \left(\left(1 - ux\right) \cdot \left(ux - 1\right)\right)\right)}\right) + 2 \cdot \left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 + {maxCos}^{2} \cdot \left({ux}^{2} \cdot \left(\left(1 - ux\right) \cdot \left(ux - 1\right)\right)\right)}\right)\right) + xi \cdot \sqrt{1 + {maxCos}^{2} \cdot \left({ux}^{2} \cdot \left(\left(1 - ux\right) \cdot \left(ux - 1\right)\right)\right)}\right) \]

      6. accelerator-lowering-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, \left(ux \cdot zi\right) \cdot \left(1 - ux\right), \color{blue}{\mathsf{fma}\left(uy, -2 \cdot \left(\left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \sqrt{1 + {maxCos}^{2} \cdot \left({ux}^{2} \cdot \left(\left(1 - ux\right) \cdot \left(ux - 1\right)\right)\right)}\right) + 2 \cdot \left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 + {maxCos}^{2} \cdot \left({ux}^{2} \cdot \left(\left(1 - ux\right) \cdot \left(ux - 1\right)\right)\right)}\right), xi \cdot \sqrt{1 + {maxCos}^{2} \cdot \left({ux}^{2} \cdot \left(\left(1 - ux\right) \cdot \left(ux - 1\right)\right)\right)}\right)}\right) \]

   7. Simplified 93.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, \left(ux \cdot zi\right) \cdot \left(1 - ux\right), \mathsf{fma}\left(uy, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \left(\pi \cdot yi\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}\right)\right)} \]

   if 0.200000003 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 95.5%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   2. Add Preprocessing

   3. Taylor expanded in yi around inf

      \[\leadsto \color{blue}{\left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)} \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)} \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

   5. Simplified 57.2%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \left(yi \cdot \sin \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)} \]

   6. Taylor expanded in maxCos around 0

      \[\leadsto \color{blue}{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. sin-lowering-sin.f32 N/A

         \[\leadsto yi \cdot \color{blue}{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      3. *-lowering-*.f32 N/A

         \[\leadsto yi \cdot \sin \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      4. *-lowering-*.f32 N/A

         \[\leadsto yi \cdot \sin \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      5. PI-lowering-PI.f32 57.2

         \[\leadsto yi \cdot \sin \left(2 \cdot \left(uy \cdot \color{blue}{\pi}\right)\right) \]

   8. Simplified 57.2%

      \[\leadsto \color{blue}{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{fma}\left(maxCos, \left(1 - ux\right) \cdot \left(ux \cdot zi\right), \mathsf{fma}\left(uy, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \left(\pi \cdot yi\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 84.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.20000000298023224:\\ \;\;\;\;\left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi + \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(2, \pi \cdot \left(uy \cdot yi\right), xi\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.20000000298023224)
   (+
    (* (* ux (* (- 1.0 ux) maxCos)) zi)
    (*
     (sqrt
      (fma (* maxCos maxCos) (* (* ux ux) (* (- 1.0 ux) (+ ux -1.0))) 1.0))
     (fma 2.0 (* PI (* uy yi)) xi)))
   (* (sin (* 2.0 (* uy PI))) yi)))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if ((2.0f * uy) <= 0.20000000298023224f) {
		tmp = ((ux * ((1.0f - ux) * maxCos)) * zi) + (sqrtf(fmaf((maxCos * maxCos), ((ux * ux) * ((1.0f - ux) * (ux + -1.0f))), 1.0f)) * fmaf(2.0f, (((float) M_PI) * (uy * yi)), xi));
	} else {
		tmp = sinf((2.0f * (uy * ((float) M_PI)))) * yi;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.20000000298023224))
		tmp = Float32(Float32(Float32(ux * Float32(Float32(Float32(1.0) - ux) * maxCos)) * zi) + Float32(sqrt(fma(Float32(maxCos * maxCos), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))), Float32(1.0))) * fma(Float32(2.0), Float32(Float32(pi) * Float32(uy * yi)), xi)));
	else
		tmp = Float32(sin(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) * yi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.200000003

   1. Initial program 99.3%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(2 \cdot \left(\left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      2. distribute-rgt-out N/A

         \[\leadsto \color{blue}{\sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)} \cdot \left(2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) + xi\right)} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)} \cdot \left(2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) + xi\right)} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   5. Simplified 89.0%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(2, \left(uy \cdot yi\right) \cdot \pi, xi\right)} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   if 0.200000003 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 95.5%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   2. Add Preprocessing

   3. Taylor expanded in yi around inf

      \[\leadsto \color{blue}{\left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)} \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)} \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

   5. Simplified 57.2%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \left(yi \cdot \sin \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)} \]

   6. Taylor expanded in maxCos around 0

      \[\leadsto \color{blue}{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. sin-lowering-sin.f32 N/A

         \[\leadsto yi \cdot \color{blue}{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      3. *-lowering-*.f32 N/A

         \[\leadsto yi \cdot \sin \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      4. *-lowering-*.f32 N/A

         \[\leadsto yi \cdot \sin \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      5. PI-lowering-PI.f32 57.2

         \[\leadsto yi \cdot \sin \left(2 \cdot \left(uy \cdot \color{blue}{\pi}\right)\right) \]

   8. Simplified 57.2%

      \[\leadsto \color{blue}{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.20000000298023224:\\ \;\;\;\;\left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi + \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(2, \pi \cdot \left(uy \cdot yi\right), xi\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 82.1% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi + \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(2, \pi \cdot \left(uy \cdot yi\right), xi\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (+
  (* (* ux (* (- 1.0 ux) maxCos)) zi)
  (*
   (sqrt (fma (* maxCos maxCos) (* (* ux ux) (* (- 1.0 ux) (+ ux -1.0))) 1.0))
   (fma 2.0 (* PI (* uy yi)) xi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return ((ux * ((1.0f - ux) * maxCos)) * zi) + (sqrtf(fmaf((maxCos * maxCos), ((ux * ux) * ((1.0f - ux) * (ux + -1.0f))), 1.0f)) * fmaf(2.0f, (((float) M_PI) * (uy * yi)), xi));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(Float32(Float32(ux * Float32(Float32(Float32(1.0) - ux) * maxCos)) * zi) + Float32(sqrt(fma(Float32(maxCos * maxCos), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))), Float32(1.0))) * fma(Float32(2.0), Float32(Float32(pi) * Float32(uy * yi)), xi)))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. Add Preprocessing

3. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\left(2 \cdot \left(\left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   2. distribute-rgt-out N/A

      \[\leadsto \color{blue}{\sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)} \cdot \left(2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) + xi\right)} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{\sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)} \cdot \left(2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) + xi\right)} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

5. Simplified 81.5%

   \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(2, \left(uy \cdot yi\right) \cdot \pi, xi\right)} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

6. Final simplification 81.5%

   \[\leadsto \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi + \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(2, \pi \cdot \left(uy \cdot yi\right), xi\right) \]
7. Add Preprocessing

Alternative 14: 82.1% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(2, \pi \cdot \left(uy \cdot yi\right), xi\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (fma
  maxCos
  (* ux (* (- 1.0 ux) zi))
  (*
   (sqrt (fma (* maxCos maxCos) (* (* ux ux) (* (- 1.0 ux) (+ ux -1.0))) 1.0))
   (fma 2.0 (* PI (* uy yi)) xi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf(maxCos, (ux * ((1.0f - ux) * zi)), (sqrtf(fmaf((maxCos * maxCos), ((ux * ux) * ((1.0f - ux) * (ux + -1.0f))), 1.0f)) * fmaf(2.0f, (((float) M_PI) * (uy * yi)), xi)));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(maxCos, Float32(ux * Float32(Float32(Float32(1.0) - ux) * zi)), Float32(sqrt(fma(Float32(maxCos * maxCos), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))), Float32(1.0))) * fma(Float32(2.0), Float32(Float32(pi) * Float32(uy * yi)), xi)))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. Add Preprocessing

3. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{2 \cdot \left(\left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) + \left(maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

5. Simplified 81.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(2, \left(uy \cdot yi\right) \cdot \pi, xi\right)\right)} \]

6. Final simplification 81.5%

   \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(2, \pi \cdot \left(uy \cdot yi\right), xi\right)\right) \]
7. Add Preprocessing

Alternative 15: 82.1% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}, \mathsf{fma}\left(2, uy \cdot \left(\pi \cdot yi\right), xi\right), maxCos \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot zi\right)\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (fma
  (sqrt (fma (* maxCos maxCos) (* (* ux ux) (* (- 1.0 ux) (+ ux -1.0))) 1.0))
  (fma 2.0 (* uy (* PI yi)) xi)
  (* maxCos (* (- 1.0 ux) (* ux zi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf(sqrtf(fmaf((maxCos * maxCos), ((ux * ux) * ((1.0f - ux) * (ux + -1.0f))), 1.0f)), fmaf(2.0f, (uy * (((float) M_PI) * yi)), xi), (maxCos * ((1.0f - ux) * (ux * zi))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(sqrt(fma(Float32(maxCos * maxCos), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))), Float32(1.0))), fma(Float32(2.0), Float32(uy * Float32(Float32(pi) * yi)), xi), Float32(maxCos * Float32(Float32(Float32(1.0) - ux) * Float32(ux * zi))))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. Add Preprocessing
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi + \left(\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi\right)} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   2. accelerator-lowering-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}, yi, \left(\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi\right)} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

4. Applied egg-rr 98.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(ux, \left(0 - \left(1 - ux\right)\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right), 1\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right), yi, \sqrt{\mathsf{fma}\left(ux, \left(0 - \left(1 - ux\right)\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right), 1\right)} \cdot \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot xi\right)\right)} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

5. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{2 \cdot \left(\left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 + {maxCos}^{2} \cdot \left({ux}^{2} \cdot \left(\left(1 - ux\right) \cdot \left(ux - 1\right)\right)\right)}\right) + \left(maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 + {maxCos}^{2} \cdot \left({ux}^{2} \cdot \left(\left(1 - ux\right) \cdot \left(ux - 1\right)\right)\right)}\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto 2 \cdot \left(\left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 + {maxCos}^{2} \cdot \left({ux}^{2} \cdot \left(\left(1 - ux\right) \cdot \left(ux - 1\right)\right)\right)}\right) + \color{blue}{\left(xi \cdot \sqrt{1 + {maxCos}^{2} \cdot \left({ux}^{2} \cdot \left(\left(1 - ux\right) \cdot \left(ux - 1\right)\right)\right)} + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)} \]

   2. associate-+r+ N/A

      \[\leadsto \color{blue}{\left(2 \cdot \left(\left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 + {maxCos}^{2} \cdot \left({ux}^{2} \cdot \left(\left(1 - ux\right) \cdot \left(ux - 1\right)\right)\right)}\right) + xi \cdot \sqrt{1 + {maxCos}^{2} \cdot \left({ux}^{2} \cdot \left(\left(1 - ux\right) \cdot \left(ux - 1\right)\right)\right)}\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]

   3. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{1 + {maxCos}^{2} \cdot \left({ux}^{2} \cdot \left(\left(1 - ux\right) \cdot \left(ux - 1\right)\right)\right)}} + xi \cdot \sqrt{1 + {maxCos}^{2} \cdot \left({ux}^{2} \cdot \left(\left(1 - ux\right) \cdot \left(ux - 1\right)\right)\right)}\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) \]

   4. distribute-rgt-out N/A

      \[\leadsto \color{blue}{\sqrt{1 + {maxCos}^{2} \cdot \left({ux}^{2} \cdot \left(\left(1 - ux\right) \cdot \left(ux - 1\right)\right)\right)} \cdot \left(2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) + xi\right)} + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) \]

7. Simplified 81.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}, \mathsf{fma}\left(2, uy \cdot \left(\pi \cdot yi\right), xi\right), maxCos \cdot \left(\left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right)\right)} \]

8. Final simplification 81.5%

   \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}, \mathsf{fma}\left(2, uy \cdot \left(\pi \cdot yi\right), xi\right), maxCos \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot zi\right)\right)\right) \]
9. Add Preprocessing

Alternative 16: 81.9% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \mathsf{fma}\left(2, \pi \cdot \left(uy \cdot yi\right), xi\right) \cdot \sqrt{1}\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (fma
  maxCos
  (* ux (* (- 1.0 ux) zi))
  (* (fma 2.0 (* PI (* uy yi)) xi) (sqrt 1.0))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf(maxCos, (ux * ((1.0f - ux) * zi)), (fmaf(2.0f, (((float) M_PI) * (uy * yi)), xi) * sqrtf(1.0f)));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(maxCos, Float32(ux * Float32(Float32(Float32(1.0) - ux) * zi)), Float32(fma(Float32(2.0), Float32(Float32(pi) * Float32(uy * yi)), xi) * sqrt(Float32(1.0))))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. Add Preprocessing

3. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{2 \cdot \left(\left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) + \left(maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

5. Simplified 81.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(2, \left(uy \cdot yi\right) \cdot \pi, xi\right)\right)} \]

6. Taylor expanded in maxCos around 0

   \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \sqrt{\color{blue}{1}} \cdot \mathsf{fma}\left(2, \left(uy \cdot yi\right) \cdot \mathsf{PI}\left(\right), xi\right)\right) \]
7. Step-by-step derivation

8. Simplified 81.4%

   \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \sqrt{\color{blue}{1}} \cdot \mathsf{fma}\left(2, \left(uy \cdot yi\right) \cdot \pi, xi\right)\right) \]

9. Final simplification 81.4%

   \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \mathsf{fma}\left(2, \pi \cdot \left(uy \cdot yi\right), xi\right) \cdot \sqrt{1}\right) \]
10. Add Preprocessing

Alternative 17: 81.9% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ xi + \mathsf{fma}\left(ux \cdot maxCos, \left(1 - ux\right) \cdot zi, \left(2 \cdot uy\right) \cdot \left(\pi \cdot yi\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (+ xi (fma (* ux maxCos) (* (- 1.0 ux) zi) (* (* 2.0 uy) (* PI yi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return xi + fmaf((ux * maxCos), ((1.0f - ux) * zi), ((2.0f * uy) * (((float) M_PI) * yi)));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(xi + fma(Float32(ux * maxCos), Float32(Float32(Float32(1.0) - ux) * zi), Float32(Float32(Float32(2.0) * uy) * Float32(Float32(pi) * yi))))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. Add Preprocessing

3. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{2 \cdot \left(\left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) + \left(maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

5. Simplified 81.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(2, \left(uy \cdot yi\right) \cdot \pi, xi\right)\right)} \]

6. Taylor expanded in maxCos around 0

   \[\leadsto \color{blue}{xi + \left(2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)} \]
7. Step-by-step derivation

   1. +-lowering-+.f32 N/A

      \[\leadsto \color{blue}{xi + \left(2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto xi + \color{blue}{\left(maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

   3. associate-*r* N/A

      \[\leadsto xi + \left(\color{blue}{\left(maxCos \cdot ux\right) \cdot \left(zi \cdot \left(1 - ux\right)\right)} + 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   4. accelerator-lowering-fma.f32 N/A

      \[\leadsto xi + \color{blue}{\mathsf{fma}\left(maxCos \cdot ux, zi \cdot \left(1 - ux\right), 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

   5. *-commutative N/A

      \[\leadsto xi + \mathsf{fma}\left(\color{blue}{ux \cdot maxCos}, zi \cdot \left(1 - ux\right), 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto xi + \mathsf{fma}\left(\color{blue}{ux \cdot maxCos}, zi \cdot \left(1 - ux\right), 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto xi + \mathsf{fma}\left(ux \cdot maxCos, \color{blue}{zi \cdot \left(1 - ux\right)}, 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   8. --lowering--.f32 N/A

      \[\leadsto xi + \mathsf{fma}\left(ux \cdot maxCos, zi \cdot \color{blue}{\left(1 - ux\right)}, 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto xi + \mathsf{fma}\left(ux \cdot maxCos, zi \cdot \left(1 - ux\right), \color{blue}{\left(2 \cdot uy\right) \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)}\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto xi + \mathsf{fma}\left(ux \cdot maxCos, zi \cdot \left(1 - ux\right), \color{blue}{\left(2 \cdot uy\right) \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)}\right) \]

   11. *-lowering-*.f32 N/A

       \[\leadsto xi + \mathsf{fma}\left(ux \cdot maxCos, zi \cdot \left(1 - ux\right), \color{blue}{\left(2 \cdot uy\right)} \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto xi + \mathsf{fma}\left(ux \cdot maxCos, zi \cdot \left(1 - ux\right), \left(2 \cdot uy\right) \cdot \color{blue}{\left(yi \cdot \mathsf{PI}\left(\right)\right)}\right) \]

   13. PI-lowering-PI.f32 81.4

       \[\leadsto xi + \mathsf{fma}\left(ux \cdot maxCos, zi \cdot \left(1 - ux\right), \left(2 \cdot uy\right) \cdot \left(yi \cdot \color{blue}{\pi}\right)\right) \]

8. Simplified 81.4%

   \[\leadsto \color{blue}{xi + \mathsf{fma}\left(ux \cdot maxCos, zi \cdot \left(1 - ux\right), \left(2 \cdot uy\right) \cdot \left(yi \cdot \pi\right)\right)} \]

9. Final simplification 81.4%

   \[\leadsto xi + \mathsf{fma}\left(ux \cdot maxCos, \left(1 - ux\right) \cdot zi, \left(2 \cdot uy\right) \cdot \left(\pi \cdot yi\right)\right) \]
10. Add Preprocessing

Alternative 18: 79.3% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ maxCos \cdot \left(ux \cdot zi\right) + \mathsf{fma}\left(2 \cdot uy, \pi \cdot yi, xi\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (+ (* maxCos (* ux zi)) (fma (* 2.0 uy) (* PI yi) xi)))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return (maxCos * (ux * zi)) + fmaf((2.0f * uy), (((float) M_PI) * yi), xi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(Float32(maxCos * Float32(ux * zi)) + fma(Float32(Float32(2.0) * uy), Float32(Float32(pi) * yi), xi))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. Add Preprocessing

3. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{2 \cdot \left(\left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) + \left(maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

5. Simplified 81.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(2, \left(uy \cdot yi\right) \cdot \pi, xi\right)\right)} \]

6. Taylor expanded in ux around 0

   \[\leadsto \color{blue}{xi + \left(2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot zi\right)\right)} \]
7. Step-by-step derivation

   1. associate-+r+ N/A

      \[\leadsto \color{blue}{\left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) + maxCos \cdot \left(ux \cdot zi\right)} \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \color{blue}{\left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) + maxCos \cdot \left(ux \cdot zi\right)} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) + xi\right)} + maxCos \cdot \left(ux \cdot zi\right) \]

   4. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(2 \cdot uy\right) \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)} + xi\right) + maxCos \cdot \left(ux \cdot zi\right) \]

   5. accelerator-lowering-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot uy, yi \cdot \mathsf{PI}\left(\right), xi\right)} + maxCos \cdot \left(ux \cdot zi\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{2 \cdot uy}, yi \cdot \mathsf{PI}\left(\right), xi\right) + maxCos \cdot \left(ux \cdot zi\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(2 \cdot uy, \color{blue}{yi \cdot \mathsf{PI}\left(\right)}, xi\right) + maxCos \cdot \left(ux \cdot zi\right) \]

   8. PI-lowering-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(2 \cdot uy, yi \cdot \color{blue}{\mathsf{PI}\left(\right)}, xi\right) + maxCos \cdot \left(ux \cdot zi\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(2 \cdot uy, yi \cdot \mathsf{PI}\left(\right), xi\right) + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]

   10. *-lowering-*.f32 78.8

       \[\leadsto \mathsf{fma}\left(2 \cdot uy, yi \cdot \pi, xi\right) + maxCos \cdot \color{blue}{\left(ux \cdot zi\right)} \]

8. Simplified 78.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot uy, yi \cdot \pi, xi\right) + maxCos \cdot \left(ux \cdot zi\right)} \]

9. Final simplification 78.8%

   \[\leadsto maxCos \cdot \left(ux \cdot zi\right) + \mathsf{fma}\left(2 \cdot uy, \pi \cdot yi, xi\right) \]
10. Add Preprocessing

Alternative 19: 74.5% accurate, 20.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(2 \cdot uy, \pi \cdot yi, xi\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (fma (* 2.0 uy) (* PI yi) xi))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf((2.0f * uy), (((float) M_PI) * yi), xi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(Float32(2.0) * uy), Float32(Float32(pi) * yi), xi)
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. Add Preprocessing

3. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{2 \cdot \left(\left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) + \left(maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

5. Simplified 81.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(2, \left(uy \cdot yi\right) \cdot \pi, xi\right)\right)} \]

6. Taylor expanded in maxCos around 0

   \[\leadsto \color{blue}{xi + 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) + xi} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(2 \cdot uy\right) \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)} + xi \]

   3. accelerator-lowering-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot uy, yi \cdot \mathsf{PI}\left(\right), xi\right)} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{2 \cdot uy}, yi \cdot \mathsf{PI}\left(\right), xi\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(2 \cdot uy, \color{blue}{yi \cdot \mathsf{PI}\left(\right)}, xi\right) \]

   6. PI-lowering-PI.f32 74.9

      \[\leadsto \mathsf{fma}\left(2 \cdot uy, yi \cdot \color{blue}{\pi}, xi\right) \]

8. Simplified 74.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot uy, yi \cdot \pi, xi\right)} \]

9. Final simplification 74.9%

   \[\leadsto \mathsf{fma}\left(2 \cdot uy, \pi \cdot yi, xi\right) \]
10. Add Preprocessing

Alternative 20: 11.8% accurate, 32.1× speedup

Math

\[\begin{array}{l} \\ \left(ux \cdot maxCos\right) \cdot zi \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos) :precision binary32 (* (* ux maxCos) zi))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return (ux * maxCos) * zi;
}

Fortran

real(4) function code(xi, yi, zi, ux, uy, maxcos)
    real(4), intent (in) :: xi
    real(4), intent (in) :: yi
    real(4), intent (in) :: zi
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = (ux * maxcos) * zi
end function

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(Float32(ux * maxCos) * zi)
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = (ux * maxCos) * zi;
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. Add Preprocessing

3. Taylor expanded in zi around inf

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) \cdot maxCos} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot ux\right)} \cdot maxCos \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{\left(zi \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot maxCos\right)} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{\left(zi \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot maxCos\right)} \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(1 - ux\right) \cdot zi\right)} \cdot \left(ux \cdot maxCos\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{\left(\left(1 - ux\right) \cdot zi\right)} \cdot \left(ux \cdot maxCos\right) \]

   7. --lowering--.f32 N/A

      \[\leadsto \left(\color{blue}{\left(1 - ux\right)} \cdot zi\right) \cdot \left(ux \cdot maxCos\right) \]

   8. *-lowering-*.f32 11.7

      \[\leadsto \left(\left(1 - ux\right) \cdot zi\right) \cdot \color{blue}{\left(ux \cdot maxCos\right)} \]

5. Simplified 11.7%

   \[\leadsto \color{blue}{\left(\left(1 - ux\right) \cdot zi\right) \cdot \left(ux \cdot maxCos\right)} \]

6. Taylor expanded in ux around 0

   \[\leadsto \color{blue}{zi} \cdot \left(ux \cdot maxCos\right) \]
7. Step-by-step derivation

8. Simplified 10.7%

   \[\leadsto \color{blue}{zi} \cdot \left(ux \cdot maxCos\right) \]

9. Final simplification 10.7%

   \[\leadsto \left(ux \cdot maxCos\right) \cdot zi \]
10. Add Preprocessing

Alternative 21: 11.8% accurate, 32.1× speedup

Math

\[\begin{array}{l} \\ maxCos \cdot \left(ux \cdot zi\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos) :precision binary32 (* maxCos (* ux zi)))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return maxCos * (ux * zi);
}

Fortran

real(4) function code(xi, yi, zi, ux, uy, maxcos)
    real(4), intent (in) :: xi
    real(4), intent (in) :: yi
    real(4), intent (in) :: zi
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = maxcos * (ux * zi)
end function

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(maxCos * Float32(ux * zi))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = maxCos * (ux * zi);
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. Add Preprocessing

3. Taylor expanded in zi around inf

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) \cdot maxCos} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot ux\right)} \cdot maxCos \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{\left(zi \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot maxCos\right)} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{\left(zi \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot maxCos\right)} \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(1 - ux\right) \cdot zi\right)} \cdot \left(ux \cdot maxCos\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{\left(\left(1 - ux\right) \cdot zi\right)} \cdot \left(ux \cdot maxCos\right) \]

   7. --lowering--.f32 N/A

      \[\leadsto \left(\color{blue}{\left(1 - ux\right)} \cdot zi\right) \cdot \left(ux \cdot maxCos\right) \]

   8. *-lowering-*.f32 11.7

      \[\leadsto \left(\left(1 - ux\right) \cdot zi\right) \cdot \color{blue}{\left(ux \cdot maxCos\right)} \]

5. Simplified 11.7%

   \[\leadsto \color{blue}{\left(\left(1 - ux\right) \cdot zi\right) \cdot \left(ux \cdot maxCos\right)} \]

6. Taylor expanded in ux around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
7. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]

   2. *-lowering-*.f32 10.7

      \[\leadsto maxCos \cdot \color{blue}{\left(ux \cdot zi\right)} \]

8. Simplified 10.7%

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024196 
(FPCore (xi yi zi ux uy maxCos)
  :name "UniformSampleCone 2"
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0)) (and (<= -10000.0 yi) (<= yi 10000.0))) (and (<= -10000.0 zi) (<= zi 10000.0))) (and (<= 2.328306437e-10 ux) (<= ux 1.0))) (and (<= 2.328306437e-10 uy) (<= uy 1.0))) (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (+ (+ (* (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (* (* (- 1.0 ux) maxCos) ux) (* (* (- 1.0 ux) maxCos) ux))))) xi) (* (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (* (* (- 1.0 ux) maxCos) ux) (* (* (- 1.0 ux) maxCos) ux))))) yi)) (* (* (* (- 1.0 ux) maxCos) ux) zi)))