UniformSampleCone 2

Percentage Accurate: 98.9% → 98.9%

Time: 19.5s

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: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

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((ux * ((maxCos / ux) - maxCos)), (ux * zi), (sqrtf((1.0f + (maxCos * ((1.0f - ux) * ((ux * ux) * (maxCos * (ux + -1.0f))))))) * ((cosf(t_0) * xi) + (sinf(t_0) * yi))));
}

Julia

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in ux around inf 99.1%

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

   1. +-commutative 99.1%

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

   2. mul-1-neg 99.1%

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

   3. unsub-neg 99.1%

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

7. Simplified 99.1%

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

Alternative 2: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

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 * (1.0f - ux)), (ux * zi), (sqrtf((1.0f + (maxCos * ((1.0f - ux) * ((ux * ux) * (maxCos * (ux + -1.0f))))))) * ((cosf(t_0) * xi) + (sinf(t_0) * yi))));
}

Julia

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Final simplification 99.1%

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

Alternative 3: 98.7% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy PI))))
   (fma maxCos (* ux (* zi (- 1.0 ux))) (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 = 2.0f * (uy * ((float) M_PI));
	return fmaf(maxCos, (ux * (zi * (1.0f - ux))), fmaf(xi, cosf(t_0), (yi * sinf(t_0))));
}

Julia

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

Derivation

1. Initial program 99.1%

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

   1. associate-+l+ 99.1%

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

   2. associate-*l* 99.1%

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

   3. fma-define 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in maxCos around 0 98.9%

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

   1. fma-define 99.0%

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

   2. *-commutative 99.0%

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

   3. fma-define 98.9%

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

7. Simplified 98.9%

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

8. Final simplification 98.9%

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

Alternative 4: 98.7% accurate, 1.4× speedup

Math

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

FPCore

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

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(xi, cosf(t_0), (yi * sinf(t_0))) + ((maxCos * zi) * (ux * (1.0f - ux)));
}

Julia

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.0%

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

5. Taylor expanded in ux around 0 98.9%

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

   1. fma-define 98.9%

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

7. Simplified 98.9%

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

   1. pow1 98.9%

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

   2. associate-*r* 98.9%

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

   3. *-commutative 98.9%

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

   4. associate-*r* 98.9%

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

   5. *-commutative 98.9%

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

   6. associate-*r* 98.9%

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

9. Applied egg-rr 98.9%

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

    1. unpow1 98.9%

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

    2. associate-*r* 98.9%

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

    3. *-commutative 98.9%

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

11. Simplified 98.9%

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

Alternative 5: 98.7% accurate, 2.0× speedup

Math

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

FPCore

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

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 (maxCos * (ux * (zi * (1.0f - ux)))) + ((yi * sinf(t_0)) + (xi * cosf(t_0)));
}

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

   1. associate-+l+ 99.1%

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

   2. associate-*l* 99.1%

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

   3. fma-define 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in maxCos around 0 98.9%

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

6. Final simplification 98.9%

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

Alternative 6: 96.0% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy PI))))
   (+ (+ (* yi (sin t_0)) (* xi (cos t_0))) (* 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));
	return ((yi * sinf(t_0)) + (xi * cosf(t_0))) + (maxCos * (ux * zi));
}

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

   1. associate-+l+ 99.1%

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

   2. associate-*l* 99.1%

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

   3. fma-define 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in ux around 0 95.2%

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

6. Final simplification 95.2%

   \[\leadsto \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) + xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) + maxCos \cdot \left(ux \cdot zi\right) \]
7. Add Preprocessing

Alternative 7: 96.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ \mathbf{if}\;uy \leq 0.001458600047044456:\\ \;\;\;\;xi + \left(maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\pi}^{2}\right)\right) + 2 \cdot \left(\pi \cdot yi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;yi \cdot \sin t\_0 + xi \cdot \cos t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy PI))))
   (if (<= uy 0.001458600047044456)
     (+
      xi
      (+
       (* maxCos (* ux (* zi (- 1.0 ux))))
       (* uy (+ (* -2.0 (* uy (* xi (pow PI 2.0)))) (* 2.0 (* PI yi))))))
     (+ (* yi (sin t_0)) (* xi (cos t_0))))))

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 tmp;
	if (uy <= 0.001458600047044456f) {
		tmp = xi + ((maxCos * (ux * (zi * (1.0f - ux)))) + (uy * ((-2.0f * (uy * (xi * powf(((float) M_PI), 2.0f)))) + (2.0f * (((float) M_PI) * yi)))));
	} else {
		tmp = (yi * sinf(t_0)) + (xi * cosf(t_0));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	tmp = Float32(0.0)
	if (uy <= Float32(0.001458600047044456))
		tmp = Float32(xi + Float32(Float32(maxCos * Float32(ux * Float32(zi * Float32(Float32(1.0) - ux)))) + Float32(uy * Float32(Float32(Float32(-2.0) * Float32(uy * Float32(xi * (Float32(pi) ^ Float32(2.0))))) + Float32(Float32(2.0) * Float32(Float32(pi) * yi))))));
	else
		tmp = Float32(Float32(yi * sin(t_0)) + Float32(xi * cos(t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = single(2.0) * (uy * single(pi));
	tmp = single(0.0);
	if (uy <= single(0.001458600047044456))
		tmp = xi + ((maxCos * (ux * (zi * (single(1.0) - ux)))) + (uy * ((single(-2.0) * (uy * (xi * (single(pi) ^ single(2.0))))) + (single(2.0) * (single(pi) * yi)))));
	else
		tmp = (yi * sin(t_0)) + (xi * cos(t_0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00145860005

   1. Initial program 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in ux around 0 99.3%

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

      1. fma-define 99.3%

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

   7. Simplified 99.3%

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

      1. add-cube-cbrt 99.0%

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), \color{blue}{\left(\sqrt[3]{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot \sqrt[3]{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}\right) \cdot \sqrt[3]{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}}\right) + \left(\left(1 - ux\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot zi \]

      2. pow3 98.9%

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

   9. Applied egg-rr 98.9%

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

   10. Taylor expanded in uy around 0 98.7%

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

   if 0.00145860005 < uy

   1. Initial program 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in ux around 0 97.8%

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

      1. fma-define 97.8%

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

   7. Simplified 97.8%

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

      1. add-cube-cbrt 97.4%

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), \color{blue}{\left(\sqrt[3]{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot \sqrt[3]{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}\right) \cdot \sqrt[3]{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}}\right) + \left(\left(1 - ux\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot zi \]

      2. pow3 97.3%

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

   9. Applied egg-rr 97.3%

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

   10. Taylor expanded in ux around -inf 97.5%

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

   11. Taylor expanded in ux around 0 91.7%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.001458600047044456:\\ \;\;\;\;xi + \left(maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\pi}^{2}\right)\right) + 2 \cdot \left(\pi \cdot yi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) + xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{fma}\left(2, yi \cdot \left(uy \cdot \pi\right), xi\right) + zi \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)\\ \mathbf{else}:\\ \;\;\;\;yi \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) + \frac{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)}{yi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.05000000074505806)
   (+ (fma 2.0 (* yi (* uy PI)) xi) (* zi (* (- 1.0 ux) (* ux maxCos))))
   (*
    yi
    (+ (sin (* 2.0 (* uy PI))) (/ (* maxCos (* ux (* zi (- 1.0 ux)))) yi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.05000000074505806f) {
		tmp = fmaf(2.0f, (yi * (uy * ((float) M_PI))), xi) + (zi * ((1.0f - ux) * (ux * maxCos)));
	} else {
		tmp = yi * (sinf((2.0f * (uy * ((float) M_PI)))) + ((maxCos * (ux * (zi * (1.0f - ux)))) / yi));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.05000000074505806))
		tmp = Float32(fma(Float32(2.0), Float32(yi * Float32(uy * Float32(pi))), xi) + Float32(zi * Float32(Float32(Float32(1.0) - ux) * Float32(ux * maxCos))));
	else
		tmp = Float32(yi * Float32(sin(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) + Float32(Float32(maxCos * Float32(ux * Float32(zi * Float32(Float32(1.0) - ux)))) / yi)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0500000007

   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. Step-by-step derivation

   3. Simplified 99.3%

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

   5. Taylor expanded in ux around 0 99.0%

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

      1. fma-define 99.1%

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

   7. Simplified 99.1%

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

      1. add-cube-cbrt 98.8%

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), \color{blue}{\left(\sqrt[3]{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot \sqrt[3]{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}\right) \cdot \sqrt[3]{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}}\right) + \left(\left(1 - ux\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot zi \]

      2. pow3 98.7%

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

   9. Applied egg-rr 98.7%

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

   10. Taylor expanded in uy around 0 91.4%

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

       1. +-commutative 91.4%

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

       2. *-commutative 91.4%

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

       3. fma-define 91.4%

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

       4. associate-*r* 91.5%

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

       5. *-commutative 91.5%

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

   12. Simplified 91.5%

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

   if 0.0500000007 < uy

   1. Initial program 98.0%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in ux around 0 97.9%

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

      1. fma-define 97.7%

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

   7. Simplified 97.7%

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

      1. add-cube-cbrt 97.4%

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), \color{blue}{\left(\sqrt[3]{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot \sqrt[3]{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}\right) \cdot \sqrt[3]{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}}\right) + \left(\left(1 - ux\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot zi \]

      2. pow3 97.2%

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

   9. Applied egg-rr 97.2%

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

   10. Taylor expanded in yi around inf 98.0%

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

       1. associate-+r+ 98.0%

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

       2. associate-/l* 97.9%

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

       3. *-commutative 97.9%

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

       4. associate-/l* 97.6%

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

   12. Simplified 97.6%

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

   13. Taylor expanded in xi around 0 65.6%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{fma}\left(2, yi \cdot \left(uy \cdot \pi\right), xi\right) + zi \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)\\ \mathbf{else}:\\ \;\;\;\;yi \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) + \frac{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)}{yi}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{fma}\left(2, yi \cdot \left(uy \cdot \pi\right), xi\right) + zi \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)\\ \mathbf{else}:\\ \;\;\;\;yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.05000000074505806)
   (+ (fma 2.0 (* yi (* uy PI)) xi) (* zi (* (- 1.0 ux) (* ux maxCos))))
   (+ (* yi (sin (* 2.0 (* uy PI)))) (* maxCos (* ux (* zi (- 1.0 ux)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.05000000074505806f) {
		tmp = fmaf(2.0f, (yi * (uy * ((float) M_PI))), xi) + (zi * ((1.0f - ux) * (ux * maxCos)));
	} else {
		tmp = (yi * sinf((2.0f * (uy * ((float) M_PI))))) + (maxCos * (ux * (zi * (1.0f - ux))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.05000000074505806))
		tmp = Float32(fma(Float32(2.0), Float32(yi * Float32(uy * Float32(pi))), xi) + Float32(zi * Float32(Float32(Float32(1.0) - ux) * Float32(ux * maxCos))));
	else
		tmp = Float32(Float32(yi * sin(Float32(Float32(2.0) * Float32(uy * Float32(pi))))) + Float32(maxCos * Float32(ux * Float32(zi * Float32(Float32(1.0) - ux)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0500000007

   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. Step-by-step derivation

   3. Simplified 99.3%

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

   5. Taylor expanded in ux around 0 99.0%

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

      1. fma-define 99.1%

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

   7. Simplified 99.1%

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

      1. add-cube-cbrt 98.8%

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), \color{blue}{\left(\sqrt[3]{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot \sqrt[3]{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}\right) \cdot \sqrt[3]{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}}\right) + \left(\left(1 - ux\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot zi \]

      2. pow3 98.7%

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

   9. Applied egg-rr 98.7%

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

   10. Taylor expanded in uy around 0 91.4%

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

       1. +-commutative 91.4%

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

       2. *-commutative 91.4%

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

       3. fma-define 91.4%

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

       4. associate-*r* 91.5%

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

       5. *-commutative 91.5%

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

   12. Simplified 91.5%

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

   if 0.0500000007 < uy

   1. Initial program 98.0%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in ux around 0 97.9%

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

      1. fma-define 97.7%

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

   7. Simplified 97.7%

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

      1. add-cube-cbrt 97.4%

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), \color{blue}{\left(\sqrt[3]{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot \sqrt[3]{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}\right) \cdot \sqrt[3]{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}}\right) + \left(\left(1 - ux\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot zi \]

      2. pow3 97.2%

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

   9. Applied egg-rr 97.2%

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

   10. Taylor expanded in xi around 0 65.5%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{fma}\left(2, yi \cdot \left(uy \cdot \pi\right), xi\right) + zi \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)\\ \mathbf{else}:\\ \;\;\;\;yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.05000000074505806:\\ \;\;\;\;\left(maxCos \cdot zi\right) \cdot \left(ux \cdot \left(1 - ux\right)\right) + \left(xi + \left(\pi \cdot yi\right) \cdot \left(uy \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.05000000074505806)
   (+ (* (* maxCos zi) (* ux (- 1.0 ux))) (+ xi (* (* PI yi) (* uy 2.0))))
   (+ (* yi (sin (* 2.0 (* uy PI)))) (* maxCos (* ux (* zi (- 1.0 ux)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.05000000074505806f) {
		tmp = ((maxCos * zi) * (ux * (1.0f - ux))) + (xi + ((((float) M_PI) * yi) * (uy * 2.0f)));
	} else {
		tmp = (yi * sinf((2.0f * (uy * ((float) M_PI))))) + (maxCos * (ux * (zi * (1.0f - ux))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.05000000074505806))
		tmp = Float32(Float32(Float32(maxCos * zi) * Float32(ux * Float32(Float32(1.0) - ux))) + Float32(xi + Float32(Float32(Float32(pi) * yi) * Float32(uy * Float32(2.0)))));
	else
		tmp = Float32(Float32(yi * sin(Float32(Float32(2.0) * Float32(uy * Float32(pi))))) + Float32(maxCos * Float32(ux * Float32(zi * Float32(Float32(1.0) - ux)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(0.05000000074505806))
		tmp = ((maxCos * zi) * (ux * (single(1.0) - ux))) + (xi + ((single(pi) * yi) * (uy * single(2.0))));
	else
		tmp = (yi * sin((single(2.0) * (uy * single(pi))))) + (maxCos * (ux * (zi * (single(1.0) - ux))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0500000007

   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. Step-by-step derivation

   3. Simplified 99.3%

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

   5. Taylor expanded in ux around 0 99.0%

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

      1. fma-define 99.1%

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

   7. Simplified 99.1%

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

      1. pow1 99.1%

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

      2. associate-*r* 99.1%

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

      3. *-commutative 99.1%

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

      4. associate-*r* 99.1%

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

      5. *-commutative 99.1%

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

      6. associate-*r* 99.1%

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

   9. Applied egg-rr 99.1%

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

       1. unpow1 99.1%

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

       2. associate-*r* 99.1%

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

       3. *-commutative 99.1%

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

   11. Simplified 99.1%

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

   12. Taylor expanded in uy around 0 91.5%

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

       1. associate-*r* 91.5%

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

   14. Simplified 91.5%

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

   if 0.0500000007 < uy

   1. Initial program 98.0%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in ux around 0 97.9%

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

      1. fma-define 97.7%

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

   7. Simplified 97.7%

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

      1. add-cube-cbrt 97.4%

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), \color{blue}{\left(\sqrt[3]{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot \sqrt[3]{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}\right) \cdot \sqrt[3]{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}}\right) + \left(\left(1 - ux\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot zi \]

      2. pow3 97.2%

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

   9. Applied egg-rr 97.2%

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

   10. Taylor expanded in xi around 0 65.5%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.05000000074505806:\\ \;\;\;\;\left(maxCos \cdot zi\right) \cdot \left(ux \cdot \left(1 - ux\right)\right) + \left(xi + \left(\pi \cdot yi\right) \cdot \left(uy \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 88.1% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

   1. associate-+l+ 99.1%

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

   2. associate-*l* 99.1%

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

   3. fma-define 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in maxCos around 0 98.9%

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

6. Taylor expanded in uy around 0 90.3%

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

Alternative 12: 84.0% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.05000000074505806:\\ \;\;\;\;\left(maxCos \cdot zi\right) \cdot \left(ux \cdot \left(1 - ux\right)\right) + \left(xi + \left(\pi \cdot yi\right) \cdot \left(uy \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) + ux \cdot \left(maxCos \cdot zi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.05000000074505806)
   (+ (* (* maxCos zi) (* ux (- 1.0 ux))) (+ xi (* (* PI yi) (* uy 2.0))))
   (+ (* yi (sin (* 2.0 (* uy PI)))) (* ux (* maxCos zi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.05000000074505806f) {
		tmp = ((maxCos * zi) * (ux * (1.0f - ux))) + (xi + ((((float) M_PI) * yi) * (uy * 2.0f)));
	} else {
		tmp = (yi * sinf((2.0f * (uy * ((float) M_PI))))) + (ux * (maxCos * zi));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.05000000074505806))
		tmp = Float32(Float32(Float32(maxCos * zi) * Float32(ux * Float32(Float32(1.0) - ux))) + Float32(xi + Float32(Float32(Float32(pi) * yi) * Float32(uy * Float32(2.0)))));
	else
		tmp = Float32(Float32(yi * sin(Float32(Float32(2.0) * Float32(uy * Float32(pi))))) + Float32(ux * Float32(maxCos * zi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(0.05000000074505806))
		tmp = ((maxCos * zi) * (ux * (single(1.0) - ux))) + (xi + ((single(pi) * yi) * (uy * single(2.0))));
	else
		tmp = (yi * sin((single(2.0) * (uy * single(pi))))) + (ux * (maxCos * zi));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0500000007

   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. Step-by-step derivation

   3. Simplified 99.3%

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

   5. Taylor expanded in ux around 0 99.0%

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

      1. fma-define 99.1%

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

   7. Simplified 99.1%

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

      1. pow1 99.1%

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

      2. associate-*r* 99.1%

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

      3. *-commutative 99.1%

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

      4. associate-*r* 99.1%

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

      5. *-commutative 99.1%

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

      6. associate-*r* 99.1%

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

   9. Applied egg-rr 99.1%

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

       1. unpow1 99.1%

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

       2. associate-*r* 99.1%

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

       3. *-commutative 99.1%

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

   11. Simplified 99.1%

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

   12. Taylor expanded in uy around 0 91.5%

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

       1. associate-*r* 91.5%

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

   14. Simplified 91.5%

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

   if 0.0500000007 < uy

   1. Initial program 98.0%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in ux around 0 97.9%

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

      1. fma-define 97.7%

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

   7. Simplified 97.7%

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

      1. pow1 97.7%

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

      2. associate-*r* 97.8%

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

      3. *-commutative 97.8%

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

      4. associate-*r* 97.7%

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

      5. *-commutative 97.7%

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

      6. associate-*r* 97.7%

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

   9. Applied egg-rr 97.7%

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

       1. unpow1 97.7%

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

       2. associate-*r* 97.7%

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

       3. *-commutative 97.7%

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

   11. Simplified 97.7%

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

   12. Taylor expanded in xi around 0 65.5%

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

   13. Taylor expanded in ux around 0 63.8%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.05000000074505806:\\ \;\;\;\;\left(maxCos \cdot zi\right) \cdot \left(ux \cdot \left(1 - ux\right)\right) + \left(xi + \left(\pi \cdot yi\right) \cdot \left(uy \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) + ux \cdot \left(maxCos \cdot zi\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 83.5% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.05000000074505806:\\ \;\;\;\;\left(maxCos \cdot zi\right) \cdot \left(ux \cdot \left(1 - ux\right)\right) + \left(xi + \left(\pi \cdot yi\right) \cdot \left(uy \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.05000000074505806)
   (+ (* (* maxCos zi) (* ux (- 1.0 ux))) (+ xi (* (* PI yi) (* uy 2.0))))
   (* (sin (* uy (* 2.0 PI))) yi)))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.05000000074505806f) {
		tmp = ((maxCos * zi) * (ux * (1.0f - ux))) + (xi + ((((float) M_PI) * yi) * (uy * 2.0f)));
	} else {
		tmp = sinf((uy * (2.0f * ((float) M_PI)))) * yi;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.05000000074505806))
		tmp = Float32(Float32(Float32(maxCos * zi) * Float32(ux * Float32(Float32(1.0) - ux))) + Float32(xi + Float32(Float32(Float32(pi) * yi) * Float32(uy * Float32(2.0)))));
	else
		tmp = Float32(sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * yi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(0.05000000074505806))
		tmp = ((maxCos * zi) * (ux * (single(1.0) - ux))) + (xi + ((single(pi) * yi) * (uy * single(2.0))));
	else
		tmp = sin((uy * (single(2.0) * single(pi)))) * yi;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0500000007

   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. Step-by-step derivation

   3. Simplified 99.3%

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

   5. Taylor expanded in ux around 0 99.0%

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

      1. fma-define 99.1%

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

   7. Simplified 99.1%

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

      1. pow1 99.1%

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

      2. associate-*r* 99.1%

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

      3. *-commutative 99.1%

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

      4. associate-*r* 99.1%

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

      5. *-commutative 99.1%

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

      6. associate-*r* 99.1%

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

   9. Applied egg-rr 99.1%

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

       1. unpow1 99.1%

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

       2. associate-*r* 99.1%

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

       3. *-commutative 99.1%

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

   11. Simplified 99.1%

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

   12. Taylor expanded in uy around 0 91.5%

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

       1. associate-*r* 91.5%

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

   14. Simplified 91.5%

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

   if 0.0500000007 < uy

   1. Initial program 98.0%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in ux around 0 97.9%

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

      1. fma-define 97.7%

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

   7. Simplified 97.7%

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

      1. add-cube-cbrt 97.4%

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), \color{blue}{\left(\sqrt[3]{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot \sqrt[3]{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}\right) \cdot \sqrt[3]{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}}\right) + \left(\left(1 - ux\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot zi \]

      2. pow3 97.2%

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

   9. Applied egg-rr 97.2%

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

   10. Taylor expanded in yi around inf 98.0%

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

       1. associate-+r+ 98.0%

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

       2. associate-/l* 97.9%

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

       3. *-commutative 97.9%

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

       4. associate-/l* 97.6%

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

   12. Simplified 97.6%

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

   13. Taylor expanded in yi around inf 62.2%

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

       1. *-commutative 62.2%

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

       2. associate-*r* 62.2%

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

       3. *-commutative 62.2%

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

   15. Simplified 62.2%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.05000000074505806:\\ \;\;\;\;\left(maxCos \cdot zi\right) \cdot \left(ux \cdot \left(1 - ux\right)\right) + \left(xi + \left(\pi \cdot yi\right) \cdot \left(uy \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.8% accurate, 17.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(maxCos \cdot zi\right) \cdot \left(ux \cdot \left(1 - ux\right)\right)\\ \mathbf{if}\;yi \leq -3.99999992980668 \cdot 10^{-13} \lor \neg \left(yi \leq 5.999999802552836 \cdot 10^{-11}\right):\\ \;\;\;\;t\_0 + \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\\ \mathbf{else}:\\ \;\;\;\;xi + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* maxCos zi) (* ux (- 1.0 ux)))))
   (if (or (<= yi -3.99999992980668e-13) (not (<= yi 5.999999802552836e-11)))
     (+ t_0 (* (* uy (* 2.0 PI)) yi))
     (+ xi t_0))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = (maxCos * zi) * (ux * (1.0f - ux));
	float tmp;
	if ((yi <= -3.99999992980668e-13f) || !(yi <= 5.999999802552836e-11f)) {
		tmp = t_0 + ((uy * (2.0f * ((float) M_PI))) * yi);
	} else {
		tmp = xi + t_0;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(maxCos * zi) * Float32(ux * Float32(Float32(1.0) - ux)))
	tmp = Float32(0.0)
	if ((yi <= Float32(-3.99999992980668e-13)) || !(yi <= Float32(5.999999802552836e-11)))
		tmp = Float32(t_0 + Float32(Float32(uy * Float32(Float32(2.0) * Float32(pi))) * yi));
	else
		tmp = Float32(xi + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = (maxCos * zi) * (ux * (single(1.0) - ux));
	tmp = single(0.0);
	if ((yi <= single(-3.99999992980668e-13)) || ~((yi <= single(5.999999802552836e-11))))
		tmp = t_0 + ((uy * (single(2.0) * single(pi))) * yi);
	else
		tmp = xi + t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if yi < -3.99999993e-13 or 5.9999998e-11 < yi

   1. Initial program 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in ux around 0 98.6%

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

      1. fma-define 98.6%

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

   7. Simplified 98.6%

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

      1. pow1 98.6%

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

      2. associate-*r* 98.6%

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

      3. *-commutative 98.6%

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

      4. associate-*r* 98.6%

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

      5. *-commutative 98.6%

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

      6. associate-*r* 98.6%

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

   9. Applied egg-rr 98.6%

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

       1. unpow1 98.6%

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

       2. associate-*r* 98.6%

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

       3. *-commutative 98.6%

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

   11. Simplified 98.6%

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

   12. Taylor expanded in xi around 0 72.7%

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

   13. Taylor expanded in uy around 0 59.9%

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

       1. *-commutative 59.9%

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

       2. associate-*r* 59.9%

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

       3. *-commutative 59.9%

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

   15. Simplified 59.9%

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

   if -3.99999993e-13 < yi < 5.9999998e-11

   1. Initial program 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in ux around 0 99.0%

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

      1. fma-define 99.0%

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

   7. Simplified 99.0%

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

      1. pow1 99.0%

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

      2. associate-*r* 99.0%

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

      3. *-commutative 99.0%

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

      4. associate-*r* 99.0%

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

      5. *-commutative 99.0%

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

      6. associate-*r* 99.0%

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

   9. Applied egg-rr 99.0%

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

       1. unpow1 99.0%

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

       2. associate-*r* 99.0%

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

       3. *-commutative 99.0%

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

   11. Simplified 99.0%

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

   12. Taylor expanded in uy around 0 68.0%

       \[\leadsto \color{blue}{xi} + \left(maxCos \cdot zi\right) \cdot \left(ux \cdot \left(1 - ux\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;yi \leq -3.99999992980668 \cdot 10^{-13} \lor \neg \left(yi \leq 5.999999802552836 \cdot 10^{-11}\right):\\ \;\;\;\;\left(maxCos \cdot zi\right) \cdot \left(ux \cdot \left(1 - ux\right)\right) + \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\\ \mathbf{else}:\\ \;\;\;\;xi + \left(maxCos \cdot zi\right) \cdot \left(ux \cdot \left(1 - ux\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 81.3% accurate, 24.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = ((maxCos * zi) * (ux * (single(1.0) - ux))) + (xi + ((single(pi) * yi) * (uy * single(2.0))));
end

Derivation

1. Initial program 99.1%

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

3. Simplified 99.0%

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

5. Taylor expanded in ux around 0 98.9%

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

   1. fma-define 98.9%

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

7. Simplified 98.9%

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

   1. pow1 98.9%

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

   2. associate-*r* 98.9%

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

   3. *-commutative 98.9%

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

   4. associate-*r* 98.9%

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

   5. *-commutative 98.9%

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

   6. associate-*r* 98.9%

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

9. Applied egg-rr 98.9%

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

    1. unpow1 98.9%

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

    2. associate-*r* 98.9%

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

    3. *-commutative 98.9%

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

11. Simplified 98.9%

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

12. Taylor expanded in uy around 0 82.5%

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

    1. associate-*r* 82.5%

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

14. Simplified 82.5%

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

15. Final simplification 82.5%

    \[\leadsto \left(maxCos \cdot zi\right) \cdot \left(ux \cdot \left(1 - ux\right)\right) + \left(xi + \left(\pi \cdot yi\right) \cdot \left(uy \cdot 2\right)\right) \]
16. Add Preprocessing

Alternative 16: 50.9% accurate, 41.9× speedup

Math

\[\begin{array}{l} \\ xi + \left(maxCos \cdot zi\right) \cdot \left(ux \cdot \left(1 - ux\right)\right) \end{array} \]

FPCore

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

C

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

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 = xi + ((maxcos * zi) * (ux * (1.0e0 - ux)))
end function

Julia

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = xi + ((maxCos * zi) * (ux * (single(1.0) - ux)));
end

Derivation

1. Initial program 99.1%

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

3. Simplified 99.0%

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

5. Taylor expanded in ux around 0 98.9%

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

   1. fma-define 98.9%

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

7. Simplified 98.9%

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

   1. pow1 98.9%

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

   2. associate-*r* 98.9%

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

   3. *-commutative 98.9%

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

   4. associate-*r* 98.9%

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

   5. *-commutative 98.9%

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

   6. associate-*r* 98.9%

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

9. Applied egg-rr 98.9%

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

    1. unpow1 98.9%

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

    2. associate-*r* 98.9%

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

    3. *-commutative 98.9%

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

11. Simplified 98.9%

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

12. Taylor expanded in uy around 0 54.2%

    \[\leadsto \color{blue}{xi} + \left(maxCos \cdot zi\right) \cdot \left(ux \cdot \left(1 - ux\right)\right) \]
13. Add Preprocessing

Alternative 17: 50.9% accurate, 41.9× speedup

Math

\[\begin{array}{l} \\ xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) \end{array} \]

FPCore

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

C

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

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 = xi + (maxcos * (ux * (zi * (1.0e0 - ux))))
end function

Julia

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = xi + (maxCos * (ux * (zi * (single(1.0) - ux))));
end

Derivation

1. Initial program 99.1%

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

3. Simplified 99.0%

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

5. Taylor expanded in ux around 0 98.9%

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

   1. fma-define 98.9%

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

7. Simplified 98.9%

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

   1. add-cube-cbrt 98.5%

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), \color{blue}{\left(\sqrt[3]{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot \sqrt[3]{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}\right) \cdot \sqrt[3]{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}}\right) + \left(\left(1 - ux\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot zi \]

   2. pow3 98.4%

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

9. Applied egg-rr 98.4%

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

10. Taylor expanded in uy around 0 54.2%

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

    1. *-commutative 54.2%

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

12. Simplified 54.2%

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

13. Final simplification 54.2%

    \[\leadsto xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) \]
14. Add Preprocessing

Alternative 18: 13.2% accurate, 51.2× speedup

Math

\[\begin{array}{l} \\ zi \cdot \left(ux \cdot \left(maxCos \cdot \left(1 - ux\right)\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (* zi (* ux (* maxCos (- 1.0 ux)))))

C

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

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 = zi * (ux * (maxcos * (1.0e0 - ux)))
end function

Julia

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = zi * (ux * (maxCos * (single(1.0) - ux)));
end

Derivation

1. Initial program 99.1%

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

3. Simplified 99.0%

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

5. Taylor expanded in ux around 0 98.9%

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

   1. fma-define 98.9%

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

7. Simplified 98.9%

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

   1. add-cube-cbrt 98.5%

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), \color{blue}{\left(\sqrt[3]{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot \sqrt[3]{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}\right) \cdot \sqrt[3]{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}}\right) + \left(\left(1 - ux\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot zi \]

   2. pow3 98.4%

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

9. Applied egg-rr 98.4%

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

10. Taylor expanded in yi around inf 98.2%

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

    1. associate-+r+ 98.3%

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

    2. associate-/l* 97.8%

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

    3. *-commutative 97.8%

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

    4. associate-/l* 97.6%

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

12. Simplified 97.6%

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

13. Taylor expanded in maxCos around inf 14.5%

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

    1. *-commutative 14.5%

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

    2. associate-*r* 14.5%

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

    3. *-commutative 14.5%

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

    4. associate-*l* 14.5%

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

    5. associate-*r* 14.5%

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

    6. associate-*l* 14.5%

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

15. Simplified 14.5%

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

16. Final simplification 14.5%

    \[\leadsto zi \cdot \left(ux \cdot \left(maxCos \cdot \left(1 - ux\right)\right)\right) \]
17. Add Preprocessing

Alternative 19: 13.2% accurate, 51.2× speedup

Math

\[\begin{array}{l} \\ maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (* maxCos (* ux (* zi (- 1.0 ux)))))

C

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

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 * (1.0e0 - ux)))
end function

Julia

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = maxCos * (ux * (zi * (single(1.0) - ux)));
end

Derivation

1. Initial program 99.1%

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

   1. associate-+l+ 99.1%

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

   2. associate-*l* 99.1%

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

   3. fma-define 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in zi around inf 14.5%

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

Alternative 20: 11.7% accurate, 92.2× speedup

Math

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

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(ux * Float32(maxCos * zi))
end

MATLAB

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

Derivation

1. Initial program 99.1%

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

   1. associate-+l+ 99.1%

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

   2. associate-*l* 99.1%

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

   3. fma-define 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in zi around inf 14.5%

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

6. Taylor expanded in ux around 0 12.7%

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

   1. *-commutative 12.7%

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

   2. associate-*r* 12.7%

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

   3. add-exp-log 8.4%

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

   4. add-exp-log 8.4%

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

   5. prod-exp 8.4%

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

8. Applied egg-rr 8.4%

   \[\leadsto \color{blue}{e^{\log \left(maxCos \cdot zi\right) + \log ux}} \]
9. Step-by-step derivation

   1. +-commutative 8.4%

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

   2. exp-sum 8.4%

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

   3. rem-exp-log 8.4%

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

   4. rem-exp-log 12.7%

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

10. Simplified 12.7%

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

Alternative 21: 11.7% accurate, 92.2× 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 99.1%

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

   1. associate-+l+ 99.1%

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

   2. associate-*l* 99.1%

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

   3. fma-define 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in zi around inf 14.5%

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

6. Taylor expanded in ux around 0 12.7%

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

Reproduce

herbie shell --seed 2024141 
(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)))