UniformSampleCone, y

Percentage Accurate: 57.6% → 98.3%

Time: 13.8s

Alternatives: 11

Speedup: 2.0×

Specification

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

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* t_0 t_0))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	return sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((1.0f - (t_0 * t_0)));
}

Julia

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

MATLAB

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

Initial Program: 57.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* t_0 t_0))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	return sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((1.0f - (t_0 * t_0)));
}

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{{\left({ux}^{2} \cdot \left(\frac{\mathsf{fma}\left(maxCos, -2, 2\right)}{ux} - {\left(1 - maxCos\right)}^{2}\right)\right)}^{1.5} \cdot {\sin \left(uy \cdot \left(2 \cdot \pi\right)\right)}^{3}} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (cbrt
  (*
   (pow
    (* (pow ux 2.0) (- (/ (fma maxCos -2.0 2.0) ux) (pow (- 1.0 maxCos) 2.0)))
    1.5)
   (pow (sin (* uy (* 2.0 PI))) 3.0))))

C

float code(float ux, float uy, float maxCos) {
	return cbrtf((powf((powf(ux, 2.0f) * ((fmaf(maxCos, -2.0f, 2.0f) / ux) - powf((1.0f - maxCos), 2.0f))), 1.5f) * powf(sinf((uy * (2.0f * ((float) M_PI)))), 3.0f)));
}

Julia

function code(ux, uy, maxCos)
	return cbrt(Float32((Float32((ux ^ Float32(2.0)) * Float32(Float32(fma(maxCos, Float32(-2.0), Float32(2.0)) / ux) - (Float32(Float32(1.0) - maxCos) ^ Float32(2.0)))) ^ Float32(1.5)) * (sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) ^ Float32(3.0))))
end

Derivation

1. Initial program 56.4%

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

3. Taylor expanded in ux around -inf 98.3%

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

   1. +-commutative 98.3%

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

   2. metadata-eval 98.3%

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

   3. cancel-sign-sub-inv 98.3%

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

   4. associate-*r/ 98.3%

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

   5. metadata-eval 98.3%

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

   6. associate-*r/ 98.3%

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

   7. div-sub 98.4%

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

   8. cancel-sign-sub-inv 98.4%

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

   9. metadata-eval 98.4%

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

   10. +-commutative 98.4%

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

   11. *-commutative 98.4%

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

   12. fma-define 98.4%

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

   13. mul-1-neg 98.4%

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

   14. sub-neg 98.4%

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

5. Simplified 98.4%

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

   1. *-commutative 98.4%

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

   2. add-cbrt-cube 98.4%

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

   3. add-cbrt-cube 98.4%

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

   4. cbrt-unprod 98.3%

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

7. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\sqrt[3]{{\left({ux}^{2} \cdot \left(\frac{\mathsf{fma}\left(maxCos, -2, 2\right)}{ux} - {\left(1 - maxCos\right)}^{2}\right)\right)}^{1.5} \cdot {\sin \left(uy \cdot \left(2 \cdot \pi\right)\right)}^{3}}} \]
8. Add Preprocessing

Alternative 2: 98.2% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sin (* PI (* 2.0 uy)))
  (sqrt
   (*
    (pow ux 2.0)
    (- (/ (fma maxCos -2.0 2.0) ux) (pow (- 1.0 maxCos) 2.0))))))

C

float code(float ux, float uy, float maxCos) {
	return sinf((((float) M_PI) * (2.0f * uy))) * sqrtf((powf(ux, 2.0f) * ((fmaf(maxCos, -2.0f, 2.0f) / ux) - powf((1.0f - maxCos), 2.0f))));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy))) * sqrt(Float32((ux ^ Float32(2.0)) * Float32(Float32(fma(maxCos, Float32(-2.0), Float32(2.0)) / ux) - (Float32(Float32(1.0) - maxCos) ^ Float32(2.0))))))
end

Derivation

1. Initial program 56.4%

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

3. Taylor expanded in ux around -inf 98.3%

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

   1. +-commutative 98.3%

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

   2. metadata-eval 98.3%

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

   3. cancel-sign-sub-inv 98.3%

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

   4. associate-*r/ 98.3%

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

   5. metadata-eval 98.3%

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

   6. associate-*r/ 98.3%

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

   7. div-sub 98.4%

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

   8. cancel-sign-sub-inv 98.4%

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

   9. metadata-eval 98.4%

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

   10. +-commutative 98.4%

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

   11. *-commutative 98.4%

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

   12. fma-define 98.4%

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

   13. mul-1-neg 98.4%

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

   14. sub-neg 98.4%

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

5. Simplified 98.4%

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

6. Final simplification 98.4%

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

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sin (* PI (* 2.0 uy)))
  (sqrt
   (*
    ux
    (+
     (+ 2.0 (- (* maxCos (- (* ux 2.0) (* ux maxCos))) ux))
     (* maxCos -2.0))))))

C

float code(float ux, float uy, float maxCos) {
	return sinf((((float) M_PI) * (2.0f * uy))) * sqrtf((ux * ((2.0f + ((maxCos * ((ux * 2.0f) - (ux * maxCos))) - ux)) + (maxCos * -2.0f))));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy))) * sqrt(Float32(ux * Float32(Float32(Float32(2.0) + Float32(Float32(maxCos * Float32(Float32(ux * Float32(2.0)) - Float32(ux * maxCos))) - ux)) + Float32(maxCos * Float32(-2.0))))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sin((single(pi) * (single(2.0) * uy))) * sqrt((ux * ((single(2.0) + ((maxCos * ((ux * single(2.0)) - (ux * maxCos))) - ux)) + (maxCos * single(-2.0)))));
end

Derivation

1. Initial program 56.4%

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

3. Taylor expanded in ux around 0 98.4%

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

   1. cancel-sign-sub-inv 98.4%

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

   2. metadata-eval 98.4%

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

   3. associate-*r* 98.4%

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

   4. mul-1-neg 98.4%

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

   5. sub-neg 98.4%

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

   6. metadata-eval 98.4%

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

   7. +-commutative 98.4%

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

   8. *-commutative 98.4%

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

5. Simplified 98.4%

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

6. Taylor expanded in maxCos around 0 98.4%

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

7. Final simplification 98.4%

   \[\leadsto \sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(\left(2 + \left(maxCos \cdot \left(ux \cdot 2 - ux \cdot maxCos\right) - ux\right)\right) + maxCos \cdot -2\right)} \]
8. Add Preprocessing

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sin (* PI (* 2.0 uy)))
  (sqrt
   (*
    ux
    (+ (* maxCos -2.0) (+ (- 2.0 ux) (* maxCos (* ux (- 2.0 maxCos)))))))))

C

float code(float ux, float uy, float maxCos) {
	return sinf((((float) M_PI) * (2.0f * uy))) * sqrtf((ux * ((maxCos * -2.0f) + ((2.0f - ux) + (maxCos * (ux * (2.0f - maxCos)))))));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy))) * sqrt(Float32(ux * Float32(Float32(maxCos * Float32(-2.0)) + Float32(Float32(Float32(2.0) - ux) + Float32(maxCos * Float32(ux * Float32(Float32(2.0) - maxCos))))))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sin((single(pi) * (single(2.0) * uy))) * sqrt((ux * ((maxCos * single(-2.0)) + ((single(2.0) - ux) + (maxCos * (ux * (single(2.0) - maxCos)))))));
end

Derivation

1. Initial program 56.4%

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

3. Taylor expanded in ux around 0 98.4%

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

   1. cancel-sign-sub-inv 98.4%

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

   2. metadata-eval 98.4%

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

   3. associate-*r* 98.4%

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

   4. mul-1-neg 98.4%

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

   5. sub-neg 98.4%

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

   6. metadata-eval 98.4%

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

   7. +-commutative 98.4%

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

   8. *-commutative 98.4%

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

5. Simplified 98.4%

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

6. Taylor expanded in maxCos around 0 98.4%

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

   1. associate-+r+ 98.4%

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

   2. mul-1-neg 98.4%

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

   3. unsub-neg 98.4%

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

   4. +-commutative 98.4%

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

   5. associate-*r* 98.4%

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

   6. distribute-rgt-out 98.4%

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

   7. mul-1-neg 98.4%

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

8. Simplified 98.4%

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

9. Final simplification 98.4%

   \[\leadsto \sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(maxCos \cdot -2 + \left(\left(2 - ux\right) + maxCos \cdot \left(ux \cdot \left(2 - maxCos\right)\right)\right)\right)} \]
10. Add Preprocessing

Alternative 5: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.4%

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

3. Taylor expanded in ux around 0 98.4%

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

   1. cancel-sign-sub-inv 98.4%

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

   2. metadata-eval 98.4%

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

   3. associate-*r* 98.4%

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

   4. mul-1-neg 98.4%

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

   5. sub-neg 98.4%

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

   6. metadata-eval 98.4%

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

   7. +-commutative 98.4%

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

   8. *-commutative 98.4%

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

5. Simplified 98.4%

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

6. Taylor expanded in maxCos around 0 98.0%

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

   1. associate-*r* 98.0%

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

   2. distribute-rgt-out 98.0%

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

8. Simplified 98.0%

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

9. Final simplification 98.0%

   \[\leadsto \sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(maxCos \cdot -2 + \left(2 + ux \cdot \left(-1 + 2 \cdot maxCos\right)\right)\right)} \]
10. Add Preprocessing

Alternative 6: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0026000000070780516:\\ \;\;\;\;ux \cdot \left(\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{-1 + \frac{2}{ux}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.0026000000070780516)
   (* ux (* (* 2.0 (* uy PI)) (sqrt (+ -1.0 (/ 2.0 ux)))))
   (* (sin (* uy (* 2.0 PI))) (sqrt (* ux 2.0)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((2.0f * uy) <= 0.0026000000070780516f) {
		tmp = ux * ((2.0f * (uy * ((float) M_PI))) * sqrtf((-1.0f + (2.0f / ux))));
	} else {
		tmp = sinf((uy * (2.0f * ((float) M_PI)))) * sqrtf((ux * 2.0f));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.0026000000070780516))
		tmp = Float32(ux * Float32(Float32(Float32(2.0) * Float32(uy * Float32(pi))) * sqrt(Float32(Float32(-1.0) + Float32(Float32(2.0) / ux)))));
	else
		tmp = Float32(sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(ux * Float32(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if ((single(2.0) * uy) <= single(0.0026000000070780516))
		tmp = ux * ((single(2.0) * (uy * single(pi))) * sqrt((single(-1.0) + (single(2.0) / ux))));
	else
		tmp = sin((uy * (single(2.0) * single(pi)))) * sqrt((ux * single(2.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.00260000001

   1. Initial program 57.1%

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

   3. Taylor expanded in ux around -inf 98.5%

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

      1. +-commutative 98.5%

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

      2. metadata-eval 98.5%

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

      3. cancel-sign-sub-inv 98.5%

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

      4. associate-*r/ 98.5%

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

      5. metadata-eval 98.5%

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

      6. associate-*r/ 98.5%

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

      7. div-sub 98.6%

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

      8. cancel-sign-sub-inv 98.6%

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

      9. metadata-eval 98.6%

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

      10. +-commutative 98.6%

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

      11. *-commutative 98.6%

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

      12. fma-define 98.6%

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

      13. mul-1-neg 98.6%

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

      14. sub-neg 98.6%

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

   5. Simplified 98.6%

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

   6. Taylor expanded in maxCos around 0 93.0%

      \[\leadsto \color{blue}{\left(ux \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \cdot \sqrt{2 \cdot \frac{1}{ux} - 1}} \]
   7. Step-by-step derivation

      1. associate-*l* 93.3%

         \[\leadsto \color{blue}{ux \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{2 \cdot \frac{1}{ux} - 1}\right)} \]

      2. sub-neg 93.3%

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

      3. associate-*r/ 93.3%

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

      4. metadata-eval 93.3%

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

      5. metadata-eval 93.3%

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

   8. Simplified 93.3%

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

   9. Taylor expanded in uy around 0 91.8%

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

   if 0.00260000001 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 54.6%

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

      1. associate-*l* 54.6%

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

      2. sub-neg 54.6%

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

      3. +-commutative 54.6%

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

      4. distribute-rgt-neg-in 54.6%

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

      5. fma-define 54.7%

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

   3. Simplified 54.7%

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

   5. Taylor expanded in maxCos around 0 53.0%

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

   6. Taylor expanded in ux around 0 79.3%

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

      1. *-commutative 79.3%

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

   8. Simplified 79.3%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0026000000070780516:\\ \;\;\;\;ux \cdot \left(\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{-1 + \frac{2}{ux}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (* (sin (* PI (* 2.0 uy))) (sqrt (* ux (+ (* maxCos -2.0) (- 2.0 ux))))))

C

float code(float ux, float uy, float maxCos) {
	return sinf((((float) M_PI) * (2.0f * uy))) * sqrtf((ux * ((maxCos * -2.0f) + (2.0f - ux))));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy))) * sqrt(Float32(ux * Float32(Float32(maxCos * Float32(-2.0)) + Float32(Float32(2.0) - ux)))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sin((single(pi) * (single(2.0) * uy))) * sqrt((ux * ((maxCos * single(-2.0)) + (single(2.0) - ux))));
end

Derivation

1. Initial program 56.4%

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

3. Taylor expanded in ux around 0 98.4%

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

   1. cancel-sign-sub-inv 98.4%

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

   2. metadata-eval 98.4%

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

   3. associate-*r* 98.4%

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

   4. mul-1-neg 98.4%

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

   5. sub-neg 98.4%

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

   6. metadata-eval 98.4%

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

   7. +-commutative 98.4%

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

   8. *-commutative 98.4%

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

5. Simplified 98.4%

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

6. Taylor expanded in maxCos around 0 97.4%

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

   1. mul-1-neg 97.4%

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

   2. unsub-neg 97.4%

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

8. Simplified 97.4%

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

9. Final simplification 97.4%

   \[\leadsto \sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(maxCos \cdot -2 + \left(2 - ux\right)\right)} \]
10. Add Preprocessing

Alternative 8: 92.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (* (sin (* uy (* 2.0 PI))) (sqrt (* ux (- 2.0 ux)))))

C

float code(float ux, float uy, float maxCos) {
	return sinf((uy * (2.0f * ((float) M_PI)))) * sqrtf((ux * (2.0f - ux)));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sin((uy * (single(2.0) * single(pi)))) * sqrt((ux * (single(2.0) - ux)));
end

Derivation

1. Initial program 56.4%

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

   1. associate-*l* 56.4%

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

   2. sub-neg 56.4%

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

   3. +-commutative 56.4%

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

   4. distribute-rgt-neg-in 56.4%

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

   5. fma-define 56.4%

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

3. Simplified 56.5%

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

5. Taylor expanded in maxCos around 0 55.0%

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

   1. +-commutative 55.0%

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

   2. associate-*r* 55.0%

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

   3. fma-define 55.0%

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

   4. mul-1-neg 55.0%

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

   5. sub-neg 55.0%

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

7. Applied egg-rr 55.0%

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

8. Taylor expanded in ux around 0 94.0%

   \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 + -1 \cdot ux\right)}} \]
9. Step-by-step derivation

   1. mul-1-neg 94.0%

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

   2. unsub-neg 94.0%

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

10. Simplified 94.0%

    \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 - ux\right)}} \]
11. Add Preprocessing

Alternative 9: 76.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ ux \cdot \left(\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{-1 + \frac{2}{ux}}\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (* ux (* (* 2.0 (* uy PI)) (sqrt (+ -1.0 (/ 2.0 ux))))))

C

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

Julia

function code(ux, uy, maxCos)
	return Float32(ux * Float32(Float32(Float32(2.0) * Float32(uy * Float32(pi))) * sqrt(Float32(Float32(-1.0) + Float32(Float32(2.0) / ux)))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = ux * ((single(2.0) * (uy * single(pi))) * sqrt((single(-1.0) + (single(2.0) / ux))));
end

Derivation

1. Initial program 56.4%

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

3. Taylor expanded in ux around -inf 98.3%

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

   1. +-commutative 98.3%

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

   2. metadata-eval 98.3%

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

   3. cancel-sign-sub-inv 98.3%

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

   4. associate-*r/ 98.3%

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

   5. metadata-eval 98.3%

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

   6. associate-*r/ 98.3%

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

   7. div-sub 98.4%

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

   8. cancel-sign-sub-inv 98.4%

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

   9. metadata-eval 98.4%

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

   10. +-commutative 98.4%

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

   11. *-commutative 98.4%

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

   12. fma-define 98.4%

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

   13. mul-1-neg 98.4%

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

   14. sub-neg 98.4%

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

5. Simplified 98.4%

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

6. Taylor expanded in maxCos around 0 93.6%

   \[\leadsto \color{blue}{\left(ux \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \cdot \sqrt{2 \cdot \frac{1}{ux} - 1}} \]
7. Step-by-step derivation

   1. associate-*l* 93.8%

      \[\leadsto \color{blue}{ux \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{2 \cdot \frac{1}{ux} - 1}\right)} \]

   2. sub-neg 93.8%

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

   3. associate-*r/ 93.8%

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

   4. metadata-eval 93.8%

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

   5. metadata-eval 93.8%

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

8. Simplified 93.8%

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

9. Taylor expanded in uy around 0 77.5%

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

10. Final simplification 77.5%

    \[\leadsto ux \cdot \left(\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{-1 + \frac{2}{ux}}\right) \]
11. Add Preprocessing

Alternative 10: 65.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right)\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (* 2.0 (* uy (* PI (sqrt (* ux (- 2.0 (* 2.0 maxCos))))))))

C

float code(float ux, float uy, float maxCos) {
	return 2.0f * (uy * (((float) M_PI) * sqrtf((ux * (2.0f - (2.0f * maxCos))))));
}

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = single(2.0) * (uy * (single(pi) * sqrt((ux * (single(2.0) - (single(2.0) * maxCos))))));
end

Derivation

1. Initial program 56.4%

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

   1. associate-*l* 56.4%

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

   2. sub-neg 56.4%

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

   3. +-commutative 56.4%

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

   4. distribute-rgt-neg-in 56.4%

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

   5. fma-define 56.4%

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

3. Simplified 56.5%

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

5. Taylor expanded in uy around 0 50.0%

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

7. Simplified 50.0%

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

8. Taylor expanded in ux around 0 66.5%

   \[\leadsto 2 \cdot \left(uy \cdot \color{blue}{\left(\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot \pi\right)}\right) \]

9. Final simplification 66.5%

   \[\leadsto 2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right)\right) \]
10. Add Preprocessing

Alternative 11: 7.1% accurate, 223.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 0.0)

C

float code(float ux, float uy, float maxCos) {
	return 0.0f;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = 0.0e0
end function

Julia

function code(ux, uy, maxCos)
	return Float32(0.0)
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = single(0.0);
end

Derivation

1. Initial program 56.4%

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

   1. associate-*l* 56.4%

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

   2. sub-neg 56.4%

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

   3. +-commutative 56.4%

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

   4. distribute-rgt-neg-in 56.4%

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

   5. fma-define 56.4%

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

3. Simplified 56.5%

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

5. Taylor expanded in uy around 0 50.0%

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

7. Simplified 50.0%

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

8. Taylor expanded in ux around 0 7.1%

   \[\leadsto 2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{1 + \left(-\color{blue}{1}\right)}\right)\right) \]

9. Taylor expanded in uy around 0 7.1%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024137 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, y"
  :precision binary32
  :pre (and (and (and (<= 2.328306437e-10 ux) (<= ux 1.0)) (and (<= 2.328306437e-10 uy) (<= uy 1.0))) (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))