UniformSampleCone, y

Percentage Accurate: 57.7% → 98.2%

Time: 14.5s

Alternatives: 14

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: 57.7% 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: 98.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.9%

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

   1. associate-*l* 61.9%

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

   2. sub-neg 61.9%

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

   3. +-commutative 61.9%

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

   4. distribute-rgt-neg-in 61.9%

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

   5. fma-define 61.9%

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

3. Simplified 62.1%

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

5. Taylor expanded in ux around inf 98.2%

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

6. Taylor expanded in ux around 0 98.2%

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

   1. associate--l+ 98.2%

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

   2. +-commutative 98.2%

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

   3. *-commutative 98.2%

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

   4. sub-neg 98.2%

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

   5. metadata-eval 98.2%

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

   6. sub-neg 98.2%

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

   7. metadata-eval 98.2%

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

   8. +-commutative 98.2%

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

   9. distribute-lft-in 98.2%

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

   10. metadata-eval 98.2%

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

   11. mul-1-neg 98.2%

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

   12. sub-neg 98.2%

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

8. Simplified 98.2%

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

9. Final simplification 98.2%

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

Alternative 2: 98.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.9%

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

3. Taylor expanded in ux around 0 98.2%

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

   1. associate--l+ 98.2%

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

   2. associate-*r* 98.2%

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

   3. mul-1-neg 98.2%

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

   4. sub-neg 98.2%

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

   5. metadata-eval 98.2%

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

   6. +-commutative 98.2%

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

5. Simplified 98.2%

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

6. Final simplification 98.2%

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

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.9%

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

   1. associate-*l* 61.9%

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

   2. sub-neg 61.9%

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

   3. +-commutative 61.9%

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

   4. distribute-rgt-neg-in 61.9%

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

   5. fma-define 61.9%

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

3. Simplified 62.1%

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

5. Taylor expanded in ux around inf 98.2%

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

6. Taylor expanded in ux around 0 98.2%

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

   1. associate--l+ 98.2%

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

   2. +-commutative 98.2%

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

   3. *-commutative 98.2%

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

   4. sub-neg 98.2%

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

   5. metadata-eval 98.2%

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

   6. sub-neg 98.2%

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

   7. metadata-eval 98.2%

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

   8. +-commutative 98.2%

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

   9. distribute-lft-in 98.2%

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

   10. metadata-eval 98.2%

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

   11. mul-1-neg 98.2%

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

   12. sub-neg 98.2%

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

8. Simplified 98.2%

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

9. Taylor expanded in uy around inf 98.2%

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

10. Final simplification 98.2%

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

Alternative 4: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.9%

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

   1. associate-*l* 61.9%

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

   2. sub-neg 61.9%

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

   3. +-commutative 61.9%

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

   4. distribute-rgt-neg-in 61.9%

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

   5. fma-define 61.9%

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

3. Simplified 62.1%

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

5. Taylor expanded in ux around inf 98.2%

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

6. Taylor expanded in ux around 0 98.2%

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

   1. associate--l+ 98.2%

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

   2. +-commutative 98.2%

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

   3. *-commutative 98.2%

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

   4. sub-neg 98.2%

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

   5. metadata-eval 98.2%

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

   6. sub-neg 98.2%

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

   7. metadata-eval 98.2%

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

   8. +-commutative 98.2%

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

   9. distribute-lft-in 98.2%

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

   10. metadata-eval 98.2%

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

   11. mul-1-neg 98.2%

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

   12. sub-neg 98.2%

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

8. Simplified 98.2%

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

9. Taylor expanded in uy around inf 98.2%

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

10. Taylor expanded in maxCos around 0 97.4%

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

    1. +-commutative 97.4%

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

    2. mul-1-neg 97.4%

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

    3. unsub-neg 97.4%

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

    4. *-commutative 97.4%

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

12. Simplified 97.4%

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

13. Final simplification 97.4%

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

Alternative 5: 95.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0003000000142492354:\\ \;\;\;\;2 \cdot \left(\sqrt{ux \cdot \left(\left(2 + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)\right) - 2 \cdot maxCos\right)} \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;ux \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{-1 + \frac{2}{ux}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.0003000000142492354)
   (*
    2.0
    (*
     (sqrt
      (*
       ux
       (- (+ 2.0 (* ux (* (- 1.0 maxCos) (+ maxCos -1.0)))) (* 2.0 maxCos))))
     (* uy PI)))
   (* ux (* (sin (* 2.0 (* uy PI))) (sqrt (+ -1.0 (/ 2.0 ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.0003000000142492354f) {
		tmp = 2.0f * (sqrtf((ux * ((2.0f + (ux * ((1.0f - maxCos) * (maxCos + -1.0f)))) - (2.0f * maxCos)))) * (uy * ((float) M_PI)));
	} else {
		tmp = ux * (sinf((2.0f * (uy * ((float) M_PI)))) * sqrtf((-1.0f + (2.0f / ux))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.0003000000142492354))
		tmp = Float32(Float32(2.0) * Float32(sqrt(Float32(ux * Float32(Float32(Float32(2.0) + Float32(ux * Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos + Float32(-1.0))))) - Float32(Float32(2.0) * maxCos)))) * Float32(uy * Float32(pi))));
	else
		tmp = Float32(ux * Float32(sin(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) * sqrt(Float32(Float32(-1.0) + Float32(Float32(2.0) / ux)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if ((uy * single(2.0)) <= single(0.0003000000142492354))
		tmp = single(2.0) * (sqrt((ux * ((single(2.0) + (ux * ((single(1.0) - maxCos) * (maxCos + single(-1.0))))) - (single(2.0) * maxCos)))) * (uy * single(pi)));
	else
		tmp = ux * (sin((single(2.0) * (uy * single(pi)))) * sqrt((single(-1.0) + (single(2.0) / ux))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 3.00000014e-4

   1. Initial program 61.1%

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

      1. associate-*l* 61.1%

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

      2. sub-neg 61.1%

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

      3. +-commutative 61.1%

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

      4. distribute-rgt-neg-in 61.1%

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

      5. fma-define 61.2%

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

   3. Simplified 61.2%

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

   5. Taylor expanded in ux around inf 98.5%

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

   6. Taylor expanded in ux around 0 98.4%

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

      1. associate--l+ 98.5%

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

      2. +-commutative 98.5%

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

      3. *-commutative 98.5%

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

      4. sub-neg 98.5%

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

      5. metadata-eval 98.5%

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

      6. sub-neg 98.5%

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

      7. metadata-eval 98.5%

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

      8. +-commutative 98.5%

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

      9. distribute-lft-in 98.5%

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

      10. metadata-eval 98.5%

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

      11. mul-1-neg 98.5%

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

      12. sub-neg 98.5%

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

   8. Simplified 98.5%

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

   9. Taylor expanded in uy around 0 98.4%

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

   if 3.00000014e-4 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 63.0%

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

      1. associate-*l* 63.0%

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

      2. sub-neg 63.0%

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

      3. +-commutative 63.0%

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

      4. distribute-rgt-neg-in 63.0%

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

      5. fma-define 63.1%

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

   3. Simplified 63.5%

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

   5. Taylor expanded in ux around inf 97.8%

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

   6. Taylor expanded in maxCos around 0 92.3%

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

      1. sub-neg 92.3%

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

      2. associate-*r/ 92.3%

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

      3. metadata-eval 92.3%

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

      4. metadata-eval 92.3%

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

   8. Simplified 92.3%

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

   9. Taylor expanded in uy around inf 92.3%

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

       1. associate-*l* 92.5%

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

       2. sub-neg 92.5%

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

       3. metadata-eval 92.5%

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

       4. +-commutative 92.5%

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

       5. associate-*r/ 92.5%

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

       6. metadata-eval 92.5%

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

   11. Simplified 92.5%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0003000000142492354:\\ \;\;\;\;2 \cdot \left(\sqrt{ux \cdot \left(\left(2 + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)\right) - 2 \cdot maxCos\right)} \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;ux \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{-1 + \frac{2}{ux}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.9%

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

   1. associate-*l* 61.9%

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

   2. sub-neg 61.9%

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

   3. +-commutative 61.9%

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

   4. distribute-rgt-neg-in 61.9%

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

   5. fma-define 61.9%

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

3. Simplified 62.1%

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

5. Taylor expanded in ux around inf 98.2%

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

6. Taylor expanded in ux around 0 98.2%

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

   1. associate--l+ 98.2%

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

   2. +-commutative 98.2%

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

   3. *-commutative 98.2%

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

   4. sub-neg 98.2%

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

   5. metadata-eval 98.2%

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

   6. sub-neg 98.2%

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

   7. metadata-eval 98.2%

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

   8. +-commutative 98.2%

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

   9. distribute-lft-in 98.2%

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

   10. metadata-eval 98.2%

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

   11. mul-1-neg 98.2%

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

   12. sub-neg 98.2%

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

8. Simplified 98.2%

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

9. Taylor expanded in uy around inf 98.2%

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

10. Taylor expanded in maxCos around 0 96.4%

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

11. Final simplification 96.4%

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

Alternative 7: 95.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0003000000142492354:\\ \;\;\;\;2 \cdot \left(\sqrt{ux \cdot \left(\left(2 + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)\right) - 2 \cdot maxCos\right)} \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.0003000000142492354)
   (*
    2.0
    (*
     (sqrt
      (*
       ux
       (- (+ 2.0 (* ux (* (- 1.0 maxCos) (+ maxCos -1.0)))) (* 2.0 maxCos))))
     (* uy PI)))
   (* (sin (* uy (* 2.0 PI))) (sqrt (* ux (- 2.0 ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.0003000000142492354f) {
		tmp = 2.0f * (sqrtf((ux * ((2.0f + (ux * ((1.0f - maxCos) * (maxCos + -1.0f)))) - (2.0f * maxCos)))) * (uy * ((float) M_PI)));
	} else {
		tmp = sinf((uy * (2.0f * ((float) M_PI)))) * sqrtf((ux * (2.0f - ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.0003000000142492354))
		tmp = Float32(Float32(2.0) * Float32(sqrt(Float32(ux * Float32(Float32(Float32(2.0) + Float32(ux * Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos + Float32(-1.0))))) - Float32(Float32(2.0) * maxCos)))) * Float32(uy * Float32(pi))));
	else
		tmp = Float32(sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if ((uy * single(2.0)) <= single(0.0003000000142492354))
		tmp = single(2.0) * (sqrt((ux * ((single(2.0) + (ux * ((single(1.0) - maxCos) * (maxCos + single(-1.0))))) - (single(2.0) * maxCos)))) * (uy * single(pi)));
	else
		tmp = sin((uy * (single(2.0) * single(pi)))) * sqrt((ux * (single(2.0) - ux)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 3.00000014e-4

   1. Initial program 61.1%

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

      1. associate-*l* 61.1%

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

      2. sub-neg 61.1%

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

      3. +-commutative 61.1%

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

      4. distribute-rgt-neg-in 61.1%

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

      5. fma-define 61.2%

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

   3. Simplified 61.2%

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

   5. Taylor expanded in ux around inf 98.5%

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

   6. Taylor expanded in ux around 0 98.4%

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

      1. associate--l+ 98.5%

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

      2. +-commutative 98.5%

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

      3. *-commutative 98.5%

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

      4. sub-neg 98.5%

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

      5. metadata-eval 98.5%

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

      6. sub-neg 98.5%

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

      7. metadata-eval 98.5%

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

      8. +-commutative 98.5%

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

      9. distribute-lft-in 98.5%

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

      10. metadata-eval 98.5%

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

      11. mul-1-neg 98.5%

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

      12. sub-neg 98.5%

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

   8. Simplified 98.5%

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

   9. Taylor expanded in uy around 0 98.4%

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

   if 3.00000014e-4 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 63.0%

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

      1. associate-*l* 63.0%

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

      2. sub-neg 63.0%

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

      3. +-commutative 63.0%

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

      4. distribute-rgt-neg-in 63.0%

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

      5. fma-define 63.1%

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

   3. Simplified 63.5%

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

   5. Taylor expanded in ux around inf 97.8%

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

   6. Taylor expanded in ux around 0 97.8%

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

      1. associate--l+ 97.8%

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

      2. +-commutative 97.8%

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

      3. *-commutative 97.8%

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

      4. sub-neg 97.8%

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

      5. metadata-eval 97.8%

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

      6. sub-neg 97.8%

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

      7. metadata-eval 97.8%

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

      8. +-commutative 97.8%

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

      9. distribute-lft-in 97.8%

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

      10. metadata-eval 97.8%

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

      11. mul-1-neg 97.8%

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

      12. sub-neg 97.8%

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

   8. Simplified 97.8%

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

   9. Taylor expanded in maxCos around 0 92.3%

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

       1. neg-mul-1 92.3%

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

       2. unsub-neg 92.3%

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

   11. Simplified 92.3%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0003000000142492354:\\ \;\;\;\;2 \cdot \left(\sqrt{ux \cdot \left(\left(2 + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)\right) - 2 \cdot maxCos\right)} \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0044999998062849045:\\ \;\;\;\;2 \cdot \left(\sqrt{ux \cdot \left(\left(2 + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)\right) - 2 \cdot maxCos\right)} \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.0044999998062849045)
   (*
    2.0
    (*
     (sqrt
      (*
       ux
       (- (+ 2.0 (* ux (* (- 1.0 maxCos) (+ maxCos -1.0)))) (* 2.0 maxCos))))
     (* uy PI)))
   (* (sin (* uy (* 2.0 PI))) (sqrt (* 2.0 ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.0044999998062849045f) {
		tmp = 2.0f * (sqrtf((ux * ((2.0f + (ux * ((1.0f - maxCos) * (maxCos + -1.0f)))) - (2.0f * maxCos)))) * (uy * ((float) M_PI)));
	} else {
		tmp = sinf((uy * (2.0f * ((float) M_PI)))) * sqrtf((2.0f * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.0044999998062849045))
		tmp = Float32(Float32(2.0) * Float32(sqrt(Float32(ux * Float32(Float32(Float32(2.0) + Float32(ux * Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos + Float32(-1.0))))) - Float32(Float32(2.0) * maxCos)))) * Float32(uy * Float32(pi))));
	else
		tmp = Float32(sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(2.0) * ux)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if ((uy * single(2.0)) <= single(0.0044999998062849045))
		tmp = single(2.0) * (sqrt((ux * ((single(2.0) + (ux * ((single(1.0) - maxCos) * (maxCos + single(-1.0))))) - (single(2.0) * maxCos)))) * (uy * single(pi)));
	else
		tmp = sin((uy * (single(2.0) * single(pi)))) * sqrt((single(2.0) * ux));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.00449999981

   1. Initial program 61.7%

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

      1. associate-*l* 61.7%

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

      2. sub-neg 61.7%

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

      3. +-commutative 61.7%

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

      4. distribute-rgt-neg-in 61.7%

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

      5. fma-define 61.6%

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

   3. Simplified 61.7%

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

   5. Taylor expanded in ux around inf 98.5%

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

   6. Taylor expanded in ux around 0 98.5%

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

      1. associate--l+ 98.5%

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

      2. +-commutative 98.5%

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

      3. *-commutative 98.5%

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

      4. sub-neg 98.5%

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

      5. metadata-eval 98.5%

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

      6. sub-neg 98.5%

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

      7. metadata-eval 98.5%

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

      8. +-commutative 98.5%

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

      9. distribute-lft-in 98.5%

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

      10. metadata-eval 98.5%

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

      11. mul-1-neg 98.5%

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

      12. sub-neg 98.5%

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

   8. Simplified 98.5%

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

   9. Taylor expanded in uy around 0 96.4%

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

   if 0.00449999981 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 62.5%

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

      1. associate-*l* 62.5%

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

      2. sub-neg 62.5%

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

      3. +-commutative 62.5%

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

      4. distribute-rgt-neg-in 62.5%

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

      5. fma-define 62.7%

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

   3. Simplified 63.3%

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

   5. Taylor expanded in maxCos around 0 58.8%

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

   6. Taylor expanded in ux around 0 70.9%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0044999998062849045:\\ \;\;\;\;2 \cdot \left(\sqrt{ux \cdot \left(\left(2 + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)\right) - 2 \cdot maxCos\right)} \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.9%

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

   1. associate-*l* 61.9%

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

   2. sub-neg 61.9%

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

   3. +-commutative 61.9%

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

   4. distribute-rgt-neg-in 61.9%

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

   5. fma-define 61.9%

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

3. Simplified 62.1%

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

5. Taylor expanded in ux around inf 98.2%

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

6. Taylor expanded in ux around 0 98.2%

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

   1. associate--l+ 98.2%

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

   2. +-commutative 98.2%

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

   3. *-commutative 98.2%

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

   4. sub-neg 98.2%

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

   5. metadata-eval 98.2%

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

   6. sub-neg 98.2%

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

   7. metadata-eval 98.2%

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

   8. +-commutative 98.2%

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

   9. distribute-lft-in 98.2%

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

   10. metadata-eval 98.2%

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

   11. mul-1-neg 98.2%

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

   12. sub-neg 98.2%

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

8. Simplified 98.2%

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

9. Taylor expanded in uy around inf 98.2%

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

10. Taylor expanded in maxCos around 0 96.2%

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

    1. mul-1-neg 96.2%

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

12. Simplified 96.2%

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

13. Final simplification 96.2%

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

Alternative 10: 81.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.9%

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

   1. associate-*l* 61.9%

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

   2. sub-neg 61.9%

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

   3. +-commutative 61.9%

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

   4. distribute-rgt-neg-in 61.9%

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

   5. fma-define 61.9%

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

3. Simplified 62.1%

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

5. Taylor expanded in ux around inf 98.2%

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

6. Taylor expanded in ux around 0 98.2%

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

   1. associate--l+ 98.2%

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

   2. +-commutative 98.2%

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

   3. *-commutative 98.2%

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

   4. sub-neg 98.2%

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

   5. metadata-eval 98.2%

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

   6. sub-neg 98.2%

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

   7. metadata-eval 98.2%

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

   8. +-commutative 98.2%

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

   9. distribute-lft-in 98.2%

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

   10. metadata-eval 98.2%

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

   11. mul-1-neg 98.2%

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

   12. sub-neg 98.2%

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

8. Simplified 98.2%

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

9. Taylor expanded in uy around 0 81.2%

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

10. Final simplification 81.2%

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

Alternative 11: 81.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 61.9%

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

   1. associate-*l* 61.9%

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

   2. sub-neg 61.9%

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

   3. +-commutative 61.9%

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

   4. distribute-rgt-neg-in 61.9%

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

   5. fma-define 61.9%

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

3. Simplified 62.1%

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

5. Taylor expanded in ux around inf 98.2%

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

6. Taylor expanded in ux around 0 98.2%

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

   1. associate--l+ 98.2%

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

   2. +-commutative 98.2%

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

   3. *-commutative 98.2%

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

   4. sub-neg 98.2%

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

   5. metadata-eval 98.2%

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

   6. sub-neg 98.2%

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

   7. metadata-eval 98.2%

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

   8. +-commutative 98.2%

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

   9. distribute-lft-in 98.2%

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

   10. metadata-eval 98.2%

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

   11. mul-1-neg 98.2%

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

   12. sub-neg 98.2%

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

8. Simplified 98.2%

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

9. Taylor expanded in uy around 0 81.2%

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

    1. associate--l+ 81.1%

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

    2. sub-neg 81.1%

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

    3. metadata-eval 81.1%

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

11. Simplified 81.1%

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

12. Final simplification 81.1%

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

Alternative 12: 76.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.9%

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

   1. associate-*l* 61.9%

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

   2. sub-neg 61.9%

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

   3. +-commutative 61.9%

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

   4. distribute-rgt-neg-in 61.9%

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

   5. fma-define 61.9%

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

3. Simplified 62.1%

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

5. Taylor expanded in ux around inf 98.2%

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

6. Taylor expanded in maxCos around 0 91.3%

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

   1. sub-neg 91.3%

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

   2. associate-*r/ 91.3%

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

   3. metadata-eval 91.3%

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

   4. metadata-eval 91.3%

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

8. Simplified 91.3%

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

9. Taylor expanded in uy around 0 76.2%

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

    1. associate-*r* 76.2%

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

    2. *-commutative 76.2%

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

11. Simplified 76.2%

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

12. Final simplification 76.2%

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

Alternative 13: 65.8% accurate, 2.0× speedup

Math

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

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(Float32(uy * 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 61.9%

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

   1. associate-*l* 61.9%

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

   2. sub-neg 61.9%

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

   3. +-commutative 61.9%

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

   4. distribute-rgt-neg-in 61.9%

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

   5. fma-define 61.9%

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

3. Simplified 62.1%

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

5. Taylor expanded in uy around 0 53.2%

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

7. Simplified 53.2%

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

8. Taylor expanded in ux around 0 63.8%

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

Alternative 14: 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 61.9%

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

   1. associate-*l* 61.9%

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

   2. sub-neg 61.9%

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

   3. +-commutative 61.9%

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

   4. distribute-rgt-neg-in 61.9%

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

   5. fma-define 61.9%

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

3. Simplified 62.1%

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

5. Taylor expanded in uy around 0 53.2%

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

7. Simplified 53.2%

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

8. Taylor expanded in ux around 0 7.1%

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

9. Taylor expanded in uy around 0 7.1%

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

Reproduce

herbie shell --seed 2024160 
(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)))))))