UniformSampleCone, y

Percentage Accurate: 57.6% → 98.4%

Time: 18.5s

Alternatives: 19

Speedup: 2.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.5%

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

   1. associate-*l* 56.5%

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

   2. sub-neg 56.5%

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

   3. +-commutative 56.5%

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

   4. distribute-rgt-neg-in 56.5%

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

   5. fma-define 56.5%

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

3. Simplified 56.6%

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

5. Taylor expanded in ux around 0 98.3%

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

   1. add-sqr-sqrt 98.3%

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

   2. associate-*r* 98.6%

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

7. Applied egg-rr 98.6%

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

8. Final simplification 98.6%

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

Alternative 2: 98.2% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sin (* uy (* 2.0 PI)))
  (sqrt
   (*
    (pow ux 2.0)
    (-
     (* (- 1.0 maxCos) (+ -1.0 maxCos))
     (/ (+ -1.0 (+ 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 - maxCos) * (-1.0f + maxCos)) - ((-1.0f + (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(Float32(1.0) - maxCos) * Float32(Float32(-1.0) + maxCos)) - Float32(Float32(Float32(-1.0) + Float32(maxCos + Float32(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) - maxCos) * (single(-1.0) + maxCos)) - ((single(-1.0) + (maxCos + (single(-1.0) + maxCos))) / ux))));
end

Derivation

1. Initial program 56.5%

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

   1. associate-*l* 56.5%

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

   2. sub-neg 56.5%

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

   3. +-commutative 56.5%

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

   4. distribute-rgt-neg-in 56.5%

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

   5. fma-define 56.5%

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

3. Simplified 56.6%

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

5. Taylor expanded in ux around -inf 98.3%

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

   1. +-commutative 98.3%

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

   2. mul-1-neg 98.3%

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

   3. unsub-neg 98.3%

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

   4. *-commutative 98.3%

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

   5. sub-neg 98.3%

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

   6. metadata-eval 98.3%

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

   7. +-commutative 98.3%

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

   8. sub-neg 98.3%

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

   9. mul-1-neg 98.3%

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

   10. unsub-neg 98.3%

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

   11. metadata-eval 98.3%

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

7. Simplified 98.3%

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

8. Final simplification 98.3%

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

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.5%

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

   1. associate-*l* 56.5%

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

   2. sub-neg 56.5%

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

   3. +-commutative 56.5%

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

   4. distribute-rgt-neg-in 56.5%

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

   5. fma-define 56.5%

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

3. Simplified 56.6%

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

5. Taylor expanded in ux around 0 98.3%

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

6. Taylor expanded in ux around -inf 98.3%

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

   1. associate-*r* 98.3%

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

   2. mul-1-neg 98.3%

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

   3. mul-1-neg 98.3%

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

   4. sub-neg 98.3%

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

   5. metadata-eval 98.3%

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

   6. mul-1-neg 98.3%

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

   7. sub-neg 98.3%

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

   8. metadata-eval 98.3%

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

   9. +-commutative 98.3%

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

   10. neg-mul-1 98.3%

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

   11. distribute-neg-in 98.3%

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

   12. metadata-eval 98.3%

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

   13. sub-neg 98.3%

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

8. Simplified 98.3%

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

9. Final simplification 98.3%

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

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.5%

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

   1. associate-*l* 56.5%

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

   2. sub-neg 56.5%

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

   3. +-commutative 56.5%

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

   4. distribute-rgt-neg-in 56.5%

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

   5. fma-define 56.5%

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

3. Simplified 56.6%

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

5. Taylor expanded in ux around 0 98.3%

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

6. Final simplification 98.3%

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

Alternative 5: 96.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.00019999999494757503:\\ \;\;\;\;2 \cdot \left(\sqrt{ux \cdot \left(ux \cdot \left(\left(\left(1 - maxCos\right) \cdot \left(-1 + maxCos\right) + 2 \cdot \frac{1}{ux}\right) - \frac{maxCos}{ux}\right) - maxCos\right)} \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(ux \cdot \left(-1 - \frac{-2}{ux}\right) - maxCos\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.00019999999494757503)
   (*
    2.0
    (*
     (sqrt
      (*
       ux
       (-
        (*
         ux
         (-
          (+ (* (- 1.0 maxCos) (+ -1.0 maxCos)) (* 2.0 (/ 1.0 ux)))
          (/ maxCos ux)))
        maxCos)))
     (* uy PI)))
   (*
    (sin (* uy (* 2.0 PI)))
    (sqrt (* ux (- (* ux (- -1.0 (/ -2.0 ux))) maxCos))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.00019999999494757503f) {
		tmp = 2.0f * (sqrtf((ux * ((ux * ((((1.0f - maxCos) * (-1.0f + maxCos)) + (2.0f * (1.0f / ux))) - (maxCos / ux))) - maxCos))) * (uy * ((float) M_PI)));
	} else {
		tmp = sinf((uy * (2.0f * ((float) M_PI)))) * sqrtf((ux * ((ux * (-1.0f - (-2.0f / ux))) - maxCos)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.00019999999494757503))
		tmp = Float32(Float32(2.0) * Float32(sqrt(Float32(ux * Float32(Float32(ux * Float32(Float32(Float32(Float32(Float32(1.0) - maxCos) * Float32(Float32(-1.0) + maxCos)) + Float32(Float32(2.0) * Float32(Float32(1.0) / ux))) - Float32(maxCos / ux))) - maxCos))) * Float32(uy * Float32(pi))));
	else
		tmp = Float32(sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(ux * Float32(Float32(-1.0) - Float32(Float32(-2.0) / ux))) - maxCos))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if ((uy * single(2.0)) <= single(0.00019999999494757503))
		tmp = single(2.0) * (sqrt((ux * ((ux * ((((single(1.0) - maxCos) * (single(-1.0) + maxCos)) + (single(2.0) * (single(1.0) / ux))) - (maxCos / ux))) - maxCos))) * (uy * single(pi)));
	else
		tmp = sin((uy * (single(2.0) * single(pi)))) * sqrt((ux * ((ux * (single(-1.0) - (single(-2.0) / ux))) - maxCos)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 1.99999995e-4

   1. Initial program 55.2%

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

      1. associate-*l* 55.2%

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

      2. sub-neg 55.2%

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

      3. +-commutative 55.2%

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

      4. distribute-rgt-neg-in 55.2%

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

      5. fma-define 55.1%

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

   3. Simplified 55.1%

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

   5. Taylor expanded in ux around 0 98.3%

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

   6. Taylor expanded in ux around -inf 98.3%

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

      1. associate-*r* 98.3%

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

      2. mul-1-neg 98.3%

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

      3. mul-1-neg 98.3%

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

      4. sub-neg 98.3%

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

      5. metadata-eval 98.3%

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

      6. mul-1-neg 98.3%

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

      7. sub-neg 98.3%

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

      8. metadata-eval 98.3%

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

      9. +-commutative 98.3%

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

      10. neg-mul-1 98.3%

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

      11. distribute-neg-in 98.3%

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

      12. metadata-eval 98.3%

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

      13. sub-neg 98.3%

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

   8. Simplified 98.3%

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

   9. Taylor expanded in uy around 0 98.3%

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

   if 1.99999995e-4 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 58.5%

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

      1. associate-*l* 58.5%

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

      2. sub-neg 58.5%

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

      3. +-commutative 58.5%

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

      4. distribute-rgt-neg-in 58.5%

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

      5. fma-define 58.7%

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

   3. Simplified 58.8%

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

   5. Taylor expanded in ux around 0 98.2%

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

   6. Taylor expanded in ux around -inf 98.3%

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

      1. associate-*r* 98.3%

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

      2. mul-1-neg 98.3%

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

      3. mul-1-neg 98.3%

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

      4. sub-neg 98.3%

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

      5. metadata-eval 98.3%

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

      6. mul-1-neg 98.3%

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

      7. sub-neg 98.3%

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

      8. metadata-eval 98.3%

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

      9. +-commutative 98.3%

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

      10. neg-mul-1 98.3%

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

      11. distribute-neg-in 98.3%

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

      12. metadata-eval 98.3%

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

      13. sub-neg 98.3%

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

   8. Simplified 98.3%

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

   9. Taylor expanded in maxCos around 0 92.7%

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

       1. mul-1-neg 92.7%

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

       2. *-commutative 92.7%

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

       3. distribute-rgt-neg-in 92.7%

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

       4. sub-neg 92.7%

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

       5. associate-*r/ 92.7%

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

       6. metadata-eval 92.7%

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

       7. distribute-neg-frac 92.7%

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

       8. metadata-eval 92.7%

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

   11. Simplified 92.7%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.00019999999494757503:\\ \;\;\;\;2 \cdot \left(\sqrt{ux \cdot \left(ux \cdot \left(\left(\left(1 - maxCos\right) \cdot \left(-1 + maxCos\right) + 2 \cdot \frac{1}{ux}\right) - \frac{maxCos}{ux}\right) - maxCos\right)} \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(ux \cdot \left(-1 - \frac{-2}{ux}\right) - maxCos\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.5%

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

   1. associate-*l* 56.5%

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

   2. sub-neg 56.5%

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

   3. +-commutative 56.5%

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

   4. distribute-rgt-neg-in 56.5%

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

   5. fma-define 56.5%

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

3. Simplified 56.6%

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

5. Taylor expanded in ux around -inf 98.3%

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

   1. +-commutative 98.3%

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

   2. mul-1-neg 98.3%

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

   3. unsub-neg 98.3%

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

   4. *-commutative 98.3%

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

   5. sub-neg 98.3%

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

   6. metadata-eval 98.3%

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

   7. +-commutative 98.3%

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

   8. sub-neg 98.3%

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

   9. mul-1-neg 98.3%

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

   10. unsub-neg 98.3%

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

   11. metadata-eval 98.3%

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

7. Simplified 98.3%

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

8. Taylor expanded in ux around 0 98.2%

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

   1. associate--l+ 98.2%

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

   2. sub-neg 98.2%

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

   3. metadata-eval 98.2%

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

   4. +-commutative 98.2%

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

   5. +-commutative 98.2%

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

10. Simplified 98.2%

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

11. Final simplification 98.2%

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

Alternative 7: 96.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.00019999999494757503:\\ \;\;\;\;2 \cdot \left(\sqrt{ux \cdot \left(ux \cdot \left(\left(\left(1 - maxCos\right) \cdot \left(-1 + maxCos\right) + 2 \cdot \frac{1}{ux}\right) - \frac{maxCos}{ux}\right) - maxCos\right)} \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(2 - ux\right) - maxCos\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.00019999999494757503)
   (*
    2.0
    (*
     (sqrt
      (*
       ux
       (-
        (*
         ux
         (-
          (+ (* (- 1.0 maxCos) (+ -1.0 maxCos)) (* 2.0 (/ 1.0 ux)))
          (/ maxCos ux)))
        maxCos)))
     (* uy PI)))
   (* (sin (* uy (* 2.0 PI))) (sqrt (* ux (- (- 2.0 ux) maxCos))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.00019999999494757503f) {
		tmp = 2.0f * (sqrtf((ux * ((ux * ((((1.0f - maxCos) * (-1.0f + maxCos)) + (2.0f * (1.0f / ux))) - (maxCos / ux))) - maxCos))) * (uy * ((float) M_PI)));
	} else {
		tmp = sinf((uy * (2.0f * ((float) M_PI)))) * sqrtf((ux * ((2.0f - ux) - maxCos)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.00019999999494757503))
		tmp = Float32(Float32(2.0) * Float32(sqrt(Float32(ux * Float32(Float32(ux * Float32(Float32(Float32(Float32(Float32(1.0) - maxCos) * Float32(Float32(-1.0) + maxCos)) + Float32(Float32(2.0) * Float32(Float32(1.0) / ux))) - Float32(maxCos / ux))) - maxCos))) * Float32(uy * Float32(pi))));
	else
		tmp = Float32(sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(Float32(2.0) - ux) - maxCos))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if ((uy * single(2.0)) <= single(0.00019999999494757503))
		tmp = single(2.0) * (sqrt((ux * ((ux * ((((single(1.0) - maxCos) * (single(-1.0) + maxCos)) + (single(2.0) * (single(1.0) / ux))) - (maxCos / ux))) - maxCos))) * (uy * single(pi)));
	else
		tmp = sin((uy * (single(2.0) * single(pi)))) * sqrt((ux * ((single(2.0) - ux) - maxCos)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 1.99999995e-4

   1. Initial program 55.2%

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

      1. associate-*l* 55.2%

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

      2. sub-neg 55.2%

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

      3. +-commutative 55.2%

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

      4. distribute-rgt-neg-in 55.2%

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

      5. fma-define 55.1%

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

   3. Simplified 55.1%

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

   5. Taylor expanded in ux around 0 98.3%

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

   6. Taylor expanded in ux around -inf 98.3%

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

      1. associate-*r* 98.3%

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

      2. mul-1-neg 98.3%

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

      3. mul-1-neg 98.3%

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

      4. sub-neg 98.3%

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

      5. metadata-eval 98.3%

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

      6. mul-1-neg 98.3%

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

      7. sub-neg 98.3%

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

      8. metadata-eval 98.3%

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

      9. +-commutative 98.3%

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

      10. neg-mul-1 98.3%

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

      11. distribute-neg-in 98.3%

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

      12. metadata-eval 98.3%

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

      13. sub-neg 98.3%

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

   8. Simplified 98.3%

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

   9. Taylor expanded in uy around 0 98.3%

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

   if 1.99999995e-4 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 58.5%

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

      1. associate-*l* 58.5%

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

      2. sub-neg 58.5%

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

      3. +-commutative 58.5%

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

      4. distribute-rgt-neg-in 58.5%

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

      5. fma-define 58.7%

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

   3. Simplified 58.8%

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

   5. Taylor expanded in ux around 0 98.2%

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

   6. Taylor expanded in maxCos around 0 92.5%

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

      1. mul-1-neg 92.5%

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

   8. Simplified 92.5%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.00019999999494757503:\\ \;\;\;\;2 \cdot \left(\sqrt{ux \cdot \left(ux \cdot \left(\left(\left(1 - maxCos\right) \cdot \left(-1 + maxCos\right) + 2 \cdot \frac{1}{ux}\right) - \frac{maxCos}{ux}\right) - maxCos\right)} \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(2 - ux\right) - maxCos\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 95.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.00019999999494757503:\\ \;\;\;\;2 \cdot \left(\sqrt{ux \cdot \left(ux \cdot \left(\left(\left(1 - maxCos\right) \cdot \left(-1 + maxCos\right) + 2 \cdot \frac{1}{ux}\right) - \frac{maxCos}{ux}\right) - maxCos\right)} \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.00019999999494757503)
   (*
    2.0
    (*
     (sqrt
      (*
       ux
       (-
        (*
         ux
         (-
          (+ (* (- 1.0 maxCos) (+ -1.0 maxCos)) (* 2.0 (/ 1.0 ux)))
          (/ maxCos ux)))
        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) <= 0.00019999999494757503f) {
		tmp = 2.0f * (sqrtf((ux * ((ux * ((((1.0f - maxCos) * (-1.0f + maxCos)) + (2.0f * (1.0f / ux))) - (maxCos / ux))) - 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(0.00019999999494757503))
		tmp = Float32(Float32(2.0) * Float32(sqrt(Float32(ux * Float32(Float32(ux * Float32(Float32(Float32(Float32(Float32(1.0) - maxCos) * Float32(Float32(-1.0) + maxCos)) + Float32(Float32(2.0) * Float32(Float32(1.0) / ux))) - Float32(maxCos / ux))) - 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(0.00019999999494757503))
		tmp = single(2.0) * (sqrt((ux * ((ux * ((((single(1.0) - maxCos) * (single(-1.0) + maxCos)) + (single(2.0) * (single(1.0) / ux))) - (maxCos / ux))) - 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)) < 1.99999995e-4

   1. Initial program 55.2%

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

      1. associate-*l* 55.2%

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

      2. sub-neg 55.2%

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

      3. +-commutative 55.2%

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

      4. distribute-rgt-neg-in 55.2%

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

      5. fma-define 55.1%

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

   3. Simplified 55.1%

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

   5. Taylor expanded in ux around 0 98.3%

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

   6. Taylor expanded in ux around -inf 98.3%

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

      1. associate-*r* 98.3%

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

      2. mul-1-neg 98.3%

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

      3. mul-1-neg 98.3%

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

      4. sub-neg 98.3%

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

      5. metadata-eval 98.3%

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

      6. mul-1-neg 98.3%

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

      7. sub-neg 98.3%

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

      8. metadata-eval 98.3%

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

      9. +-commutative 98.3%

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

      10. neg-mul-1 98.3%

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

      11. distribute-neg-in 98.3%

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

      12. metadata-eval 98.3%

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

      13. sub-neg 98.3%

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

   8. Simplified 98.3%

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

   9. Taylor expanded in uy around 0 98.3%

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

   if 1.99999995e-4 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 58.5%

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

      1. associate-*l* 58.5%

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

      2. sub-neg 58.5%

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

      3. +-commutative 58.5%

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

      4. distribute-rgt-neg-in 58.5%

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

      5. fma-define 58.7%

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

   3. Simplified 58.8%

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

   5. Taylor expanded in ux around 0 98.2%

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

      1. add-sqr-sqrt 98.2%

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

      2. associate-*r* 98.3%

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

   7. Applied egg-rr 98.3%

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

   8. Taylor expanded in maxCos around 0 91.8%

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

      1. mul-1-neg 91.8%

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

      2. unsub-neg 91.8%

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

      3. unpow2 91.8%

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

      4. rem-square-sqrt 92.0%

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

   10. Simplified 92.0%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.00019999999494757503:\\ \;\;\;\;2 \cdot \left(\sqrt{ux \cdot \left(ux \cdot \left(\left(\left(1 - maxCos\right) \cdot \left(-1 + maxCos\right) + 2 \cdot \frac{1}{ux}\right) - \frac{maxCos}{ux}\right) - maxCos\right)} \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0031999999191612005:\\ \;\;\;\;2 \cdot \left(\sqrt{ux \cdot \left(ux \cdot \left(\left(\left(1 - maxCos\right) \cdot \left(-1 + maxCos\right) + 2 \cdot \frac{1}{ux}\right) - \frac{maxCos}{ux}\right) - maxCos\right)} \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.0031999999191612005)
   (*
    2.0
    (*
     (sqrt
      (*
       ux
       (-
        (*
         ux
         (-
          (+ (* (- 1.0 maxCos) (+ -1.0 maxCos)) (* 2.0 (/ 1.0 ux)))
          (/ maxCos ux)))
        maxCos)))
     (* uy PI)))
   (* (sin (* 2.0 (* uy PI))) (sqrt (* 2.0 ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.0031999999191612005f) {
		tmp = 2.0f * (sqrtf((ux * ((ux * ((((1.0f - maxCos) * (-1.0f + maxCos)) + (2.0f * (1.0f / ux))) - (maxCos / ux))) - maxCos))) * (uy * ((float) M_PI)));
	} else {
		tmp = sinf((2.0f * (uy * ((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(0.0031999999191612005))
		tmp = Float32(Float32(2.0) * Float32(sqrt(Float32(ux * Float32(Float32(ux * Float32(Float32(Float32(Float32(Float32(1.0) - maxCos) * Float32(Float32(-1.0) + maxCos)) + Float32(Float32(2.0) * Float32(Float32(1.0) / ux))) - Float32(maxCos / ux))) - maxCos))) * Float32(uy * Float32(pi))));
	else
		tmp = Float32(sin(Float32(Float32(2.0) * Float32(uy * 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(0.0031999999191612005))
		tmp = single(2.0) * (sqrt((ux * ((ux * ((((single(1.0) - maxCos) * (single(-1.0) + maxCos)) + (single(2.0) * (single(1.0) / ux))) - (maxCos / ux))) - maxCos))) * (uy * single(pi)));
	else
		tmp = sin((single(2.0) * (uy * 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)) < 0.00319999992

   1. Initial program 55.6%

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

      1. associate-*l* 55.6%

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

      2. sub-neg 55.6%

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

      3. +-commutative 55.6%

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

      4. distribute-rgt-neg-in 55.6%

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

      5. fma-define 55.6%

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

   3. Simplified 55.7%

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

   5. Taylor expanded in ux around 0 98.3%

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

   6. Taylor expanded in ux around -inf 98.2%

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

      1. associate-*r* 98.2%

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

      2. mul-1-neg 98.2%

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

      3. mul-1-neg 98.2%

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

      4. sub-neg 98.2%

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

      5. metadata-eval 98.2%

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

      6. mul-1-neg 98.2%

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

      7. sub-neg 98.2%

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

      8. metadata-eval 98.2%

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

      9. +-commutative 98.2%

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

      10. neg-mul-1 98.2%

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

      11. distribute-neg-in 98.2%

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

      12. metadata-eval 98.2%

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

      13. sub-neg 98.2%

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

   8. Simplified 98.2%

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

   9. Taylor expanded in uy around 0 96.7%

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

   if 0.00319999992 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 58.4%

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

   3. Taylor expanded in ux around 0 78.0%

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

   4. Taylor expanded in maxCos around 0 75.6%

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

      1. *-commutative 75.6%

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

   6. Simplified 75.6%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0031999999191612005:\\ \;\;\;\;2 \cdot \left(\sqrt{ux \cdot \left(ux \cdot \left(\left(\left(1 - maxCos\right) \cdot \left(-1 + maxCos\right) + 2 \cdot \frac{1}{ux}\right) - \frac{maxCos}{ux}\right) - maxCos\right)} \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.5%

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

   1. associate-*l* 56.5%

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

   2. sub-neg 56.5%

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

   3. +-commutative 56.5%

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

   4. distribute-rgt-neg-in 56.5%

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

   5. fma-define 56.5%

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

3. Simplified 56.6%

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

5. Taylor expanded in ux around 0 98.3%

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

6. Taylor expanded in ux around -inf 98.3%

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

   1. associate-*r* 98.3%

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

   2. mul-1-neg 98.3%

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

   3. mul-1-neg 98.3%

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

   4. sub-neg 98.3%

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

   5. metadata-eval 98.3%

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

   6. mul-1-neg 98.3%

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

   7. sub-neg 98.3%

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

   8. metadata-eval 98.3%

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

   9. +-commutative 98.3%

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

   10. neg-mul-1 98.3%

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

   11. distribute-neg-in 98.3%

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

   12. metadata-eval 98.3%

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

   13. sub-neg 98.3%

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

8. Simplified 98.3%

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

9. Taylor expanded in uy around 0 80.5%

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

10. Final simplification 80.5%

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

Alternative 11: 76.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.0001500000071246177:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(\left(1 + \left(1 - maxCos\right)\right) - maxCos\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 + \left(1 + ux \cdot \left(-1 + maxCos\right)\right) \cdot \left(-1 + ux \cdot \left(1 - maxCos\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.0001500000071246177)
   (* 2.0 (* (* uy PI) (sqrt (* ux (- (+ 1.0 (- 1.0 maxCos)) maxCos)))))
   (*
    2.0
    (*
     (* uy PI)
     (sqrt
      (+
       1.0
       (* (+ 1.0 (* ux (+ -1.0 maxCos))) (+ -1.0 (* ux (- 1.0 maxCos))))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.0001500000071246177f) {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf((ux * ((1.0f + (1.0f - maxCos)) - maxCos))));
	} else {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf((1.0f + ((1.0f + (ux * (-1.0f + maxCos))) * (-1.0f + (ux * (1.0f - maxCos)))))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.0001500000071246177))
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(ux * Float32(Float32(Float32(1.0) + Float32(Float32(1.0) - maxCos)) - maxCos)))));
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) + Float32(ux * Float32(Float32(-1.0) + maxCos))) * Float32(Float32(-1.0) + Float32(ux * Float32(Float32(1.0) - maxCos))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.0001500000071246177))
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((ux * ((single(1.0) + (single(1.0) - maxCos)) - maxCos))));
	else
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((single(1.0) + ((single(1.0) + (ux * (single(-1.0) + maxCos))) * (single(-1.0) + (ux * (single(1.0) - maxCos)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.50000007e-4

   1. Initial program 36.0%

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

      1. associate-*l* 36.0%

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

      2. sub-neg 36.0%

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

      3. +-commutative 36.0%

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

      4. distribute-rgt-neg-in 36.0%

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

      5. fma-define 36.0%

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

   3. Simplified 36.0%

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

   5. Taylor expanded in uy around 0 32.7%

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

   6. Taylor expanded in ux around 0 76.6%

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

   if 1.50000007e-4 < ux

   1. Initial program 87.5%

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

      1. associate-*l* 87.5%

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

      2. sub-neg 87.5%

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

      3. +-commutative 87.5%

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

      4. distribute-rgt-neg-in 87.5%

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

      5. fma-define 87.6%

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

   3. Simplified 87.7%

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

   5. Taylor expanded in uy around 0 74.8%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.0001500000071246177:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(\left(1 + \left(1 - maxCos\right)\right) - maxCos\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 + \left(1 + ux \cdot \left(-1 + maxCos\right)\right) \cdot \left(-1 + ux \cdot \left(1 - maxCos\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.5%

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

   1. associate-*l* 56.5%

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

   2. sub-neg 56.5%

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

   3. +-commutative 56.5%

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

   4. distribute-rgt-neg-in 56.5%

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

   5. fma-define 56.5%

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

3. Simplified 56.6%

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

5. Taylor expanded in uy around 0 49.5%

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

6. Taylor expanded in ux around 0 80.5%

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

7. Final simplification 80.5%

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

Alternative 13: 81.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.5%

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

   1. associate-*l* 56.5%

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

   2. sub-neg 56.5%

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

   3. +-commutative 56.5%

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

   4. distribute-rgt-neg-in 56.5%

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

   5. fma-define 56.5%

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

3. Simplified 56.6%

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

5. Taylor expanded in ux around -inf 98.3%

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

   1. +-commutative 98.3%

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

   2. mul-1-neg 98.3%

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

   3. unsub-neg 98.3%

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

   4. *-commutative 98.3%

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

   5. sub-neg 98.3%

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

   6. metadata-eval 98.3%

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

   7. +-commutative 98.3%

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

   8. sub-neg 98.3%

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

   9. mul-1-neg 98.3%

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

   10. unsub-neg 98.3%

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

   11. metadata-eval 98.3%

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

7. Simplified 98.3%

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

8. Taylor expanded in uy around 0 80.5%

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

   1. associate-*r* 80.5%

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

   2. associate-*r* 80.5%

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

   3. +-commutative 80.5%

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

   4. associate--l+ 80.5%

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

   5. sub-neg 80.5%

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

   6. metadata-eval 80.5%

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

   7. associate-*r/ 80.5%

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

   8. metadata-eval 80.5%

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

   9. associate-*r/ 80.5%

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

   10. div-sub 80.5%

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

   11. cancel-sign-sub-inv 80.5%

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

   12. metadata-eval 80.5%

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

10. Simplified 80.5%

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

11. Final simplification 80.5%

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

Alternative 14: 75.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00023799999326001853:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(\left(1 + \left(1 - maxCos\right)\right) - maxCos\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{1 + \left(ux + -1\right) \cdot \left(1 - ux\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00023799999326001853)
   (* 2.0 (* (* uy PI) (sqrt (* ux (- (+ 1.0 (- 1.0 maxCos)) maxCos)))))
   (* 2.0 (* uy (* PI (sqrt (+ 1.0 (* (+ ux -1.0) (- 1.0 ux)))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00023799999326001853f) {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf((ux * ((1.0f + (1.0f - maxCos)) - maxCos))));
	} else {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf((1.0f + ((ux + -1.0f) * (1.0f - ux))))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00023799999326001853))
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(ux * Float32(Float32(Float32(1.0) + Float32(Float32(1.0) - maxCos)) - maxCos)))));
	else
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(Float32(1.0) + Float32(Float32(ux + Float32(-1.0)) * Float32(Float32(1.0) - ux)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00023799999326001853))
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((ux * ((single(1.0) + (single(1.0) - maxCos)) - maxCos))));
	else
		tmp = single(2.0) * (uy * (single(pi) * sqrt((single(1.0) + ((ux + single(-1.0)) * (single(1.0) - ux))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 2.37999993e-4

   1. Initial program 37.3%

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

      1. associate-*l* 37.3%

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

      2. sub-neg 37.3%

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

      3. +-commutative 37.3%

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

      4. distribute-rgt-neg-in 37.3%

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

      5. fma-define 37.4%

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

   3. Simplified 37.5%

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

   5. Taylor expanded in uy around 0 34.2%

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

   6. Taylor expanded in ux around 0 76.3%

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

   if 2.37999993e-4 < ux

   1. Initial program 88.5%

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

      1. associate-*l* 88.5%

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

      2. sub-neg 88.5%

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

      3. +-commutative 88.5%

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

      4. distribute-rgt-neg-in 88.5%

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

      5. fma-define 88.5%

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

   3. Simplified 88.5%

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

   5. Taylor expanded in uy around 0 75.0%

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

   6. Taylor expanded in maxCos around 0 71.9%

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

      1. associate-*l* 72.0%

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

      2. mul-1-neg 72.0%

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

      3. sub-neg 72.0%

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

      4. metadata-eval 72.0%

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

   8. Simplified 72.0%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00023799999326001853:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(\left(1 + \left(1 - maxCos\right)\right) - maxCos\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{1 + \left(ux + -1\right) \cdot \left(1 - ux\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 75.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00023799999326001853:\\ \;\;\;\;\left(uy \cdot \pi\right) \cdot \left(2 \cdot \sqrt{ux \cdot \left(2 + maxCos \cdot -2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{1 + \left(ux + -1\right) \cdot \left(1 - ux\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00023799999326001853)
   (* (* uy PI) (* 2.0 (sqrt (* ux (+ 2.0 (* maxCos -2.0))))))
   (* 2.0 (* uy (* PI (sqrt (+ 1.0 (* (+ ux -1.0) (- 1.0 ux)))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00023799999326001853f) {
		tmp = (uy * ((float) M_PI)) * (2.0f * sqrtf((ux * (2.0f + (maxCos * -2.0f)))));
	} else {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf((1.0f + ((ux + -1.0f) * (1.0f - ux))))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00023799999326001853))
		tmp = Float32(Float32(uy * Float32(pi)) * Float32(Float32(2.0) * sqrt(Float32(ux * Float32(Float32(2.0) + Float32(maxCos * Float32(-2.0)))))));
	else
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(Float32(1.0) + Float32(Float32(ux + Float32(-1.0)) * Float32(Float32(1.0) - ux)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00023799999326001853))
		tmp = (uy * single(pi)) * (single(2.0) * sqrt((ux * (single(2.0) + (maxCos * single(-2.0))))));
	else
		tmp = single(2.0) * (uy * (single(pi) * sqrt((single(1.0) + ((ux + single(-1.0)) * (single(1.0) - ux))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 2.37999993e-4

   1. Initial program 37.3%

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

   3. Taylor expanded in ux around 0 91.9%

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

   4. Taylor expanded in uy around 0 76.3%

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

      1. associate-*r* 76.3%

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

      2. sub-neg 76.3%

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

      3. distribute-rgt-in 76.3%

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

      4. distribute-lft-neg-in 76.3%

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

      5. metadata-eval 76.3%

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

      6. associate-*r* 76.3%

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

      7. +-commutative 76.3%

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

      8. associate-*r* 76.3%

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

      9. metadata-eval 76.3%

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

      10. distribute-lft-neg-in 76.3%

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

      11. distribute-rgt-out 76.3%

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

      12. *-commutative 76.3%

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

      13. distribute-rgt-neg-in 76.3%

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

      14. metadata-eval 76.3%

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

   6. Simplified 76.3%

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

   if 2.37999993e-4 < ux

   1. Initial program 88.5%

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

      1. associate-*l* 88.5%

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

      2. sub-neg 88.5%

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

      3. +-commutative 88.5%

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

      4. distribute-rgt-neg-in 88.5%

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

      5. fma-define 88.5%

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

   3. Simplified 88.5%

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

   5. Taylor expanded in uy around 0 75.0%

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

   6. Taylor expanded in maxCos around 0 71.9%

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

      1. associate-*l* 72.0%

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

      2. mul-1-neg 72.0%

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

      3. sub-neg 72.0%

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

      4. metadata-eval 72.0%

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

   8. Simplified 72.0%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00023799999326001853:\\ \;\;\;\;\left(uy \cdot \pi\right) \cdot \left(2 \cdot \sqrt{ux \cdot \left(2 + maxCos \cdot -2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{1 + \left(ux + -1\right) \cdot \left(1 - ux\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 75.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00023799999326001853:\\ \;\;\;\;\left(uy \cdot \pi\right) \cdot \left(2 \cdot \sqrt{ux \cdot \left(2 + maxCos \cdot -2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 + \left(ux + -1\right) \cdot \left(1 - ux\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00023799999326001853)
   (* (* uy PI) (* 2.0 (sqrt (* ux (+ 2.0 (* maxCos -2.0))))))
   (* 2.0 (* (* uy PI) (sqrt (+ 1.0 (* (+ ux -1.0) (- 1.0 ux))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00023799999326001853f) {
		tmp = (uy * ((float) M_PI)) * (2.0f * sqrtf((ux * (2.0f + (maxCos * -2.0f)))));
	} else {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf((1.0f + ((ux + -1.0f) * (1.0f - ux)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00023799999326001853))
		tmp = Float32(Float32(uy * Float32(pi)) * Float32(Float32(2.0) * sqrt(Float32(ux * Float32(Float32(2.0) + Float32(maxCos * Float32(-2.0)))))));
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(Float32(1.0) + Float32(Float32(ux + Float32(-1.0)) * Float32(Float32(1.0) - ux))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00023799999326001853))
		tmp = (uy * single(pi)) * (single(2.0) * sqrt((ux * (single(2.0) + (maxCos * single(-2.0))))));
	else
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((single(1.0) + ((ux + single(-1.0)) * (single(1.0) - ux)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 2.37999993e-4

   1. Initial program 37.3%

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

   3. Taylor expanded in ux around 0 91.9%

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

   4. Taylor expanded in uy around 0 76.3%

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

      1. associate-*r* 76.3%

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

      2. sub-neg 76.3%

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

      3. distribute-rgt-in 76.3%

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

      4. distribute-lft-neg-in 76.3%

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

      5. metadata-eval 76.3%

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

      6. associate-*r* 76.3%

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

      7. +-commutative 76.3%

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

      8. associate-*r* 76.3%

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

      9. metadata-eval 76.3%

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

      10. distribute-lft-neg-in 76.3%

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

      11. distribute-rgt-out 76.3%

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

      12. *-commutative 76.3%

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

      13. distribute-rgt-neg-in 76.3%

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

      14. metadata-eval 76.3%

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

   6. Simplified 76.3%

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

   if 2.37999993e-4 < ux

   1. Initial program 88.5%

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

      1. associate-*l* 88.5%

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

      2. sub-neg 88.5%

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

      3. +-commutative 88.5%

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

      4. distribute-rgt-neg-in 88.5%

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

      5. fma-define 88.5%

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

   3. Simplified 88.5%

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

   5. Taylor expanded in uy around 0 75.0%

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

      1. add-cube-cbrt 74.9%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\left(\sqrt[3]{uy \cdot \pi} \cdot \sqrt[3]{uy \cdot \pi}\right) \cdot \sqrt[3]{uy \cdot \pi}\right)} \cdot \sqrt{1 + \left(1 + ux \cdot \left(maxCos - 1\right)\right) \cdot \left(ux \cdot \left(1 - maxCos\right) - 1\right)}\right) \]

      2. pow3 75.0%

         \[\leadsto 2 \cdot \left(\color{blue}{{\left(\sqrt[3]{uy \cdot \pi}\right)}^{3}} \cdot \sqrt{1 + \left(1 + ux \cdot \left(maxCos - 1\right)\right) \cdot \left(ux \cdot \left(1 - maxCos\right) - 1\right)}\right) \]

   7. Applied egg-rr 75.0%

      \[\leadsto 2 \cdot \left(\color{blue}{{\left(\sqrt[3]{uy \cdot \pi}\right)}^{3}} \cdot \sqrt{1 + \left(1 + ux \cdot \left(maxCos - 1\right)\right) \cdot \left(ux \cdot \left(1 - maxCos\right) - 1\right)}\right) \]

   8. Taylor expanded in maxCos around 0 71.9%

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

      1. mul-1-neg 71.9%

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

      2. unsub-neg 71.9%

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

      3. sub-neg 71.9%

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

      4. metadata-eval 71.9%

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

   10. Simplified 71.9%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00023799999326001853:\\ \;\;\;\;\left(uy \cdot \pi\right) \cdot \left(2 \cdot \sqrt{ux \cdot \left(2 + maxCos \cdot -2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 + \left(ux + -1\right) \cdot \left(1 - ux\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 65.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in ux around 0 77.5%

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

4. Taylor expanded in uy around 0 66.2%

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

   1. associate-*r* 66.2%

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

   2. sub-neg 66.2%

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

   3. distribute-rgt-in 66.2%

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

   4. distribute-lft-neg-in 66.2%

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

   5. metadata-eval 66.2%

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

   6. associate-*r* 66.2%

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

   7. +-commutative 66.2%

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

   8. associate-*r* 66.2%

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

   9. metadata-eval 66.2%

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

   10. distribute-lft-neg-in 66.2%

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

   11. distribute-rgt-out 66.2%

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

   12. *-commutative 66.2%

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

   13. distribute-rgt-neg-in 66.2%

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

   14. metadata-eval 66.2%

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

6. Simplified 66.2%

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

7. Final simplification 66.2%

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

Alternative 18: 63.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in ux around 0 77.5%

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

4. Taylor expanded in uy around 0 66.2%

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

   1. associate-*r* 66.2%

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

   2. sub-neg 66.2%

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

   3. distribute-rgt-in 66.2%

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

   4. distribute-lft-neg-in 66.2%

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

   5. metadata-eval 66.2%

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

   6. associate-*r* 66.2%

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

   7. +-commutative 66.2%

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

   8. associate-*r* 66.2%

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

   9. metadata-eval 66.2%

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

   10. distribute-lft-neg-in 66.2%

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

   11. distribute-rgt-out 66.2%

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

   12. *-commutative 66.2%

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

   13. distribute-rgt-neg-in 66.2%

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

   14. metadata-eval 66.2%

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

6. Simplified 66.2%

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

7. Taylor expanded in maxCos around 0 63.4%

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

8. Final simplification 63.4%

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

Alternative 19: 7.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{0}\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (* 2.0 (* (* uy PI) (sqrt 0.0))))

C

float code(float ux, float uy, float maxCos) {
	return 2.0f * ((uy * ((float) M_PI)) * sqrtf(0.0f));
}

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(0.0))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = single(2.0) * ((uy * single(pi)) * sqrt(single(0.0)));
end

Derivation

1. Initial program 56.5%

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

   1. associate-*l* 56.5%

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

   2. sub-neg 56.5%

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

   3. +-commutative 56.5%

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

   4. distribute-rgt-neg-in 56.5%

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

   5. fma-define 56.5%

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

3. Simplified 56.6%

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

5. Taylor expanded in uy around 0 49.5%

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

6. Taylor expanded in ux around 0 7.1%

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

7. Final simplification 7.1%

   \[\leadsto 2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{0}\right) \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024095 
(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)))))))