UniformSampleCone, y

Percentage Accurate: 57.2% → 98.1%

Time: 12.6s

Alternatives: 9

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: 57.2% 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: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 54.3%

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

   1. associate-*l* 54.3%

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

   2. sub-neg 54.3%

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

   3. +-commutative 54.3%

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

   4. distribute-rgt-neg-in 54.3%

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

   5. fma-define 54.3%

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

3. Simplified 54.4%

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

5. Taylor expanded in ux around inf 98.3%

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

6. Taylor expanded in maxCos around inf 88.7%

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

7. Taylor expanded in ux around 0 98.5%

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

8. Final simplification 98.5%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 54.3%

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

   1. associate-*l* 54.3%

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

   2. sub-neg 54.3%

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

   3. +-commutative 54.3%

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

   4. distribute-rgt-neg-in 54.3%

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

   5. fma-define 54.3%

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

3. Simplified 54.4%

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

5. Taylor expanded in ux around inf 98.3%

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

6. Taylor expanded in ux around 0 98.5%

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

   1. associate--l+ 98.4%

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

   2. sub-neg 98.4%

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

   3. metadata-eval 98.4%

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

   4. +-commutative 98.4%

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

   5. distribute-lft-in 98.4%

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

   6. metadata-eval 98.4%

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

   7. neg-mul-1 98.4%

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

   8. sub-neg 98.4%

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

   9. associate-*r* 98.4%

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

   10. sub-neg 98.4%

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

   11. metadata-eval 98.4%

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

8. Simplified 98.4%

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

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 54.3%

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

   1. associate-*l* 54.3%

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

   2. sub-neg 54.3%

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

   3. +-commutative 54.3%

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

   4. distribute-rgt-neg-in 54.3%

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

   5. fma-define 54.3%

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

3. Simplified 54.4%

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

5. Taylor expanded in ux around inf 98.3%

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

6. Taylor expanded in maxCos around 0 98.3%

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

   1. +-commutative 98.3%

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

   2. mul-1-neg 98.3%

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

   3. unsub-neg 98.3%

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

   4. div-sub 98.3%

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

8. Simplified 98.3%

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

9. Taylor expanded in ux around 0 98.4%

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

10. Final simplification 98.4%

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

Alternative 4: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 54.3%

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

3. Taylor expanded in ux around 0 98.4%

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

   1. associate--l+ 98.4%

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

   2. associate-*r* 98.4%

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

   3. mul-1-neg 98.4%

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

   4. sub-neg 98.4%

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

   5. metadata-eval 98.4%

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

   6. +-commutative 98.4%

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

5. Simplified 98.4%

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

6. Taylor expanded in maxCos around 0 97.6%

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

7. Final simplification 97.6%

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

Alternative 5: 95.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(uy \cdot 2\right)\\ \mathbf{if}\;uy \cdot 2 \leq 0.0003000000142492354:\\ \;\;\;\;\sqrt{ux \cdot \left(1 + \left(\left(\left(1 - maxCos\right) + \left(ux \cdot \left(1 - maxCos\right)\right) \cdot \left(maxCos + -1\right)\right) - maxCos\right)\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* PI (* uy 2.0))))
   (if (<= (* uy 2.0) 0.0003000000142492354)
     (*
      (sqrt
       (*
        ux
        (+
         1.0
         (-
          (+ (- 1.0 maxCos) (* (* ux (- 1.0 maxCos)) (+ maxCos -1.0)))
          maxCos))))
      t_0)
     (* (sin t_0) (sqrt (* ux (- 2.0 ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = ((float) M_PI) * (uy * 2.0f);
	float tmp;
	if ((uy * 2.0f) <= 0.0003000000142492354f) {
		tmp = sqrtf((ux * (1.0f + (((1.0f - maxCos) + ((ux * (1.0f - maxCos)) * (maxCos + -1.0f))) - maxCos)))) * t_0;
	} else {
		tmp = sinf(t_0) * sqrtf((ux * (2.0f - ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(pi) * Float32(uy * Float32(2.0)))
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.0003000000142492354))
		tmp = Float32(sqrt(Float32(ux * Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - maxCos) + Float32(Float32(ux * Float32(Float32(1.0) - maxCos)) * Float32(maxCos + Float32(-1.0)))) - maxCos)))) * t_0);
	else
		tmp = Float32(sin(t_0) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = single(pi) * (uy * single(2.0));
	tmp = single(0.0);
	if ((uy * single(2.0)) <= single(0.0003000000142492354))
		tmp = sqrt((ux * (single(1.0) + (((single(1.0) - maxCos) + ((ux * (single(1.0) - maxCos)) * (maxCos + single(-1.0)))) - maxCos)))) * t_0;
	else
		tmp = sin(t_0) * sqrt((ux * (single(2.0) - ux)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 54.3%

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

      1. associate-*l* 54.3%

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

      2. sub-neg 54.3%

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

      3. +-commutative 54.3%

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

      4. distribute-rgt-neg-in 54.3%

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

      5. fma-define 54.5%

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

   3. Simplified 54.5%

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

   5. Taylor expanded in ux around inf 98.4%

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

   6. Taylor expanded in ux around 0 98.7%

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

      1. associate--l+ 98.6%

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

      2. sub-neg 98.6%

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

      3. metadata-eval 98.6%

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

      4. +-commutative 98.6%

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

      5. distribute-lft-in 98.6%

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

      6. metadata-eval 98.6%

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

      7. neg-mul-1 98.6%

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

      8. sub-neg 98.6%

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

      9. associate-*r* 98.6%

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

      10. sub-neg 98.6%

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

      11. metadata-eval 98.6%

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

   8. Simplified 98.6%

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

   9. Taylor expanded in uy around 0 98.4%

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

       1. associate-*r* 98.4%

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

   11. Simplified 98.4%

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

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

   1. Initial program 54.2%

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

   3. Taylor expanded in ux around 0 98.1%

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

      1. associate--l+ 98.1%

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

      2. associate-*r* 98.1%

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

      3. mul-1-neg 98.1%

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

      4. sub-neg 98.1%

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

      5. metadata-eval 98.1%

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

      6. +-commutative 98.1%

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

   5. Simplified 98.1%

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

   6. Taylor expanded in maxCos around 0 92.3%

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

      1. neg-mul-1 92.3%

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

      2. unsub-neg 92.3%

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

   8. Simplified 92.3%

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

4. Final simplification 96.2%

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

Alternative 6: 89.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.01600000075995922:\\ \;\;\;\;\sqrt{ux \cdot \left(1 + \left(\left(\left(1 - maxCos\right) + \left(ux \cdot \left(1 - maxCos\right)\right) \cdot \left(maxCos + -1\right)\right) - maxCos\right)\right)} \cdot \left(\pi \cdot \left(uy \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.01600000075995922)
   (*
    (sqrt
     (*
      ux
      (+
       1.0
       (-
        (+ (- 1.0 maxCos) (* (* ux (- 1.0 maxCos)) (+ maxCos -1.0)))
        maxCos))))
    (* PI (* uy 2.0)))
   (* (sin (* uy (* 2.0 PI))) (sqrt (* 2.0 ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.01600000075995922f) {
		tmp = sqrtf((ux * (1.0f + (((1.0f - maxCos) + ((ux * (1.0f - maxCos)) * (maxCos + -1.0f))) - maxCos)))) * (((float) M_PI) * (uy * 2.0f));
	} else {
		tmp = sinf((uy * (2.0f * ((float) M_PI)))) * sqrtf((2.0f * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.01600000075995922))
		tmp = Float32(sqrt(Float32(ux * Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - maxCos) + Float32(Float32(ux * Float32(Float32(1.0) - maxCos)) * Float32(maxCos + Float32(-1.0)))) - maxCos)))) * Float32(Float32(pi) * Float32(uy * Float32(2.0))));
	else
		tmp = Float32(sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(2.0) * ux)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if ((uy * single(2.0)) <= single(0.01600000075995922))
		tmp = sqrt((ux * (single(1.0) + (((single(1.0) - maxCos) + ((ux * (single(1.0) - maxCos)) * (maxCos + single(-1.0)))) - maxCos)))) * (single(pi) * (uy * single(2.0)));
	else
		tmp = sin((uy * (single(2.0) * single(pi)))) * sqrt((single(2.0) * ux));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.0160000008

   1. Initial program 56.2%

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

      1. associate-*l* 56.2%

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

      2. sub-neg 56.2%

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

      3. +-commutative 56.2%

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

      4. distribute-rgt-neg-in 56.2%

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

      5. fma-define 56.3%

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

   3. Simplified 56.4%

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

   5. Taylor expanded in ux around inf 98.3%

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

   6. Taylor expanded in ux around 0 98.6%

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

      1. associate--l+ 98.5%

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

      2. sub-neg 98.5%

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

      3. metadata-eval 98.5%

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

      4. +-commutative 98.5%

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

      5. distribute-lft-in 98.5%

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

      6. metadata-eval 98.5%

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

      7. neg-mul-1 98.5%

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

      8. sub-neg 98.5%

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

      9. associate-*r* 98.5%

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

      10. sub-neg 98.5%

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

      11. metadata-eval 98.5%

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

   8. Simplified 98.5%

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

   9. Taylor expanded in uy around 0 95.5%

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

       1. associate-*r* 95.5%

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

   11. Simplified 95.5%

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

   if 0.0160000008 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 47.9%

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

      1. associate-*l* 47.9%

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

      2. sub-neg 47.9%

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

      3. +-commutative 47.9%

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

      4. distribute-rgt-neg-in 47.9%

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

      5. fma-define 47.8%

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

   3. Simplified 48.0%

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

   5. Taylor expanded in maxCos around 0 47.5%

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

   6. Taylor expanded in ux around 0 77.0%

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

      1. *-commutative 77.0%

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

   8. Simplified 77.0%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.01600000075995922:\\ \;\;\;\;\sqrt{ux \cdot \left(1 + \left(\left(\left(1 - maxCos\right) + \left(ux \cdot \left(1 - maxCos\right)\right) \cdot \left(maxCos + -1\right)\right) - maxCos\right)\right)} \cdot \left(\pi \cdot \left(uy \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 54.3%

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

   1. associate-*l* 54.3%

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

   2. sub-neg 54.3%

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

   3. +-commutative 54.3%

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

   4. distribute-rgt-neg-in 54.3%

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

   5. fma-define 54.3%

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

3. Simplified 54.4%

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

5. Taylor expanded in ux around inf 98.3%

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

6. Taylor expanded in ux around 0 98.5%

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

   1. associate--l+ 98.4%

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

   2. sub-neg 98.4%

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

   3. metadata-eval 98.4%

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

   4. +-commutative 98.4%

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

   5. distribute-lft-in 98.4%

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

   6. metadata-eval 98.4%

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

   7. neg-mul-1 98.4%

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

   8. sub-neg 98.4%

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

   9. associate-*r* 98.4%

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

   10. sub-neg 98.4%

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

   11. metadata-eval 98.4%

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

8. Simplified 98.4%

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

9. Taylor expanded in uy around 0 82.2%

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

    1. associate-*r* 82.2%

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

11. Simplified 82.2%

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

12. Final simplification 82.2%

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

Alternative 8: 76.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 54.3%

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

3. Taylor expanded in ux around 0 98.4%

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

   1. associate--l+ 98.4%

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

   2. associate-*r* 98.4%

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

   3. mul-1-neg 98.4%

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

   4. sub-neg 98.4%

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

   5. metadata-eval 98.4%

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

   6. +-commutative 98.4%

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

5. Simplified 98.4%

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

6. Taylor expanded in maxCos around 0 92.6%

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

   1. neg-mul-1 92.6%

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

   2. unsub-neg 92.6%

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

8. Simplified 92.6%

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

9. Taylor expanded in uy around 0 78.1%

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

    1. associate-*r* 78.1%

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

11. Simplified 78.1%

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

    1. pow1 78.1%

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

    2. associate-*l* 78.1%

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

13. Applied egg-rr 78.1%

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

    1. unpow1 78.1%

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

    2. *-commutative 78.1%

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

15. Simplified 78.1%

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

16. Final simplification 78.1%

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

Alternative 9: 63.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 54.3%

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

3. Taylor expanded in ux around 0 98.4%

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

   1. associate--l+ 98.4%

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

   2. associate-*r* 98.4%

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

   3. mul-1-neg 98.4%

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

   4. sub-neg 98.4%

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

   5. metadata-eval 98.4%

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

   6. +-commutative 98.4%

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

5. Simplified 98.4%

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

6. Taylor expanded in maxCos around 0 92.6%

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

   1. neg-mul-1 92.6%

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

   2. unsub-neg 92.6%

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

8. Simplified 92.6%

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

9. Taylor expanded in uy around 0 78.1%

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

    1. associate-*r* 78.1%

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

11. Simplified 78.1%

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

12. Taylor expanded in ux around 0 65.3%

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

13. Final simplification 65.3%

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

Reproduce

herbie shell --seed 2024130 
(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)))))))