UniformSampleCone, y

Percentage Accurate: 57.7% → 98.4%

Time: 17.9s

Alternatives: 16

Speedup: 1.5×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.7%

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

   1. associate-*l* 55.7%

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

   2. sub-neg 55.7%

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

   3. +-commutative 55.7%

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

   4. distribute-rgt-neg-in 55.7%

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

   5. fma-def 55.9%

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

3. Simplified 55.9%

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

4. Taylor expanded in ux around -inf 98.3%

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

   1. +-commutative 98.3%

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

   2. mul-1-neg 98.3%

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

   3. unsub-neg 98.3%

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

   4. *-commutative 98.3%

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

   5. mul-1-neg 98.3%

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

   6. sub-neg 98.3%

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

   7. sub-neg 98.3%

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

   8. metadata-eval 98.3%

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

   9. +-commutative 98.3%

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

   10. sub-neg 98.3%

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

   11. mul-1-neg 98.3%

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

   12. unsub-neg 98.3%

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

   13. mul-1-neg 98.3%

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

   14. sub-neg 98.3%

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

   15. metadata-eval 98.3%

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

6. Simplified 98.3%

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

8. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{\sqrt[3]{{\left({ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(-1 + maxCos\right)\right) - ux \cdot \left(-1 + \left(\left(-1 + maxCos\right) + maxCos\right)\right)\right)}^{1.5} \cdot {\sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}^{3}}} \]

9. Final simplification 98.4%

   \[\leadsto \sqrt[3]{{\left({ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right) + ux \cdot \left(\left(\left(1 - maxCos\right) - maxCos\right) - -1\right)\right)}^{1.5} \cdot {\sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}^{3}} \]

Alternative 2: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right) \cdot \sqrt{{ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right) + ux \cdot \left(\left(\left(1 - maxCos\right) - maxCos\right) - -1\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (log1p (expm1 (sin (* 2.0 (* uy PI)))))
  (sqrt
   (+
    (* (pow ux 2.0) (* (- 1.0 maxCos) (+ maxCos -1.0)))
    (* ux (- (- (- 1.0 maxCos) maxCos) -1.0))))))

C

float code(float ux, float uy, float maxCos) {
	return log1pf(expm1f(sinf((2.0f * (uy * ((float) M_PI)))))) * sqrtf(((powf(ux, 2.0f) * ((1.0f - maxCos) * (maxCos + -1.0f))) + (ux * (((1.0f - maxCos) - maxCos) - -1.0f))));
}

Julia

function code(ux, uy, maxCos)
	return Float32(log1p(expm1(sin(Float32(Float32(2.0) * Float32(uy * Float32(pi)))))) * sqrt(Float32(Float32((ux ^ Float32(2.0)) * Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos + Float32(-1.0)))) + Float32(ux * Float32(Float32(Float32(Float32(1.0) - maxCos) - maxCos) - Float32(-1.0))))))
end

Derivation

1. Initial program 55.7%

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

   1. associate-*l* 55.7%

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

   2. sub-neg 55.7%

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

   3. +-commutative 55.7%

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

   4. distribute-rgt-neg-in 55.7%

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

   5. fma-def 55.9%

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

3. Simplified 55.9%

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

4. Taylor expanded in ux around -inf 98.3%

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

   1. +-commutative 98.3%

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

   2. mul-1-neg 98.3%

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

   3. unsub-neg 98.3%

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

   4. *-commutative 98.3%

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

   5. mul-1-neg 98.3%

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

   6. sub-neg 98.3%

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

   7. sub-neg 98.3%

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

   8. metadata-eval 98.3%

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

   9. +-commutative 98.3%

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

   10. sub-neg 98.3%

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

   11. mul-1-neg 98.3%

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

   12. unsub-neg 98.3%

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

   13. mul-1-neg 98.3%

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

   14. sub-neg 98.3%

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

   15. metadata-eval 98.3%

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

6. Simplified 98.3%

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

   1. log1p-expm1-u 98.3%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right)} \cdot \sqrt{{ux}^{2} \cdot \left(\left(-1 + maxCos\right) \cdot \left(1 - maxCos\right)\right) - ux \cdot \left(\left(maxCos - \left(1 - maxCos\right)\right) + -1\right)} \]

   2. *-commutative 98.3%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \color{blue}{\left(\left(2 \cdot \pi\right) \cdot uy\right)}\right)\right) \cdot \sqrt{{ux}^{2} \cdot \left(\left(-1 + maxCos\right) \cdot \left(1 - maxCos\right)\right) - ux \cdot \left(\left(maxCos - \left(1 - maxCos\right)\right) + -1\right)} \]

   3. associate-*l* 98.3%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \color{blue}{\left(2 \cdot \left(\pi \cdot uy\right)\right)}\right)\right) \cdot \sqrt{{ux}^{2} \cdot \left(\left(-1 + maxCos\right) \cdot \left(1 - maxCos\right)\right) - ux \cdot \left(\left(maxCos - \left(1 - maxCos\right)\right) + -1\right)} \]

8. Applied egg-rr 98.3%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(2 \cdot \left(\pi \cdot uy\right)\right)\right)\right)} \cdot \sqrt{{ux}^{2} \cdot \left(\left(-1 + maxCos\right) \cdot \left(1 - maxCos\right)\right) - ux \cdot \left(\left(maxCos - \left(1 - maxCos\right)\right) + -1\right)} \]

9. Final simplification 98.3%

   \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right) \cdot \sqrt{{ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right) + ux \cdot \left(\left(\left(1 - maxCos\right) - maxCos\right) - -1\right)} \]

Alternative 3: 98.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.7%

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

   1. associate-*l* 55.7%

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

   2. sub-neg 55.7%

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

   3. +-commutative 55.7%

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

   4. distribute-rgt-neg-in 55.7%

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

   5. fma-def 55.9%

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

3. Simplified 55.9%

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

4. Taylor expanded in ux around -inf 98.3%

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

   1. +-commutative 98.3%

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

   2. mul-1-neg 98.3%

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

   3. unsub-neg 98.3%

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

   4. *-commutative 98.3%

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

   5. mul-1-neg 98.3%

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

   6. sub-neg 98.3%

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

   7. sub-neg 98.3%

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

   8. metadata-eval 98.3%

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

   9. +-commutative 98.3%

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

   10. sub-neg 98.3%

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

   11. mul-1-neg 98.3%

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

   12. unsub-neg 98.3%

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

   13. mul-1-neg 98.3%

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

   14. sub-neg 98.3%

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

   15. metadata-eval 98.3%

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

6. Simplified 98.3%

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

7. Taylor expanded in uy around inf 98.3%

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

8. Final simplification 98.3%

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

Alternative 4: 96.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.7%

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

   1. associate-*l* 55.7%

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

   2. sub-neg 55.7%

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

   3. +-commutative 55.7%

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

   4. distribute-rgt-neg-in 55.7%

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

   5. fma-def 55.9%

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

3. Simplified 55.9%

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

4. Taylor expanded in ux around -inf 98.3%

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

   1. +-commutative 98.3%

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

   2. mul-1-neg 98.3%

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

   3. unsub-neg 98.3%

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

   4. *-commutative 98.3%

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

   5. mul-1-neg 98.3%

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

   6. sub-neg 98.3%

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

   7. sub-neg 98.3%

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

   8. metadata-eval 98.3%

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

   9. +-commutative 98.3%

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

   10. sub-neg 98.3%

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

   11. mul-1-neg 98.3%

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

   12. unsub-neg 98.3%

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

   13. mul-1-neg 98.3%

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

   14. sub-neg 98.3%

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

   15. metadata-eval 98.3%

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

6. Simplified 98.3%

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

7. Taylor expanded in maxCos around 0 96.4%

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

   1. mul-1-neg 96.4%

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

9. Simplified 96.4%

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

10. Final simplification 96.4%

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

Alternative 5: 95.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 1.500000053056283 \cdot 10^{-6}:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot 2 - {ux}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right) - {ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(1 - maxCos\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 1.500000053056283e-6)
   (* (sin (* PI (* 2.0 uy))) (sqrt (- (* ux 2.0) (pow ux 2.0))))
   (*
    2.0
    (*
     (* uy PI)
     (sqrt
      (-
       (* ux (- 2.0 (* 2.0 maxCos)))
       (* (pow ux 2.0) (* (- 1.0 maxCos) (- 1.0 maxCos)))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 1.500000053056283e-6f) {
		tmp = sinf((((float) M_PI) * (2.0f * uy))) * sqrtf(((ux * 2.0f) - powf(ux, 2.0f)));
	} else {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf(((ux * (2.0f - (2.0f * maxCos))) - (powf(ux, 2.0f) * ((1.0f - maxCos) * (1.0f - maxCos))))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(1.500000053056283e-6))
		tmp = Float32(sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy))) * sqrt(Float32(Float32(ux * Float32(2.0)) - (ux ^ Float32(2.0)))));
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos))) - Float32((ux ^ Float32(2.0)) * Float32(Float32(Float32(1.0) - maxCos) * Float32(Float32(1.0) - maxCos)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (maxCos <= single(1.500000053056283e-6))
		tmp = sin((single(pi) * (single(2.0) * uy))) * sqrt(((ux * single(2.0)) - (ux ^ single(2.0))));
	else
		tmp = single(2.0) * ((uy * single(pi)) * sqrt(((ux * (single(2.0) - (single(2.0) * maxCos))) - ((ux ^ single(2.0)) * ((single(1.0) - maxCos) * (single(1.0) - maxCos))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.50000005e-6

   1. Initial program 54.9%

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

      1. associate-*l* 54.9%

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

      2. sub-neg 54.9%

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

      3. +-commutative 54.9%

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

      4. distribute-rgt-neg-in 54.9%

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

      5. fma-def 54.9%

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

   3. Simplified 54.9%

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

   4. Taylor expanded in ux around -inf 98.3%

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

      1. +-commutative 98.3%

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

      2. mul-1-neg 98.3%

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

      3. unsub-neg 98.3%

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

      4. *-commutative 98.3%

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

      5. mul-1-neg 98.3%

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

      6. sub-neg 98.3%

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

      7. sub-neg 98.3%

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

      8. metadata-eval 98.3%

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

      9. +-commutative 98.3%

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

      10. sub-neg 98.3%

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

      11. mul-1-neg 98.3%

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

      12. unsub-neg 98.3%

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

      13. mul-1-neg 98.3%

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

      14. sub-neg 98.3%

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

      15. metadata-eval 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in maxCos around 0 98.2%

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

      1. associate-*r* 98.2%

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

      2. cancel-sign-sub-inv 98.2%

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

      3. metadata-eval 98.2%

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

      4. +-commutative 98.2%

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

      5. mul-1-neg 98.2%

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

      6. unsub-neg 98.2%

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

   9. Simplified 98.2%

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

   if 1.50000005e-6 < maxCos

   1. Initial program 59.4%

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

      1. associate-*l* 59.4%

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

      2. sub-neg 59.4%

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

      3. +-commutative 59.4%

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

      4. distribute-rgt-neg-in 59.4%

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

      5. fma-def 60.0%

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

   3. Simplified 60.4%

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

   4. Taylor expanded in ux around -inf 98.4%

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

      1. +-commutative 98.4%

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

      2. mul-1-neg 98.4%

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

      3. unsub-neg 98.4%

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

      4. *-commutative 98.4%

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

      5. mul-1-neg 98.4%

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

      6. sub-neg 98.4%

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

      7. sub-neg 98.4%

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

      8. metadata-eval 98.4%

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

      9. +-commutative 98.4%

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

      10. sub-neg 98.4%

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

      11. mul-1-neg 98.4%

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

      12. unsub-neg 98.4%

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

      13. mul-1-neg 98.4%

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

      14. sub-neg 98.4%

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

      15. metadata-eval 98.4%

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

   6. Simplified 98.4%

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

   7. Taylor expanded in uy around 0 79.9%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 1.500000053056283 \cdot 10^{-6}:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot 2 - {ux}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right) - {ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(1 - maxCos\right)\right)}\right)\\ \end{array} \]

Alternative 6: 91.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ t_1 := \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\\ \mathbf{if}\;t_0 \leq 0.9998300075531006:\\ \;\;\;\;t_1 \cdot \sqrt{1 + t_0 \cdot \left(\left(ux + -1\right) - ux \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))) (t_1 (sin (* PI (* 2.0 uy)))))
   (if (<= t_0 0.9998300075531006)
     (* t_1 (sqrt (+ 1.0 (* t_0 (- (+ ux -1.0) (* ux maxCos))))))
     (* t_1 (sqrt (* ux (- 2.0 (* 2.0 maxCos))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	float t_1 = sinf((((float) M_PI) * (2.0f * uy)));
	float tmp;
	if (t_0 <= 0.9998300075531006f) {
		tmp = t_1 * sqrtf((1.0f + (t_0 * ((ux + -1.0f) - (ux * maxCos)))));
	} else {
		tmp = t_1 * sqrtf((ux * (2.0f - (2.0f * maxCos))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	t_1 = sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9998300075531006))
		tmp = Float32(t_1 * sqrt(Float32(Float32(1.0) + Float32(t_0 * Float32(Float32(ux + Float32(-1.0)) - Float32(ux * maxCos))))));
	else
		tmp = Float32(t_1 * sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = (single(1.0) - ux) + (ux * maxCos);
	t_1 = sin((single(pi) * (single(2.0) * uy)));
	tmp = single(0.0);
	if (t_0 <= single(0.9998300075531006))
		tmp = t_1 * sqrt((single(1.0) + (t_0 * ((ux + single(-1.0)) - (ux * maxCos)))));
	else
		tmp = t_1 * sqrt((ux * (single(2.0) - (single(2.0) * maxCos))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 (-.f32 1 ux) (*.f32 ux maxCos)) < 0.999830008

   1. Initial program 88.5%

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

   if 0.999830008 < (+.f32 (-.f32 1 ux) (*.f32 ux maxCos))

   1. Initial program 36.4%

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

   2. Taylor expanded in ux around 0 92.9%

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

      1. *-commutative 92.9%

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

   4. Simplified 92.9%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - ux\right) + ux \cdot maxCos \leq 0.9998300075531006:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 + \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(ux + -1\right) - ux \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \]

Alternative 7: 90.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := ux \cdot \left(2 - 2 \cdot maxCos\right)\\ \mathbf{if}\;2 \cdot uy \leq 0.003000000026077032:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{{ux}^{2} \cdot \left(-1 + 2 \cdot maxCos\right) + t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* ux (- 2.0 (* 2.0 maxCos)))))
   (if (<= (* 2.0 uy) 0.003000000026077032)
     (*
      2.0
      (* (* uy PI) (sqrt (+ (* (pow ux 2.0) (+ -1.0 (* 2.0 maxCos))) t_0))))
     (* (sin (* PI (* 2.0 uy))) (sqrt t_0)))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = ux * (2.0f - (2.0f * maxCos));
	float tmp;
	if ((2.0f * uy) <= 0.003000000026077032f) {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf(((powf(ux, 2.0f) * (-1.0f + (2.0f * maxCos))) + t_0)));
	} else {
		tmp = sinf((((float) M_PI) * (2.0f * uy))) * sqrtf(t_0);
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.003000000026077032))
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(Float32((ux ^ Float32(2.0)) * Float32(Float32(-1.0) + Float32(Float32(2.0) * maxCos))) + t_0))));
	else
		tmp = Float32(sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy))) * sqrt(t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = ux * (single(2.0) - (single(2.0) * maxCos));
	tmp = single(0.0);
	if ((single(2.0) * uy) <= single(0.003000000026077032))
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((((ux ^ single(2.0)) * (single(-1.0) + (single(2.0) * maxCos))) + t_0)));
	else
		tmp = sin((single(pi) * (single(2.0) * uy))) * sqrt(t_0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy 2) < 0.00300000003

   1. Initial program 57.6%

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

      1. associate-*l* 57.6%

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

      2. sub-neg 57.6%

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

      3. +-commutative 57.6%

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

      4. distribute-rgt-neg-in 57.6%

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

      5. fma-def 57.6%

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

   3. Simplified 57.7%

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

   4. Taylor expanded in ux around -inf 98.6%

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

      1. +-commutative 98.6%

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

      2. mul-1-neg 98.6%

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

      3. unsub-neg 98.6%

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

      4. *-commutative 98.6%

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

      5. mul-1-neg 98.6%

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

      6. sub-neg 98.6%

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

      7. sub-neg 98.6%

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

      8. metadata-eval 98.6%

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

      9. +-commutative 98.6%

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

      10. sub-neg 98.6%

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

      11. mul-1-neg 98.6%

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

      12. unsub-neg 98.6%

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

      13. mul-1-neg 98.6%

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

      14. sub-neg 98.6%

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

      15. metadata-eval 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in uy around 0 96.9%

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

   8. Taylor expanded in maxCos around 0 96.1%

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

   if 0.00300000003 < (*.f32 uy 2)

   1. Initial program 52.4%

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

   2. Taylor expanded in ux around 0 78.9%

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

      1. *-commutative 78.9%

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

   4. Simplified 78.9%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.003000000026077032:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{{ux}^{2} \cdot \left(-1 + 2 \cdot maxCos\right) + ux \cdot \left(2 - 2 \cdot maxCos\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \]

Alternative 8: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := ux \cdot \left(2 - 2 \cdot maxCos\right)\\ \mathbf{if}\;2 \cdot uy \leq 0.003000000026077032:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{t_0 - {ux}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* ux (- 2.0 (* 2.0 maxCos)))))
   (if (<= (* 2.0 uy) 0.003000000026077032)
     (* 2.0 (* (* uy PI) (sqrt (- t_0 (pow ux 2.0)))))
     (* (sin (* PI (* 2.0 uy))) (sqrt t_0)))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = ux * (2.0f - (2.0f * maxCos));
	float tmp;
	if ((2.0f * uy) <= 0.003000000026077032f) {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf((t_0 - powf(ux, 2.0f))));
	} else {
		tmp = sinf((((float) M_PI) * (2.0f * uy))) * sqrtf(t_0);
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.003000000026077032))
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(t_0 - (ux ^ Float32(2.0))))));
	else
		tmp = Float32(sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy))) * sqrt(t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = ux * (single(2.0) - (single(2.0) * maxCos));
	tmp = single(0.0);
	if ((single(2.0) * uy) <= single(0.003000000026077032))
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((t_0 - (ux ^ single(2.0)))));
	else
		tmp = sin((single(pi) * (single(2.0) * uy))) * sqrt(t_0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy 2) < 0.00300000003

   1. Initial program 57.6%

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

      1. associate-*l* 57.6%

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

      2. sub-neg 57.6%

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

      3. +-commutative 57.6%

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

      4. distribute-rgt-neg-in 57.6%

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

      5. fma-def 57.6%

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

   3. Simplified 57.7%

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

   4. Taylor expanded in ux around -inf 98.6%

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

      1. +-commutative 98.6%

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

      2. mul-1-neg 98.6%

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

      3. unsub-neg 98.6%

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

      4. *-commutative 98.6%

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

      5. mul-1-neg 98.6%

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

      6. sub-neg 98.6%

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

      7. sub-neg 98.6%

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

      8. metadata-eval 98.6%

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

      9. +-commutative 98.6%

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

      10. sub-neg 98.6%

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

      11. mul-1-neg 98.6%

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

      12. unsub-neg 98.6%

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

      13. mul-1-neg 98.6%

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

      14. sub-neg 98.6%

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

      15. metadata-eval 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in uy around 0 96.9%

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

   8. Taylor expanded in maxCos around 0 95.3%

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

   if 0.00300000003 < (*.f32 uy 2)

   1. Initial program 52.4%

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

   2. Taylor expanded in ux around 0 78.9%

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

      1. *-commutative 78.9%

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

   4. Simplified 78.9%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.003000000026077032:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right) - {ux}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \]

Alternative 9: 86.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.003000000026077032:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot 2 - {ux}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.003000000026077032)
   (* 2.0 (* (* uy PI) (sqrt (- (* ux 2.0) (pow ux 2.0)))))
   (* (sin (* PI (* 2.0 uy))) (sqrt (* ux (- 2.0 (* 2.0 maxCos)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((2.0f * uy) <= 0.003000000026077032f) {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf(((ux * 2.0f) - powf(ux, 2.0f))));
	} else {
		tmp = sinf((((float) M_PI) * (2.0f * uy))) * sqrtf((ux * (2.0f - (2.0f * maxCos))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.003000000026077032))
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(Float32(ux * Float32(2.0)) - (ux ^ Float32(2.0))))));
	else
		tmp = Float32(sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy))) * sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if ((single(2.0) * uy) <= single(0.003000000026077032))
		tmp = single(2.0) * ((uy * single(pi)) * sqrt(((ux * single(2.0)) - (ux ^ single(2.0)))));
	else
		tmp = sin((single(pi) * (single(2.0) * uy))) * sqrt((ux * (single(2.0) - (single(2.0) * maxCos))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy 2) < 0.00300000003

   1. Initial program 57.6%

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

      1. associate-*l* 57.6%

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

      2. sub-neg 57.6%

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

      3. +-commutative 57.6%

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

      4. distribute-rgt-neg-in 57.6%

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

      5. fma-def 57.6%

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

   3. Simplified 57.7%

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

   4. Taylor expanded in ux around -inf 98.6%

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

      1. +-commutative 98.6%

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

      2. mul-1-neg 98.6%

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

      3. unsub-neg 98.6%

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

      4. *-commutative 98.6%

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

      5. mul-1-neg 98.6%

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

      6. sub-neg 98.6%

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

      7. sub-neg 98.6%

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

      8. metadata-eval 98.6%

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

      9. +-commutative 98.6%

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

      10. sub-neg 98.6%

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

      11. mul-1-neg 98.6%

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

      12. unsub-neg 98.6%

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

      13. mul-1-neg 98.6%

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

      14. sub-neg 98.6%

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

      15. metadata-eval 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in uy around 0 96.9%

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

   8. Taylor expanded in maxCos around 0 89.7%

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

      1. cancel-sign-sub-inv 89.7%

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

      2. metadata-eval 89.7%

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

      3. mul-1-neg 89.7%

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

   10. Simplified 89.7%

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

   if 0.00300000003 < (*.f32 uy 2)

   1. Initial program 52.4%

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

   2. Taylor expanded in ux around 0 78.9%

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

      1. *-commutative 78.9%

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

   4. Simplified 78.9%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.003000000026077032:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot 2 - {ux}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \]

Alternative 10: 89.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\\ \mathbf{if}\;ux \leq 0.0002500000118743628:\\ \;\;\;\;t_0 \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \sqrt{1 + \left(1 - ux\right) \cdot \left(ux + -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (sin (* PI (* 2.0 uy)))))
   (if (<= ux 0.0002500000118743628)
     (* t_0 (sqrt (* ux (- 2.0 (* 2.0 maxCos)))))
     (* t_0 (sqrt (+ 1.0 (* (- 1.0 ux) (+ ux -1.0))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = sinf((((float) M_PI) * (2.0f * uy)));
	float tmp;
	if (ux <= 0.0002500000118743628f) {
		tmp = t_0 * sqrtf((ux * (2.0f - (2.0f * maxCos))));
	} else {
		tmp = t_0 * sqrtf((1.0f + ((1.0f - ux) * (ux + -1.0f))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy)))
	tmp = Float32(0.0)
	if (ux <= Float32(0.0002500000118743628))
		tmp = Float32(t_0 * sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))));
	else
		tmp = Float32(t_0 * sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = sin((single(pi) * (single(2.0) * uy)));
	tmp = single(0.0);
	if (ux <= single(0.0002500000118743628))
		tmp = t_0 * sqrt((ux * (single(2.0) - (single(2.0) * maxCos))));
	else
		tmp = t_0 * sqrt((single(1.0) + ((single(1.0) - ux) * (ux + single(-1.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 2.50000012e-4

   1. Initial program 37.5%

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

   2. Taylor expanded in ux around 0 92.3%

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

      1. *-commutative 92.3%

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

   4. Simplified 92.3%

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

   if 2.50000012e-4 < ux

   1. Initial program 89.3%

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

      1. associate-*l* 89.3%

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

      2. sub-neg 89.3%

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

      3. +-commutative 89.3%

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

      4. distribute-rgt-neg-in 89.3%

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

      5. fma-def 89.6%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in uy around inf 89.2%

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

   5. Taylor expanded in maxCos around 0 82.2%

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

      1. associate-*r* 82.2%

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

      2. *-commutative 82.2%

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

      3. *-commutative 82.2%

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

      4. *-commutative 82.2%

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

      5. mul-1-neg 82.2%

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

      6. unsub-neg 82.2%

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

      7. mul-1-neg 82.2%

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

      8. sub-neg 82.2%

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

   7. Simplified 82.2%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.0002500000118743628:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 + \left(1 - ux\right) \cdot \left(ux + -1\right)}\\ \end{array} \]

Alternative 11: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.003000000026077032:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot 2 - {ux}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.003000000026077032)
   (* 2.0 (* (* uy PI) (sqrt (- (* ux 2.0) (pow ux 2.0)))))
   (* (sin (* uy (* 2.0 PI))) (sqrt (* ux 2.0)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((2.0f * uy) <= 0.003000000026077032f) {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf(((ux * 2.0f) - powf(ux, 2.0f))));
	} else {
		tmp = sinf((uy * (2.0f * ((float) M_PI)))) * sqrtf((ux * 2.0f));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.003000000026077032))
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(Float32(ux * Float32(2.0)) - (ux ^ Float32(2.0))))));
	else
		tmp = Float32(sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(ux * Float32(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if ((single(2.0) * uy) <= single(0.003000000026077032))
		tmp = single(2.0) * ((uy * single(pi)) * sqrt(((ux * single(2.0)) - (ux ^ single(2.0)))));
	else
		tmp = sin((uy * (single(2.0) * single(pi)))) * sqrt((ux * single(2.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy 2) < 0.00300000003

   1. Initial program 57.6%

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

      1. associate-*l* 57.6%

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

      2. sub-neg 57.6%

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

      3. +-commutative 57.6%

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

      4. distribute-rgt-neg-in 57.6%

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

      5. fma-def 57.6%

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

   3. Simplified 57.7%

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

   4. Taylor expanded in ux around -inf 98.6%

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

      1. +-commutative 98.6%

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

      2. mul-1-neg 98.6%

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

      3. unsub-neg 98.6%

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

      4. *-commutative 98.6%

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

      5. mul-1-neg 98.6%

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

      6. sub-neg 98.6%

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

      7. sub-neg 98.6%

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

      8. metadata-eval 98.6%

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

      9. +-commutative 98.6%

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

      10. sub-neg 98.6%

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

      11. mul-1-neg 98.6%

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

      12. unsub-neg 98.6%

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

      13. mul-1-neg 98.6%

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

      14. sub-neg 98.6%

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

      15. metadata-eval 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in uy around 0 96.9%

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

   8. Taylor expanded in maxCos around 0 89.7%

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

      1. cancel-sign-sub-inv 89.7%

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

      2. metadata-eval 89.7%

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

      3. mul-1-neg 89.7%

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

   10. Simplified 89.7%

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

   if 0.00300000003 < (*.f32 uy 2)

   1. Initial program 52.4%

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

      1. associate-*l* 52.4%

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

      2. sub-neg 52.4%

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

      3. +-commutative 52.4%

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

      4. distribute-rgt-neg-in 52.4%

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

      5. fma-def 52.7%

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

   3. Simplified 52.9%

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

   4. Taylor expanded in maxCos around 0 49.0%

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

   5. Taylor expanded in ux around 0 73.5%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.003000000026077032:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot 2 - {ux}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot 2}\\ \end{array} \]

Alternative 12: 82.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 4.5000000682193786 \cdot 10^{-5}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 + \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux + \left(-1 - ux \cdot maxCos\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 4.5000000682193786e-5)
   (* (sin (* uy (* 2.0 PI))) (sqrt (* ux 2.0)))
   (*
    2.0
    (*
     (* uy PI)
     (sqrt
      (+
       1.0
       (* (+ 1.0 (* ux (+ maxCos -1.0))) (+ ux (- -1.0 (* ux maxCos))))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 4.5000000682193786e-5f) {
		tmp = sinf((uy * (2.0f * ((float) M_PI)))) * sqrtf((ux * 2.0f));
	} else {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf((1.0f + ((1.0f + (ux * (maxCos + -1.0f))) * (ux + (-1.0f - (ux * maxCos)))))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(4.5000000682193786e-5))
		tmp = Float32(sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(ux * Float32(2.0))));
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) + Float32(ux * Float32(maxCos + Float32(-1.0)))) * Float32(ux + Float32(Float32(-1.0) - Float32(ux * maxCos))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(4.5000000682193786e-5))
		tmp = sin((uy * (single(2.0) * single(pi)))) * sqrt((ux * single(2.0)));
	else
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((single(1.0) + ((single(1.0) + (ux * (maxCos + single(-1.0)))) * (ux + (single(-1.0) - (ux * maxCos)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 4.50000007e-5

   1. Initial program 34.1%

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

      1. associate-*l* 34.1%

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

      2. sub-neg 34.1%

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

      3. +-commutative 34.1%

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

      4. distribute-rgt-neg-in 34.1%

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

      5. fma-def 34.1%

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

   3. Simplified 34.4%

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

   4. Taylor expanded in maxCos around 0 34.1%

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

   5. Taylor expanded in ux around 0 88.3%

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

   if 4.50000007e-5 < ux

   1. Initial program 86.7%

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

      1. associate-*l* 86.7%

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

      2. sub-neg 86.7%

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

      3. +-commutative 86.7%

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

      4. distribute-rgt-neg-in 86.7%

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

      5. fma-def 87.1%

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

   3. Simplified 87.0%

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

   4. Taylor expanded in uy around 0 71.7%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 4.5000000682193786 \cdot 10^{-5}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 + \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux + \left(-1 - ux \cdot maxCos\right)\right)}\right)\\ \end{array} \]

Alternative 13: 77.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00016999999934341758:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(--2\right) - 2 \cdot \left(ux \cdot maxCos\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 + \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux + \left(-1 - ux \cdot maxCos\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00016999999934341758)
   (* 2.0 (* (* uy PI) (sqrt (- (* ux (- -2.0)) (* 2.0 (* ux maxCos))))))
   (*
    2.0
    (*
     (* uy PI)
     (sqrt
      (+
       1.0
       (* (+ 1.0 (* ux (+ maxCos -1.0))) (+ ux (- -1.0 (* ux maxCos))))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00016999999934341758f) {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf(((ux * -(-2.0f)) - (2.0f * (ux * maxCos)))));
	} else {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf((1.0f + ((1.0f + (ux * (maxCos + -1.0f))) * (ux + (-1.0f - (ux * maxCos)))))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00016999999934341758))
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(Float32(ux * Float32(-Float32(-2.0))) - Float32(Float32(2.0) * Float32(ux * 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(maxCos + Float32(-1.0)))) * Float32(ux + Float32(Float32(-1.0) - Float32(ux * maxCos))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00016999999934341758))
		tmp = single(2.0) * ((uy * single(pi)) * sqrt(((ux * -single(-2.0)) - (single(2.0) * (ux * maxCos)))));
	else
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((single(1.0) + ((single(1.0) + (ux * (maxCos + single(-1.0)))) * (ux + (single(-1.0) - (ux * maxCos)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.69999999e-4

   1. Initial program 36.4%

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

      1. associate-*l* 36.4%

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

      2. sub-neg 36.4%

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

      3. +-commutative 36.4%

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

      4. distribute-rgt-neg-in 36.4%

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

      5. fma-def 36.5%

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

   3. Simplified 36.7%

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

   4. Taylor expanded in uy around 0 33.7%

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

   5. Taylor expanded in ux around 0 73.5%

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

   6. Taylor expanded in maxCos around 0 73.5%

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

   if 1.69999999e-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-def 88.7%

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

   3. Simplified 88.5%

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

   4. Taylor expanded in uy around 0 72.0%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00016999999934341758:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(--2\right) - 2 \cdot \left(ux \cdot maxCos\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 + \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux + \left(-1 - ux \cdot maxCos\right)\right)}\right)\\ \end{array} \]

Alternative 14: 75.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.0002500000118743628:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(--2\right) - 2 \cdot \left(ux \cdot maxCos\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 + \left(1 - ux\right) \cdot \left(ux + -1\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.0002500000118743628)
   (* 2.0 (* (* uy PI) (sqrt (- (* ux (- -2.0)) (* 2.0 (* ux maxCos))))))
   (* 2.0 (* (* uy PI) (sqrt (+ 1.0 (* (- 1.0 ux) (+ ux -1.0))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.0002500000118743628f) {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf(((ux * -(-2.0f)) - (2.0f * (ux * maxCos)))));
	} else {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf((1.0f + ((1.0f - ux) * (ux + -1.0f)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.0002500000118743628))
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(Float32(ux * Float32(-Float32(-2.0))) - Float32(Float32(2.0) * Float32(ux * maxCos))))));
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.0002500000118743628))
		tmp = single(2.0) * ((uy * single(pi)) * sqrt(((ux * -single(-2.0)) - (single(2.0) * (ux * maxCos)))));
	else
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((single(1.0) + ((single(1.0) - ux) * (ux + single(-1.0))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 2.50000012e-4

   1. Initial program 37.5%

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

      1. associate-*l* 37.5%

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

      2. sub-neg 37.5%

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

      3. +-commutative 37.5%

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

      4. distribute-rgt-neg-in 37.5%

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

      5. fma-def 37.6%

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

   3. Simplified 37.8%

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

   4. Taylor expanded in uy around 0 34.6%

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

   5. Taylor expanded in ux around 0 73.1%

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

   6. Taylor expanded in maxCos around 0 73.1%

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

   if 2.50000012e-4 < ux

   1. Initial program 89.3%

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

      1. associate-*l* 89.3%

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

      2. sub-neg 89.3%

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

      3. +-commutative 89.3%

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

      4. distribute-rgt-neg-in 89.3%

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

      5. fma-def 89.6%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in uy around 0 72.4%

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

   5. Taylor expanded in maxCos around 0 68.3%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.0002500000118743628:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(--2\right) - 2 \cdot \left(ux \cdot maxCos\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 + \left(1 - ux\right) \cdot \left(ux + -1\right)}\right)\\ \end{array} \]

Alternative 15: 66.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.7%

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

   1. associate-*l* 55.7%

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

   2. sub-neg 55.7%

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

   3. +-commutative 55.7%

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

   4. distribute-rgt-neg-in 55.7%

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

   5. fma-def 55.9%

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

3. Simplified 55.9%

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

4. Taylor expanded in uy around 0 47.9%

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

5. Taylor expanded in ux around 0 64.6%

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

6. Final simplification 64.6%

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

Alternative 16: 63.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.7%

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

   1. associate-*l* 55.7%

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

   2. sub-neg 55.7%

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

   3. +-commutative 55.7%

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

   4. distribute-rgt-neg-in 55.7%

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

   5. fma-def 55.9%

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

3. Simplified 55.9%

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

4. Taylor expanded in uy around 0 47.9%

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

5. Taylor expanded in ux around 0 64.6%

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

6. Taylor expanded in maxCos around 0 61.6%

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

   1. *-commutative 61.6%

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

8. Simplified 61.6%

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

9. Final simplification 61.6%

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

Reproduce

herbie shell --seed 2023332 
(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)))))))