UniformSampleCone, y

Percentage Accurate: 57.0% → 98.4%

Time: 22.1s

Alternatives: 15

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: 57.0% 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, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.2%

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

   1. associate-*l* 57.2%

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

   2. sub-neg 57.2%

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

   3. +-commutative 57.2%

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

   4. distribute-rgt-neg-in 57.2%

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

   5. fma-def 57.3%

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

3. Simplified 57.4%

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

4. Taylor expanded in ux around 0 98.2%

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

   1. add-sqr-sqrt 98.5%

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

   2. pow2 98.5%

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

   3. +-commutative 98.5%

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

   4. fma-def 98.5%

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

   5. sub-neg 98.5%

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

   6. metadata-eval 98.5%

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

   7. +-commutative 98.5%

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

6. Applied egg-rr 98.5%

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

7. Final simplification 98.5%

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

Alternative 2: 98.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.2%

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

   1. associate-*l* 57.2%

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

   2. sub-neg 57.2%

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

   3. +-commutative 57.2%

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

   4. distribute-rgt-neg-in 57.2%

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

   5. fma-def 57.3%

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

3. Simplified 57.4%

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

4. Taylor expanded in ux around -inf 98.2%

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

   1. +-commutative 98.2%

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

   2. mul-1-neg 98.2%

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

   3. unsub-neg 98.2%

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

   4. *-commutative 98.2%

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

   5. mul-1-neg 98.2%

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

   6. sub-neg 98.2%

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

   7. sub-neg 98.2%

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

   8. metadata-eval 98.2%

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

   9. +-commutative 98.2%

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

   10. sub-neg 98.2%

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

   11. mul-1-neg 98.2%

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

   12. unsub-neg 98.2%

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

   13. mul-1-neg 98.2%

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

   14. sub-neg 98.2%

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

   15. metadata-eval 98.2%

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

6. Simplified 98.2%

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

   1. add-cbrt-cube 98.2%

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

   2. pow1/3 96.0%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \color{blue}{{\left(\left(\sqrt{{ux}^{2} \cdot \left(\left(-1 + maxCos\right) \cdot \left(1 - maxCos\right)\right) - ux \cdot \left(\left(maxCos - \left(1 - maxCos\right)\right) + -1\right)} \cdot \sqrt{{ux}^{2} \cdot \left(\left(-1 + maxCos\right) \cdot \left(1 - maxCos\right)\right) - ux \cdot \left(\left(maxCos - \left(1 - maxCos\right)\right) + -1\right)}\right) \cdot \sqrt{{ux}^{2} \cdot \left(\left(-1 + maxCos\right) \cdot \left(1 - maxCos\right)\right) - ux \cdot \left(\left(maxCos - \left(1 - maxCos\right)\right) + -1\right)}\right)}^{0.3333333333333333}} \]

8. Applied egg-rr 96.1%

   \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \color{blue}{{\left({\left(\left({ux}^{2} \cdot \left(1 - maxCos\right)\right) \cdot \left(-1 + maxCos\right) - ux \cdot \left(-1 + \left(\left(-1 + maxCos\right) + maxCos\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
9. Step-by-step derivation

   1. unpow1/3 98.3%

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

10. Simplified 98.3%

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

11. Final simplification 98.3%

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

Alternative 3: 95.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.00018780000391416252:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{\mathsf{fma}\left(ux, 1 + \left(\left(1 - maxCos\right) - maxCos\right), \left(-1 + maxCos\right) \cdot \left({ux}^{2} \cdot \left(1 - maxCos\right)\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux - {ux}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.00018780000391416252)
   (*
    2.0
    (*
     uy
     (*
      PI
      (sqrt
       (fma
        ux
        (+ 1.0 (- (- 1.0 maxCos) maxCos))
        (* (+ -1.0 maxCos) (* (pow ux 2.0) (- 1.0 maxCos))))))))
   (* (sin (* uy (* 2.0 PI))) (sqrt (- (* 2.0 ux) (pow ux 2.0))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.00018780000391416252f) {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf(fmaf(ux, (1.0f + ((1.0f - maxCos) - maxCos)), ((-1.0f + maxCos) * (powf(ux, 2.0f) * (1.0f - maxCos)))))));
	} else {
		tmp = sinf((uy * (2.0f * ((float) M_PI)))) * sqrtf(((2.0f * ux) - powf(ux, 2.0f)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.00018780000391416252))
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(fma(ux, Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - maxCos) - maxCos)), Float32(Float32(Float32(-1.0) + maxCos) * Float32((ux ^ Float32(2.0)) * Float32(Float32(1.0) - maxCos))))))));
	else
		tmp = Float32(sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(Float32(2.0) * ux) - (ux ^ Float32(2.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy 2) < 1.87800004e-4

   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 59.6%

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

   3. Simplified 59.8%

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

   4. Taylor expanded in ux around 0 98.4%

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

   5. Taylor expanded in uy around 0 98.3%

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

   7. Applied egg-rr 17.9%

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

   9. Simplified 98.4%

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

   if 1.87800004e-4 < (*.f32 uy 2)

   1. Initial program 53.8%

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

      1. associate-*l* 53.8%

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

      2. sub-neg 53.8%

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

      3. +-commutative 53.8%

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

      4. distribute-rgt-neg-in 53.8%

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

      5. fma-def 53.9%

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

   3. Simplified 53.9%

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

   4. Taylor expanded in ux around -inf 98.0%

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

      1. +-commutative 98.0%

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

      2. mul-1-neg 98.0%

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

      3. unsub-neg 98.0%

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

      4. *-commutative 98.0%

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

      5. mul-1-neg 98.0%

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

      6. sub-neg 98.0%

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

      7. sub-neg 98.0%

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

      8. metadata-eval 98.0%

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

      9. +-commutative 98.0%

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

      10. sub-neg 98.0%

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

      11. mul-1-neg 98.0%

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

      12. unsub-neg 98.0%

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

      13. mul-1-neg 98.0%

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

      14. sub-neg 98.0%

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

      15. metadata-eval 98.0%

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

   6. Simplified 98.0%

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

   7. Taylor expanded in maxCos around 0 94.5%

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

      1. associate-*r* 94.5%

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

      2. *-commutative 94.5%

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

      3. associate-*l* 94.5%

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

      4. cancel-sign-sub-inv 94.5%

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

      5. metadata-eval 94.5%

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

      6. +-commutative 94.5%

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

      7. mul-1-neg 94.5%

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

      8. unsub-neg 94.5%

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

   9. Simplified 94.5%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.00018780000391416252:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{\mathsf{fma}\left(ux, 1 + \left(\left(1 - maxCos\right) - maxCos\right), \left(-1 + maxCos\right) \cdot \left({ux}^{2} \cdot \left(1 - maxCos\right)\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux - {ux}^{2}}\\ \end{array} \]

Alternative 4: 98.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.2%

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

   1. associate-*l* 57.2%

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

   2. sub-neg 57.2%

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

   3. +-commutative 57.2%

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

   4. distribute-rgt-neg-in 57.2%

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

   5. fma-def 57.3%

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

3. Simplified 57.4%

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

4. Taylor expanded in ux around 0 98.2%

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

5. Taylor expanded in maxCos around 0 98.2%

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

   1. *-commutative 98.2%

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

7. Simplified 98.2%

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

8. Final simplification 98.2%

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

Alternative 5: 95.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.00018780000391416252:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{{ux}^{2} \cdot \left(\left(-1 + maxCos\right) \cdot \left(1 - maxCos\right)\right) - ux \cdot \left(2 \cdot maxCos - 2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux - {ux}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.00018780000391416252)
   (*
    2.0
    (*
     (* uy PI)
     (sqrt
      (-
       (* (pow ux 2.0) (* (+ -1.0 maxCos) (- 1.0 maxCos)))
       (* ux (- (* 2.0 maxCos) 2.0))))))
   (* (sin (* uy (* 2.0 PI))) (sqrt (- (* 2.0 ux) (pow ux 2.0))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.00018780000391416252f) {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf(((powf(ux, 2.0f) * ((-1.0f + maxCos) * (1.0f - maxCos))) - (ux * ((2.0f * maxCos) - 2.0f)))));
	} else {
		tmp = sinf((uy * (2.0f * ((float) M_PI)))) * sqrtf(((2.0f * ux) - powf(ux, 2.0f)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.00018780000391416252))
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(Float32((ux ^ Float32(2.0)) * Float32(Float32(Float32(-1.0) + maxCos) * Float32(Float32(1.0) - maxCos))) - Float32(ux * Float32(Float32(Float32(2.0) * maxCos) - Float32(2.0)))))));
	else
		tmp = Float32(sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(Float32(2.0) * ux) - (ux ^ Float32(2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if ((uy * single(2.0)) <= single(0.00018780000391416252))
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((((ux ^ single(2.0)) * ((single(-1.0) + maxCos) * (single(1.0) - maxCos))) - (ux * ((single(2.0) * maxCos) - single(2.0))))));
	else
		tmp = sin((uy * (single(2.0) * single(pi)))) * sqrt(((single(2.0) * ux) - (ux ^ single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy 2) < 1.87800004e-4

   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 59.6%

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

   3. Simplified 59.8%

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

   4. Taylor expanded in 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 98.4%

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

   if 1.87800004e-4 < (*.f32 uy 2)

   1. Initial program 53.8%

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

      1. associate-*l* 53.8%

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

      2. sub-neg 53.8%

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

      3. +-commutative 53.8%

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

      4. distribute-rgt-neg-in 53.8%

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

      5. fma-def 53.9%

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

   3. Simplified 53.9%

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

   4. Taylor expanded in ux around -inf 98.0%

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

      1. +-commutative 98.0%

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

      2. mul-1-neg 98.0%

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

      3. unsub-neg 98.0%

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

      4. *-commutative 98.0%

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

      5. mul-1-neg 98.0%

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

      6. sub-neg 98.0%

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

      7. sub-neg 98.0%

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

      8. metadata-eval 98.0%

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

      9. +-commutative 98.0%

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

      10. sub-neg 98.0%

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

      11. mul-1-neg 98.0%

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

      12. unsub-neg 98.0%

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

      13. mul-1-neg 98.0%

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

      14. sub-neg 98.0%

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

      15. metadata-eval 98.0%

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

   6. Simplified 98.0%

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

   7. Taylor expanded in maxCos around 0 94.5%

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

      1. associate-*r* 94.5%

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

      2. *-commutative 94.5%

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

      3. associate-*l* 94.5%

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

      4. cancel-sign-sub-inv 94.5%

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

      5. metadata-eval 94.5%

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

      6. +-commutative 94.5%

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

      7. mul-1-neg 94.5%

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

      8. unsub-neg 94.5%

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

   9. Simplified 94.5%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.00018780000391416252:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{{ux}^{2} \cdot \left(\left(-1 + maxCos\right) \cdot \left(1 - maxCos\right)\right) - ux \cdot \left(2 \cdot maxCos - 2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux - {ux}^{2}}\\ \end{array} \]

Alternative 6: 91.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ t_1 := \sin \left(\pi \cdot \left(uy \cdot 2\right)\right)\\ \mathbf{if}\;t_0 \leq 0.9998599886894226:\\ \;\;\;\;t_1 \cdot \sqrt{1 + t_0 \cdot \left(\left(ux + -1\right) - ux \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))) (t_1 (sin (* PI (* uy 2.0)))))
   (if (<= t_0 0.9998599886894226)
     (* 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) * (uy * 2.0f)));
	float tmp;
	if (t_0 <= 0.9998599886894226f) {
		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(uy * Float32(2.0))))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9998599886894226))
		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) * (uy * single(2.0))));
	tmp = single(0.0);
	if (t_0 <= single(0.9998599886894226))
		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)) < 0.999859989

   1. Initial program 88.2%

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

   if 0.999859989 < (+.f32 (-.f32 1 ux) (*.f32 ux maxCos))

   1. Initial program 38.9%

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

   2. Taylor expanded in ux around 0 91.2%

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

      1. *-commutative 91.2%

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

   4. Simplified 91.2%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - ux\right) + ux \cdot maxCos \leq 0.9998599886894226:\\ \;\;\;\;\sin \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{1 + \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(ux + -1\right) - ux \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \]

Alternative 7: 90.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0024999999441206455:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{{ux}^{2} \cdot \left(\left(-1 + maxCos\right) \cdot \left(1 - maxCos\right)\right) - ux \cdot \left(2 \cdot maxCos - 2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(2 - maxCos\right) - maxCos\right)} \cdot \sin \left(\pi \cdot \left(uy \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.0024999999441206455)
   (*
    2.0
    (*
     (* uy PI)
     (sqrt
      (-
       (* (pow ux 2.0) (* (+ -1.0 maxCos) (- 1.0 maxCos)))
       (* ux (- (* 2.0 maxCos) 2.0))))))
   (* (sqrt (* ux (- (- 2.0 maxCos) maxCos))) (sin (* PI (* uy 2.0))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.0024999999441206455f) {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf(((powf(ux, 2.0f) * ((-1.0f + maxCos) * (1.0f - maxCos))) - (ux * ((2.0f * maxCos) - 2.0f)))));
	} else {
		tmp = sqrtf((ux * ((2.0f - maxCos) - maxCos))) * sinf((((float) M_PI) * (uy * 2.0f)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.0024999999441206455))
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(Float32((ux ^ Float32(2.0)) * Float32(Float32(Float32(-1.0) + maxCos) * Float32(Float32(1.0) - maxCos))) - Float32(ux * Float32(Float32(Float32(2.0) * maxCos) - Float32(2.0)))))));
	else
		tmp = Float32(sqrt(Float32(ux * Float32(Float32(Float32(2.0) - maxCos) - maxCos))) * sin(Float32(Float32(pi) * Float32(uy * Float32(2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if ((uy * single(2.0)) <= single(0.0024999999441206455))
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((((ux ^ single(2.0)) * ((single(-1.0) + maxCos) * (single(1.0) - maxCos))) - (ux * ((single(2.0) * maxCos) - single(2.0))))));
	else
		tmp = sqrt((ux * ((single(2.0) - maxCos) - maxCos))) * sin((single(pi) * (uy * single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy 2) < 0.00249999994

   1. Initial program 58.6%

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

      1. associate-*l* 58.6%

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

      2. sub-neg 58.6%

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

      3. +-commutative 58.6%

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

      4. distribute-rgt-neg-in 58.6%

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

      5. fma-def 58.7%

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

   3. Simplified 58.9%

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

   4. Taylor expanded in ux around -inf 98.4%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{-1 \cdot \left(ux \cdot \left(\left(maxCos + -1 \cdot \left(1 + -1 \cdot maxCos\right)\right) - 1\right)\right) + {ux}^{2} \cdot \left(\left(1 + -1 \cdot maxCos\right) \cdot \left(maxCos - 1\right)\right)}} \]
   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 97.0%

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

   if 0.00249999994 < (*.f32 uy 2)

   1. Initial program 53.5%

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

      1. associate-*l* 53.5%

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

      2. sub-neg 53.5%

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

      3. +-commutative 53.5%

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

      4. distribute-rgt-neg-in 53.5%

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

      5. fma-def 53.5%

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

   3. Simplified 53.6%

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

   4. Taylor expanded in ux around 0 80.4%

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

      1. associate-*r* 80.4%

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

      2. add-exp-log 80.3%

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

      3. associate-*r* 80.3%

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

      4. *-commutative 80.3%

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

      5. associate-*l* 80.3%

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

   6. Applied egg-rr 80.3%

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

      1. rem-exp-log 80.4%

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

      2. associate-*r* 80.4%

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

      3. *-commutative 80.4%

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

      4. add-cbrt-cube 80.4%

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

      5. add-cbrt-cube 80.5%

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

      6. cbrt-unprod 80.5%

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

   8. Applied egg-rr 80.3%

      \[\leadsto \color{blue}{\sqrt[3]{{\sin \left(\pi \cdot \left(2 \cdot uy\right)\right)}^{3} \cdot {\left(\left(\mathsf{fma}\left(-1, -1 + maxCos, 1\right) - maxCos\right) \cdot ux\right)}^{1.5}}} \]
   9. Step-by-step derivation

      1. *-commutative 80.3%

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

      2. associate-*r* 80.3%

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

      3. *-commutative 80.3%

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

      4. fma-udef 80.3%

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

      5. neg-mul-1 80.3%

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

      6. distribute-neg-in 80.3%

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

      7. metadata-eval 80.3%

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

      8. sub-neg 80.3%

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

      9. +-commutative 80.3%

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

      10. associate-+r- 80.3%

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

   10. Simplified 80.3%

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

       1. expm1-log1p-u 80.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{\sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}^{3} \cdot {\left(ux \cdot \left(1 + \left(\left(1 - maxCos\right) - maxCos\right)\right)\right)}^{1.5}}\right)\right)} \]

       2. expm1-udef 62.1%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{{\sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}^{3} \cdot {\left(ux \cdot \left(1 + \left(\left(1 - maxCos\right) - maxCos\right)\right)\right)}^{1.5}}\right)} - 1} \]

   12. Applied egg-rr 62.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{ux \cdot \left(1 + \left(\left(1 - maxCos\right) - maxCos\right)\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 80.4%

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

       2. expm1-log1p 80.4%

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

       3. *-commutative 80.4%

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

       4. associate-*r* 80.4%

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

       5. *-commutative 80.4%

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

       6. *-commutative 80.4%

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

       7. *-commutative 80.4%

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

       8. associate-+r- 80.4%

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

       9. associate-+r- 80.4%

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

       10. metadata-eval 80.4%

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

   14. Simplified 80.4%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0024999999441206455:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{{ux}^{2} \cdot \left(\left(-1 + maxCos\right) \cdot \left(1 - maxCos\right)\right) - ux \cdot \left(2 \cdot maxCos - 2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(2 - maxCos\right) - maxCos\right)} \cdot \sin \left(\pi \cdot \left(uy \cdot 2\right)\right)\\ \end{array} \]

Alternative 8: 86.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0024999999441206455:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{2 \cdot ux - {ux}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(2 - maxCos\right) - maxCos\right)} \cdot \sin \left(\pi \cdot \left(uy \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.0024999999441206455)
   (* 2.0 (* uy (* PI (sqrt (- (* 2.0 ux) (pow ux 2.0))))))
   (* (sqrt (* ux (- (- 2.0 maxCos) maxCos))) (sin (* PI (* uy 2.0))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.0024999999441206455f) {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf(((2.0f * ux) - powf(ux, 2.0f)))));
	} else {
		tmp = sqrtf((ux * ((2.0f - maxCos) - maxCos))) * sinf((((float) M_PI) * (uy * 2.0f)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.0024999999441206455))
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(Float32(Float32(2.0) * ux) - (ux ^ Float32(2.0)))))));
	else
		tmp = Float32(sqrt(Float32(ux * Float32(Float32(Float32(2.0) - maxCos) - maxCos))) * sin(Float32(Float32(pi) * Float32(uy * Float32(2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if ((uy * single(2.0)) <= single(0.0024999999441206455))
		tmp = single(2.0) * (uy * (single(pi) * sqrt(((single(2.0) * ux) - (ux ^ single(2.0))))));
	else
		tmp = sqrt((ux * ((single(2.0) - maxCos) - maxCos))) * sin((single(pi) * (uy * single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy 2) < 0.00249999994

   1. Initial program 58.6%

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

      1. associate-*l* 58.6%

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

      2. sub-neg 58.6%

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

      3. +-commutative 58.6%

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

      4. distribute-rgt-neg-in 58.6%

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

      5. fma-def 58.7%

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

   3. Simplified 58.9%

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

   4. Taylor expanded in ux around 0 98.4%

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

   5. Taylor expanded in uy around 0 96.9%

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

   6. Taylor expanded in maxCos around 0 90.5%

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

      1. associate-*l* 90.6%

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

      2. +-commutative 90.6%

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

      3. mul-1-neg 90.6%

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

      4. unsub-neg 90.6%

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

   8. Simplified 90.6%

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

   if 0.00249999994 < (*.f32 uy 2)

   1. Initial program 53.5%

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

      1. associate-*l* 53.5%

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

      2. sub-neg 53.5%

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

      3. +-commutative 53.5%

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

      4. distribute-rgt-neg-in 53.5%

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

      5. fma-def 53.5%

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

   3. Simplified 53.6%

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

   4. Taylor expanded in ux around 0 80.4%

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

      1. associate-*r* 80.4%

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

      2. add-exp-log 80.3%

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

      3. associate-*r* 80.3%

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

      4. *-commutative 80.3%

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

      5. associate-*l* 80.3%

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

   6. Applied egg-rr 80.3%

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

      1. rem-exp-log 80.4%

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

      2. associate-*r* 80.4%

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

      3. *-commutative 80.4%

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

      4. add-cbrt-cube 80.4%

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

      5. add-cbrt-cube 80.5%

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

      6. cbrt-unprod 80.5%

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

   8. Applied egg-rr 80.3%

      \[\leadsto \color{blue}{\sqrt[3]{{\sin \left(\pi \cdot \left(2 \cdot uy\right)\right)}^{3} \cdot {\left(\left(\mathsf{fma}\left(-1, -1 + maxCos, 1\right) - maxCos\right) \cdot ux\right)}^{1.5}}} \]
   9. Step-by-step derivation

      1. *-commutative 80.3%

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

      2. associate-*r* 80.3%

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

      3. *-commutative 80.3%

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

      4. fma-udef 80.3%

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

      5. neg-mul-1 80.3%

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

      6. distribute-neg-in 80.3%

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

      7. metadata-eval 80.3%

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

      8. sub-neg 80.3%

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

      9. +-commutative 80.3%

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

      10. associate-+r- 80.3%

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

   10. Simplified 80.3%

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

       1. expm1-log1p-u 80.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{\sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}^{3} \cdot {\left(ux \cdot \left(1 + \left(\left(1 - maxCos\right) - maxCos\right)\right)\right)}^{1.5}}\right)\right)} \]

       2. expm1-udef 62.1%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{{\sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}^{3} \cdot {\left(ux \cdot \left(1 + \left(\left(1 - maxCos\right) - maxCos\right)\right)\right)}^{1.5}}\right)} - 1} \]

   12. Applied egg-rr 62.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{ux \cdot \left(1 + \left(\left(1 - maxCos\right) - maxCos\right)\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 80.4%

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

       2. expm1-log1p 80.4%

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

       3. *-commutative 80.4%

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

       4. associate-*r* 80.4%

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

       5. *-commutative 80.4%

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

       6. *-commutative 80.4%

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

       7. *-commutative 80.4%

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

       8. associate-+r- 80.4%

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

       9. associate-+r- 80.4%

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

       10. metadata-eval 80.4%

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

   14. Simplified 80.4%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0024999999441206455:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{2 \cdot ux - {ux}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(2 - maxCos\right) - maxCos\right)} \cdot \sin \left(\pi \cdot \left(uy \cdot 2\right)\right)\\ \end{array} \]

Alternative 9: 89.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.0003000000142492354f) {
		tmp = sinf((((float) M_PI) * (uy * 2.0f))) * sqrtf((ux * (2.0f - (2.0f * maxCos))));
	} else {
		tmp = sinf((uy * (2.0f * ((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(0.0003000000142492354))
		tmp = Float32(sin(Float32(Float32(pi) * Float32(uy * Float32(2.0)))) * sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))));
	else
		tmp = Float32(sin(Float32(uy * Float32(Float32(2.0) * 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(0.0003000000142492354))
		tmp = sin((single(pi) * (uy * single(2.0)))) * sqrt((ux * (single(2.0) - (single(2.0) * maxCos))));
	else
		tmp = sin((uy * (single(2.0) * 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 < 3.00000014e-4

   1. Initial program 40.5%

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

   2. Taylor expanded in ux around 0 90.3%

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

      1. *-commutative 90.3%

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

   4. Simplified 90.3%

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

   if 3.00000014e-4 < ux

   1. Initial program 89.1%

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

      1. associate-*l* 89.1%

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

      2. sub-neg 89.1%

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

      3. +-commutative 89.1%

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

      4. distribute-rgt-neg-in 89.1%

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

      5. fma-def 89.5%

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

   3. Simplified 89.7%

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

   4. Taylor expanded in maxCos around 0 82.7%

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

4. Final simplification 87.7%

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

Alternative 10: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0024999999441206455:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{2 \cdot ux - {ux}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.0024999999441206455)
   (* 2.0 (* uy (* PI (sqrt (- (* 2.0 ux) (pow ux 2.0))))))
   (* (sin (* uy (* 2.0 PI))) (sqrt (* 2.0 ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.0024999999441206455f) {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf(((2.0f * ux) - powf(ux, 2.0f)))));
	} else {
		tmp = sinf((uy * (2.0f * ((float) M_PI)))) * sqrtf((2.0f * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.0024999999441206455))
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(Float32(Float32(2.0) * ux) - (ux ^ Float32(2.0)))))));
	else
		tmp = Float32(sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(2.0) * ux)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if ((uy * single(2.0)) <= single(0.0024999999441206455))
		tmp = single(2.0) * (uy * (single(pi) * sqrt(((single(2.0) * ux) - (ux ^ single(2.0))))));
	else
		tmp = sin((uy * (single(2.0) * single(pi)))) * sqrt((single(2.0) * ux));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy 2) < 0.00249999994

   1. Initial program 58.6%

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

      1. associate-*l* 58.6%

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

      2. sub-neg 58.6%

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

      3. +-commutative 58.6%

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

      4. distribute-rgt-neg-in 58.6%

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

      5. fma-def 58.7%

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

   3. Simplified 58.9%

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

   4. Taylor expanded in ux around 0 98.4%

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

   5. Taylor expanded in uy around 0 96.9%

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

   6. Taylor expanded in maxCos around 0 90.5%

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

      1. associate-*l* 90.6%

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

      2. +-commutative 90.6%

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

      3. mul-1-neg 90.6%

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

      4. unsub-neg 90.6%

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

   8. Simplified 90.6%

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

   if 0.00249999994 < (*.f32 uy 2)

   1. Initial program 53.5%

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

      1. associate-*l* 53.5%

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

      2. sub-neg 53.5%

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

      3. +-commutative 53.5%

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

      4. distribute-rgt-neg-in 53.5%

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

      5. fma-def 53.5%

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

   3. Simplified 53.6%

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

   4. Taylor expanded in ux around 0 80.4%

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

   5. Taylor expanded in maxCos around 0 77.1%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0024999999441206455:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{2 \cdot ux - {ux}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]

Alternative 11: 82.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00018000000272877514:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 + \left(1 + ux \cdot \left(-1 + maxCos\right)\right) \cdot \left(ux + \left(-1 - ux \cdot maxCos\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00018000000272877514)
   (* (sin (* uy (* 2.0 PI))) (sqrt (* 2.0 ux)))
   (*
    2.0
    (*
     (* uy PI)
     (sqrt
      (+
       1.0
       (* (+ 1.0 (* ux (+ -1.0 maxCos))) (+ ux (- -1.0 (* ux maxCos))))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00018000000272877514f) {
		tmp = sinf((uy * (2.0f * ((float) M_PI)))) * sqrtf((2.0f * ux));
	} else {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf((1.0f + ((1.0f + (ux * (-1.0f + maxCos))) * (ux + (-1.0f - (ux * maxCos)))))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00018000000272877514))
		tmp = Float32(sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(2.0) * ux)));
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) + Float32(ux * Float32(Float32(-1.0) + maxCos))) * Float32(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(0.00018000000272877514))
		tmp = sin((uy * (single(2.0) * single(pi)))) * sqrt((single(2.0) * ux));
	else
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((single(1.0) + ((single(1.0) + (ux * (single(-1.0) + maxCos))) * (ux + (single(-1.0) - (ux * maxCos)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.80000003e-4

   1. Initial program 39.1%

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

      1. associate-*l* 39.1%

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

      2. sub-neg 39.1%

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

      3. +-commutative 39.1%

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

      4. distribute-rgt-neg-in 39.1%

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

      5. fma-def 39.2%

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

   3. Simplified 39.4%

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

   4. Taylor expanded in ux around 0 91.0%

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

   5. Taylor expanded in maxCos around 0 87.6%

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

   if 1.80000003e-4 < ux

   1. Initial program 88.3%

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

      1. associate-*l* 88.3%

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

      2. sub-neg 88.3%

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

      3. +-commutative 88.3%

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

      4. distribute-rgt-neg-in 88.3%

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

      5. fma-def 88.4%

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

   3. Simplified 88.6%

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

   4. Taylor expanded in uy around 0 77.8%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00018000000272877514:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 + \left(1 + ux \cdot \left(-1 + maxCos\right)\right) \cdot \left(ux + \left(-1 - ux \cdot maxCos\right)\right)}\right)\\ \end{array} \]

Alternative 12: 76.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00014000000373926014:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{ux \cdot \left(\left(2 - maxCos\right) - maxCos\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 + \left(1 + ux \cdot \left(-1 + maxCos\right)\right) \cdot \left(ux + \left(-1 - ux \cdot maxCos\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00014000000373926014)
   (* 2.0 (* uy (* PI (sqrt (* ux (- (- 2.0 maxCos) maxCos))))))
   (*
    2.0
    (*
     (* uy PI)
     (sqrt
      (+
       1.0
       (* (+ 1.0 (* ux (+ -1.0 maxCos))) (+ ux (- -1.0 (* ux maxCos))))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00014000000373926014f) {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf((ux * ((2.0f - maxCos) - maxCos)))));
	} else {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf((1.0f + ((1.0f + (ux * (-1.0f + maxCos))) * (ux + (-1.0f - (ux * maxCos)))))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00014000000373926014))
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(ux * Float32(Float32(Float32(2.0) - maxCos) - maxCos))))));
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) + Float32(ux * Float32(Float32(-1.0) + maxCos))) * Float32(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(0.00014000000373926014))
		tmp = single(2.0) * (uy * (single(pi) * sqrt((ux * ((single(2.0) - maxCos) - maxCos)))));
	else
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((single(1.0) + ((single(1.0) + (ux * (single(-1.0) + maxCos))) * (ux + (single(-1.0) - (ux * maxCos)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.40000004e-4

   1. Initial program 38.7%

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

      1. associate-*l* 38.7%

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

      2. sub-neg 38.7%

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

      3. +-commutative 38.7%

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

      4. distribute-rgt-neg-in 38.7%

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

      5. fma-def 38.8%

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

   3. Simplified 39.0%

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

   4. Taylor expanded in ux around 0 91.2%

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

   5. Taylor expanded in uy around 0 76.4%

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

      1. *-commutative 76.4%

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

      2. associate--l+ 76.4%

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

      3. sub-neg 76.4%

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

      4. metadata-eval 76.4%

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

      5. +-commutative 76.4%

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

      6. neg-mul-1 76.4%

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

      7. distribute-neg-in 76.4%

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

      8. metadata-eval 76.4%

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

      9. sub-neg 76.4%

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

   7. Simplified 76.4%

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

      1. expm1-log1p-u 76.4%

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

      2. expm1-udef 24.4%

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

   9. Applied egg-rr 24.4%

      \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(1 + \left(\left(1 - maxCos\right) - maxCos\right)\right)}\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 76.4%

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

       2. expm1-log1p 76.4%

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

       3. associate-*l* 76.5%

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

       4. associate-+r- 76.5%

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

       5. associate-+r- 76.4%

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

       6. metadata-eval 76.4%

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

   11. Simplified 76.4%

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

   if 1.40000004e-4 < ux

   1. Initial program 88.0%

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

      1. associate-*l* 88.0%

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

      2. sub-neg 88.0%

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

      3. +-commutative 88.0%

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

      4. distribute-rgt-neg-in 88.0%

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

      5. fma-def 88.1%

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

   3. Simplified 88.2%

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

   4. Taylor expanded in uy around 0 77.7%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00014000000373926014:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{ux \cdot \left(\left(2 - maxCos\right) - maxCos\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 + \left(1 + ux \cdot \left(-1 + maxCos\right)\right) \cdot \left(ux + \left(-1 - ux \cdot maxCos\right)\right)}\right)\\ \end{array} \]

Alternative 13: 75.5% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.0003000000142492354)
   (* 2.0 (* uy (* PI (sqrt (* ux (- (- 2.0 maxCos) 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 <= 0.0003000000142492354f) {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf((ux * ((2.0f - maxCos) - 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(0.0003000000142492354))
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(ux * Float32(Float32(Float32(2.0) - maxCos) - 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(0.0003000000142492354))
		tmp = single(2.0) * (uy * (single(pi) * sqrt((ux * ((single(2.0) - maxCos) - 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 < 3.00000014e-4

   1. Initial program 40.5%

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

      1. associate-*l* 40.5%

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

      2. sub-neg 40.5%

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

      3. +-commutative 40.5%

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

      4. distribute-rgt-neg-in 40.5%

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

      5. fma-def 40.4%

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

   3. Simplified 40.6%

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

   4. Taylor expanded in ux around 0 90.3%

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

   5. Taylor expanded in uy around 0 76.2%

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

      1. *-commutative 76.2%

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

      2. associate--l+ 76.2%

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

      3. sub-neg 76.2%

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

      4. metadata-eval 76.2%

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

      5. +-commutative 76.2%

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

      6. neg-mul-1 76.2%

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

      7. distribute-neg-in 76.2%

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

      8. metadata-eval 76.2%

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

      9. sub-neg 76.2%

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

   7. Simplified 76.2%

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

      1. expm1-log1p-u 76.2%

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

      2. expm1-udef 24.1%

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

   9. Applied egg-rr 24.1%

      \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(1 + \left(\left(1 - maxCos\right) - maxCos\right)\right)}\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 76.2%

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

       2. expm1-log1p 76.2%

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

       3. associate-*l* 76.2%

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

       4. associate-+r- 76.2%

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

       5. associate-+r- 76.2%

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

       6. metadata-eval 76.2%

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

   11. Simplified 76.2%

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

   if 3.00000014e-4 < ux

   1. Initial program 89.1%

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

      1. associate-*l* 89.1%

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

      2. sub-neg 89.1%

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

      3. +-commutative 89.1%

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

      4. distribute-rgt-neg-in 89.1%

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

      5. fma-def 89.5%

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

   3. Simplified 89.7%

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

   4. Taylor expanded in uy around 0 78.0%

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

   5. Taylor expanded in maxCos around 0 72.5%

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

4. Final simplification 74.9%

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

Alternative 14: 66.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.2%

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

   1. associate-*l* 57.2%

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

   2. sub-neg 57.2%

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

   3. +-commutative 57.2%

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

   4. distribute-rgt-neg-in 57.2%

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

   5. fma-def 57.3%

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

3. Simplified 57.4%

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

4. Taylor expanded in ux around 0 77.4%

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

5. Taylor expanded in uy around 0 66.5%

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

   1. *-commutative 66.5%

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

   2. associate--l+ 66.5%

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

   3. sub-neg 66.5%

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

   4. metadata-eval 66.5%

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

   5. +-commutative 66.5%

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

   6. neg-mul-1 66.5%

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

   7. distribute-neg-in 66.5%

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

   8. metadata-eval 66.5%

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

   9. sub-neg 66.5%

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

7. Simplified 66.5%

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

   1. expm1-log1p-u 66.5%

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

   2. expm1-udef 26.4%

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

9. Applied egg-rr 26.4%

   \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(1 + \left(\left(1 - maxCos\right) - maxCos\right)\right)}\right)} - 1\right)} \]
10. Step-by-step derivation

    1. expm1-def 66.5%

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

    2. expm1-log1p 66.5%

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

    3. associate-*l* 66.5%

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

    4. associate-+r- 66.5%

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

    5. associate-+r- 66.5%

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

    6. metadata-eval 66.5%

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

11. Simplified 66.5%

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

12. Final simplification 66.5%

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

Alternative 15: 63.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.2%

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

   1. associate-*l* 57.2%

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

   2. sub-neg 57.2%

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

   3. +-commutative 57.2%

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

   4. distribute-rgt-neg-in 57.2%

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

   5. fma-def 57.3%

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

3. Simplified 57.4%

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

4. Taylor expanded in ux around 0 77.4%

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

5. Taylor expanded in uy around 0 66.5%

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

   1. *-commutative 66.5%

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

   2. associate--l+ 66.5%

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

   3. sub-neg 66.5%

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

   4. metadata-eval 66.5%

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

   5. +-commutative 66.5%

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

   6. neg-mul-1 66.5%

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

   7. distribute-neg-in 66.5%

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

   8. metadata-eval 66.5%

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

   9. sub-neg 66.5%

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

7. Simplified 66.5%

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

8. Taylor expanded in maxCos around 0 64.1%

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

9. Final simplification 64.1%

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

Reproduce

herbie shell --seed 2023319 
(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)))))))