UniformSampleCone, y

Percentage Accurate: 58.2% → 98.7%

Time: 22.2s

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: 58.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right) - ux \cdot \left(ux \cdot {\left(1 - maxCos\right)}^{2}\right)\\ t_1 := \sin \left(uy \cdot \left(2 \cdot \pi\right)\right)\\ \sqrt[3]{\left(\sin \left(uy \cdot \left(2 \cdot \left(\sqrt[3]{\pi} \cdot \left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right)\right)\right)\right) \cdot \sqrt{t_0}\right) \cdot \left(t_0 \cdot \left(t_1 \cdot t_1\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0
         (-
          (* ux (fma maxCos -2.0 2.0))
          (* ux (* ux (pow (- 1.0 maxCos) 2.0)))))
        (t_1 (sin (* uy (* 2.0 PI)))))
   (cbrt
    (*
     (* (sin (* uy (* 2.0 (* (cbrt PI) (* (cbrt PI) (cbrt PI)))))) (sqrt t_0))
     (* t_0 (* t_1 t_1))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (ux * fmaf(maxCos, -2.0f, 2.0f)) - (ux * (ux * powf((1.0f - maxCos), 2.0f)));
	float t_1 = sinf((uy * (2.0f * ((float) M_PI))));
	return cbrtf(((sinf((uy * (2.0f * (cbrtf(((float) M_PI)) * (cbrtf(((float) M_PI)) * cbrtf(((float) M_PI))))))) * sqrtf(t_0)) * (t_0 * (t_1 * t_1))));
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(ux * fma(maxCos, Float32(-2.0), Float32(2.0))) - Float32(ux * Float32(ux * (Float32(Float32(1.0) - maxCos) ^ Float32(2.0)))))
	t_1 = sin(Float32(uy * Float32(Float32(2.0) * Float32(pi))))
	return cbrt(Float32(Float32(sin(Float32(uy * Float32(Float32(2.0) * Float32(cbrt(Float32(pi)) * Float32(cbrt(Float32(pi)) * cbrt(Float32(pi))))))) * sqrt(t_0)) * Float32(t_0 * Float32(t_1 * t_1))))
end

Derivation

1. Initial program 56.1%

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

   1. associate-*l* 56.1%

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

   2. +-commutative 56.1%

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

   3. associate-+r- 56.1%

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

   4. fma-def 56.1%

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

   5. +-commutative 56.1%

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

   6. associate-+r- 56.0%

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

   7. fma-def 56.0%

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

3. Simplified 56.0%

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

4. Taylor expanded in ux around -inf 58.7%

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

   1. +-commutative 58.7%

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

   2. mul-1-neg 58.7%

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

   3. unsub-neg 58.7%

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

   4. unpow2 58.7%

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

   5. mul-1-neg 58.7%

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

   6. sub-neg 58.7%

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

   7. *-commutative 58.7%

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

   8. fma-def 58.7%

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

6. Simplified 58.7%

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

   1. add-cbrt-cube 58.7%

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

8. Applied egg-rr 98.2%

   \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{0 - \left(\left(ux \cdot ux\right) \cdot {\left(1 - maxCos\right)}^{2} - ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\right) \cdot \left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{0 - \left(\left(ux \cdot ux\right) \cdot {\left(1 - maxCos\right)}^{2} - ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\right)\right) \cdot \left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{0 - \left(\left(ux \cdot ux\right) \cdot {\left(1 - maxCos\right)}^{2} - ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\right)}} \]
9. Step-by-step derivation

10. Simplified 98.3%

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

    1. add-cube-cbrt 98.7%

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

12. Applied egg-rr 98.7%

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

13. Final simplification 98.7%

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

Alternative 2: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(uy \cdot \left(2 \cdot \pi\right)\right)\\ t_1 := ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right) - ux \cdot \left(ux \cdot {\left(1 - maxCos\right)}^{2}\right)\\ \sqrt[3]{\left(t_1 \cdot \left(t_0 \cdot t_0\right)\right) \cdot \left(\sqrt{t_1} \cdot t_0\right)} \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (sin (* uy (* 2.0 PI))))
        (t_1
         (-
          (* ux (fma maxCos -2.0 2.0))
          (* ux (* ux (pow (- 1.0 maxCos) 2.0))))))
   (cbrt (* (* t_1 (* t_0 t_0)) (* (sqrt t_1) t_0)))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = sinf((uy * (2.0f * ((float) M_PI))));
	float t_1 = (ux * fmaf(maxCos, -2.0f, 2.0f)) - (ux * (ux * powf((1.0f - maxCos), 2.0f)));
	return cbrtf(((t_1 * (t_0 * t_0)) * (sqrtf(t_1) * t_0)));
}

Julia

function code(ux, uy, maxCos)
	t_0 = sin(Float32(uy * Float32(Float32(2.0) * Float32(pi))))
	t_1 = Float32(Float32(ux * fma(maxCos, Float32(-2.0), Float32(2.0))) - Float32(ux * Float32(ux * (Float32(Float32(1.0) - maxCos) ^ Float32(2.0)))))
	return cbrt(Float32(Float32(t_1 * Float32(t_0 * t_0)) * Float32(sqrt(t_1) * t_0)))
end

Derivation

1. Initial program 56.1%

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

   1. associate-*l* 56.1%

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

   2. +-commutative 56.1%

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

   3. associate-+r- 56.1%

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

   4. fma-def 56.1%

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

   5. +-commutative 56.1%

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

   6. associate-+r- 56.0%

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

   7. fma-def 56.0%

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

3. Simplified 56.0%

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

4. Taylor expanded in ux around -inf 58.7%

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

   1. +-commutative 58.7%

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

   2. mul-1-neg 58.7%

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

   3. unsub-neg 58.7%

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

   4. unpow2 58.7%

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

   5. mul-1-neg 58.7%

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

   6. sub-neg 58.7%

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

   7. *-commutative 58.7%

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

   8. fma-def 58.7%

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

6. Simplified 58.7%

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

   1. add-cbrt-cube 58.7%

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

8. Applied egg-rr 98.2%

   \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{0 - \left(\left(ux \cdot ux\right) \cdot {\left(1 - maxCos\right)}^{2} - ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\right) \cdot \left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{0 - \left(\left(ux \cdot ux\right) \cdot {\left(1 - maxCos\right)}^{2} - ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\right)\right) \cdot \left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{0 - \left(\left(ux \cdot ux\right) \cdot {\left(1 - maxCos\right)}^{2} - ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\right)}} \]
9. Step-by-step derivation

10. Simplified 98.3%

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

11. Final simplification 98.3%

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

Alternative 3: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := ux \cdot \mathsf{fma}\left(-2, maxCos, 2\right) - {\left(1 - maxCos\right)}^{2} \cdot \left(ux \cdot ux\right)\\ \sqrt[3]{\sqrt{t_0} \cdot \left(t_0 \cdot {\sin \left(uy \cdot \left(2 \cdot \pi\right)\right)}^{3}\right)} \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0
         (-
          (* ux (fma -2.0 maxCos 2.0))
          (* (pow (- 1.0 maxCos) 2.0) (* ux ux)))))
   (cbrt (* (sqrt t_0) (* t_0 (pow (sin (* uy (* 2.0 PI))) 3.0))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (ux * fmaf(-2.0f, maxCos, 2.0f)) - (powf((1.0f - maxCos), 2.0f) * (ux * ux));
	return cbrtf((sqrtf(t_0) * (t_0 * powf(sinf((uy * (2.0f * ((float) M_PI)))), 3.0f))));
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(ux * fma(Float32(-2.0), maxCos, Float32(2.0))) - Float32((Float32(Float32(1.0) - maxCos) ^ Float32(2.0)) * Float32(ux * ux)))
	return cbrt(Float32(sqrt(t_0) * Float32(t_0 * (sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) ^ Float32(3.0)))))
end

Derivation

1. Initial program 56.1%

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

   1. associate-*l* 56.1%

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

   2. +-commutative 56.1%

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

   3. associate-+r- 56.1%

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

   4. fma-def 56.1%

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

   5. +-commutative 56.1%

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

   6. associate-+r- 56.0%

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

   7. fma-def 56.0%

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

3. Simplified 56.0%

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

4. Taylor expanded in ux around -inf 58.7%

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

   1. +-commutative 58.7%

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

   2. mul-1-neg 58.7%

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

   3. unsub-neg 58.7%

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

   4. unpow2 58.7%

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

   5. mul-1-neg 58.7%

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

   6. sub-neg 58.7%

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

   7. *-commutative 58.7%

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

   8. fma-def 58.7%

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

6. Simplified 58.7%

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

   1. add-cbrt-cube 58.7%

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

8. Applied egg-rr 98.2%

   \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{0 - \left(\left(ux \cdot ux\right) \cdot {\left(1 - maxCos\right)}^{2} - ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\right) \cdot \left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{0 - \left(\left(ux \cdot ux\right) \cdot {\left(1 - maxCos\right)}^{2} - ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\right)\right) \cdot \left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{0 - \left(\left(ux \cdot ux\right) \cdot {\left(1 - maxCos\right)}^{2} - ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\right)}} \]
9. Step-by-step derivation

10. Simplified 98.3%

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

11. Taylor expanded in uy around inf 98.3%

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

    1. +-commutative 98.3%

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

    2. fma-def 98.3%

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

    3. unpow2 98.3%

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

13. Simplified 98.3%

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

14. Final simplification 98.3%

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

Alternative 4: 98.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.1%

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

   1. associate-*l* 56.1%

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

   2. sub-neg 56.1%

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

   3. +-commutative 56.1%

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

   4. distribute-rgt-neg-in 56.1%

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

   5. fma-def 56.2%

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

   6. +-commutative 56.2%

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

   7. associate-+r- 56.2%

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

   8. fma-def 56.2%

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

   9. neg-sub0 56.2%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{0 - \left(\left(1 - ux\right) + ux \cdot maxCos\right)}, 1\right)} \]

   10. +-commutative 56.2%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}, 1\right)} \]

   11. associate-+r- 56.1%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)}, 1\right)} \]

   12. associate--r- 56.1%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(ux \cdot maxCos + 1\right)\right) + ux}, 1\right)} \]

   13. neg-sub0 56.1%

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

   14. +-commutative 56.1%

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

   15. sub-neg 56.1%

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

   16. fma-def 56.1%

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

3. Simplified 56.1%

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

4. Taylor expanded in ux around 0 98.3%

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

   1. associate-*r* 98.3%

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

   2. add-cbrt-cube 98.2%

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

   3. add-cbrt-cube 98.2%

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

   4. cbrt-unprod 98.2%

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

   5. pow3 98.3%

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

   6. pow3 98.3%

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

6. Applied egg-rr 98.3%

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

7. Final simplification 98.3%

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

Alternative 5: 98.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot {\left(\mathsf{fma}\left(maxCos + -1, \left(1 - maxCos\right) \cdot \left(ux \cdot ux\right), ux \cdot \left(\left(1 + \left(1 - maxCos\right)\right) - maxCos\right)\right)\right)}^{0.5} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sin (* uy (* 2.0 PI)))
  (pow
   (fma
    (+ maxCos -1.0)
    (* (- 1.0 maxCos) (* ux ux))
    (* ux (- (+ 1.0 (- 1.0 maxCos)) maxCos)))
   0.5)))

C

float code(float ux, float uy, float maxCos) {
	return sinf((uy * (2.0f * ((float) M_PI)))) * powf(fmaf((maxCos + -1.0f), ((1.0f - maxCos) * (ux * ux)), (ux * ((1.0f + (1.0f - maxCos)) - maxCos))), 0.5f);
}

Julia

function code(ux, uy, maxCos)
	return Float32(sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * (fma(Float32(maxCos + Float32(-1.0)), Float32(Float32(Float32(1.0) - maxCos) * Float32(ux * ux)), Float32(ux * Float32(Float32(Float32(1.0) + Float32(Float32(1.0) - maxCos)) - maxCos))) ^ Float32(0.5)))
end

Derivation

1. Initial program 56.1%

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

   1. associate-*l* 56.1%

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

   2. sub-neg 56.1%

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

   3. +-commutative 56.1%

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

   4. distribute-rgt-neg-in 56.1%

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

   5. fma-def 56.2%

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

   6. +-commutative 56.2%

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

   7. associate-+r- 56.2%

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

   8. fma-def 56.2%

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

   9. neg-sub0 56.2%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{0 - \left(\left(1 - ux\right) + ux \cdot maxCos\right)}, 1\right)} \]

   10. +-commutative 56.2%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}, 1\right)} \]

   11. associate-+r- 56.1%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)}, 1\right)} \]

   12. associate--r- 56.1%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(ux \cdot maxCos + 1\right)\right) + ux}, 1\right)} \]

   13. neg-sub0 56.1%

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

   14. +-commutative 56.1%

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

   15. sub-neg 56.1%

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

   16. fma-def 56.1%

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

3. Simplified 56.1%

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

4. Taylor expanded in ux around 0 98.3%

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

   1. pow1/2 98.3%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \color{blue}{{\left(\left(maxCos - 1\right) \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right) + ux \cdot \left(\left(1 + -1 \cdot \left(maxCos - 1\right)\right) - maxCos\right)\right)}^{0.5}} \]

   2. fma-def 98.3%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot {\color{blue}{\left(\mathsf{fma}\left(maxCos - 1, \left(1 - maxCos\right) \cdot {ux}^{2}, ux \cdot \left(\left(1 + -1 \cdot \left(maxCos - 1\right)\right) - maxCos\right)\right)\right)}}^{0.5} \]

   3. sub-neg 98.3%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot {\left(\mathsf{fma}\left(\color{blue}{maxCos + \left(-1\right)}, \left(1 - maxCos\right) \cdot {ux}^{2}, ux \cdot \left(\left(1 + -1 \cdot \left(maxCos - 1\right)\right) - maxCos\right)\right)\right)}^{0.5} \]

   4. metadata-eval 98.3%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot {\left(\mathsf{fma}\left(maxCos + \color{blue}{-1}, \left(1 - maxCos\right) \cdot {ux}^{2}, ux \cdot \left(\left(1 + -1 \cdot \left(maxCos - 1\right)\right) - maxCos\right)\right)\right)}^{0.5} \]

   5. pow2 98.3%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot {\left(\mathsf{fma}\left(maxCos + -1, \left(1 - maxCos\right) \cdot \color{blue}{\left(ux \cdot ux\right)}, ux \cdot \left(\left(1 + -1 \cdot \left(maxCos - 1\right)\right) - maxCos\right)\right)\right)}^{0.5} \]

   6. mul-1-neg 98.3%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot {\left(\mathsf{fma}\left(maxCos + -1, \left(1 - maxCos\right) \cdot \left(ux \cdot ux\right), ux \cdot \left(\left(1 + \color{blue}{\left(-\left(maxCos - 1\right)\right)}\right) - maxCos\right)\right)\right)}^{0.5} \]

   7. sub-neg 98.3%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot {\left(\mathsf{fma}\left(maxCos + -1, \left(1 - maxCos\right) \cdot \left(ux \cdot ux\right), ux \cdot \left(\left(1 + \left(-\color{blue}{\left(maxCos + \left(-1\right)\right)}\right)\right) - maxCos\right)\right)\right)}^{0.5} \]

   8. metadata-eval 98.3%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot {\left(\mathsf{fma}\left(maxCos + -1, \left(1 - maxCos\right) \cdot \left(ux \cdot ux\right), ux \cdot \left(\left(1 + \left(-\left(maxCos + \color{blue}{-1}\right)\right)\right) - maxCos\right)\right)\right)}^{0.5} \]

6. Applied egg-rr 98.3%

   \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \color{blue}{{\left(\mathsf{fma}\left(maxCos + -1, \left(1 - maxCos\right) \cdot \left(ux \cdot ux\right), ux \cdot \left(\left(1 + \left(-\left(maxCos + -1\right)\right)\right) - maxCos\right)\right)\right)}^{0.5}} \]

7. Final simplification 98.3%

   \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot {\left(\mathsf{fma}\left(maxCos + -1, \left(1 - maxCos\right) \cdot \left(ux \cdot ux\right), ux \cdot \left(\left(1 + \left(1 - maxCos\right)\right) - maxCos\right)\right)\right)}^{0.5} \]

Alternative 6: 98.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.1%

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

   1. associate-*l* 56.1%

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

   2. sub-neg 56.1%

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

   3. +-commutative 56.1%

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

   4. distribute-rgt-neg-in 56.1%

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

   5. fma-def 56.2%

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

   6. +-commutative 56.2%

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

   7. associate-+r- 56.2%

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

   8. fma-def 56.2%

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

   9. neg-sub0 56.2%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{0 - \left(\left(1 - ux\right) + ux \cdot maxCos\right)}, 1\right)} \]

   10. +-commutative 56.2%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}, 1\right)} \]

   11. associate-+r- 56.1%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)}, 1\right)} \]

   12. associate--r- 56.1%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(ux \cdot maxCos + 1\right)\right) + ux}, 1\right)} \]

   13. neg-sub0 56.1%

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

   14. +-commutative 56.1%

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

   15. sub-neg 56.1%

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

   16. fma-def 56.1%

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

3. Simplified 56.1%

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

4. Taylor expanded in ux around 0 98.3%

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

   1. add-sqr-sqrt 94.7%

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

   2. pow2 94.7%

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

6. Applied egg-rr 94.7%

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

7. Taylor expanded in uy around inf 98.3%

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

   1. *-commutative 98.3%

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

   2. *-commutative 98.3%

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

   3. associate-*r* 98.3%

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

   4. *-commutative 98.3%

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

   5. fma-def 98.3%

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

   6. sub-neg 98.3%

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

   7. metadata-eval 98.3%

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

   8. unpow2 98.3%

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

   9. associate-*r* 98.3%

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

9. Simplified 98.3%

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

10. Final simplification 98.3%

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

Alternative 7: 98.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.1%

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

   1. associate-*l* 56.1%

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

   2. sub-neg 56.1%

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

   3. +-commutative 56.1%

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

   4. distribute-rgt-neg-in 56.1%

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

   5. fma-def 56.2%

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

   6. +-commutative 56.2%

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

   7. associate-+r- 56.2%

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

   8. fma-def 56.2%

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

   9. neg-sub0 56.2%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{0 - \left(\left(1 - ux\right) + ux \cdot maxCos\right)}, 1\right)} \]

   10. +-commutative 56.2%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}, 1\right)} \]

   11. associate-+r- 56.1%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)}, 1\right)} \]

   12. associate--r- 56.1%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(ux \cdot maxCos + 1\right)\right) + ux}, 1\right)} \]

   13. neg-sub0 56.1%

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

   14. +-commutative 56.1%

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

   15. sub-neg 56.1%

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

   16. fma-def 56.1%

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

3. Simplified 56.1%

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

4. Taylor expanded in ux around 0 98.3%

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

   1. add-cbrt-cube 98.2%

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

6. Applied egg-rr 98.3%

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

8. Simplified 98.2%

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

9. Taylor expanded in uy around inf 98.2%

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

11. Simplified 98.2%

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

12. Final simplification 98.2%

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

Alternative 8: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.1%

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

   1. associate-*l* 56.1%

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

   2. +-commutative 56.1%

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

   3. associate-+r- 56.1%

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

   4. fma-def 56.1%

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

   5. +-commutative 56.1%

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

   6. associate-+r- 56.0%

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

   7. fma-def 56.0%

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

3. Simplified 56.0%

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

4. Taylor expanded in ux around -inf 58.7%

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

   1. +-commutative 58.7%

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

   2. mul-1-neg 58.7%

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

   3. unsub-neg 58.7%

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

   4. unpow2 58.7%

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

   5. mul-1-neg 58.7%

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

   6. sub-neg 58.7%

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

   7. *-commutative 58.7%

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

   8. fma-def 58.7%

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

6. Simplified 58.7%

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

7. Taylor expanded in maxCos around -inf 98.2%

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

   1. +-commutative 98.2%

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

   2. mul-1-neg 98.2%

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

   3. unsub-neg 98.2%

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

   4. mul-1-neg 98.2%

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

   5. unsub-neg 98.2%

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

   6. unpow2 98.2%

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

   7. unpow2 98.2%

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

   8. unswap-sqr 98.2%

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

   9. distribute-lft-out-- 98.2%

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

   10. unpow2 98.2%

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

   11. unpow2 98.2%

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

9. Simplified 98.2%

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

10. Final simplification 98.2%

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

Alternative 9: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.0010000000474974513:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{2 \cdot ux - ux \cdot ux}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.0010000000474974513)
   (* 2.0 (* uy (* PI (sqrt (- (* 2.0 ux) (* ux ux))))))
   (* (sin (* uy (* 2.0 PI))) (sqrt (* ux (- 2.0 (* 2.0 maxCos)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0010000000474974513f) {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf(((2.0f * ux) - (ux * ux)))));
	} else {
		tmp = sinf((uy * (2.0f * ((float) M_PI)))) * sqrtf((ux * (2.0f - (2.0f * maxCos))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.0010000000474974513))
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(Float32(Float32(2.0) * ux) - Float32(ux * ux))))));
	else
		tmp = Float32(sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(0.0010000000474974513))
		tmp = single(2.0) * (uy * (single(pi) * sqrt(((single(2.0) * ux) - (ux * ux)))));
	else
		tmp = sin((uy * (single(2.0) * single(pi)))) * sqrt((ux * (single(2.0) - (single(2.0) * maxCos))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00100000005

   1. Initial program 57.0%

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

      1. associate-*l* 57.0%

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

      2. sub-neg 57.0%

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

      3. +-commutative 57.0%

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

      4. distribute-rgt-neg-in 57.0%

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

      5. fma-def 57.0%

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

      6. +-commutative 57.0%

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

      7. associate-+r- 57.0%

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

      8. fma-def 57.0%

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

      9. neg-sub0 57.0%

         \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{0 - \left(\left(1 - ux\right) + ux \cdot maxCos\right)}, 1\right)} \]

      10. +-commutative 57.0%

          \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}, 1\right)} \]

      11. associate-+r- 56.9%

          \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)}, 1\right)} \]

      12. associate--r- 56.9%

          \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(ux \cdot maxCos + 1\right)\right) + ux}, 1\right)} \]

      13. neg-sub0 56.9%

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

      14. +-commutative 56.9%

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

      15. sub-neg 56.9%

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

      16. fma-def 56.9%

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

   3. Simplified 56.9%

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

   4. Taylor expanded in ux around 0 98.5%

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

      1. add-sqr-sqrt 98.0%

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

      2. pow2 98.0%

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

   6. Applied egg-rr 98.0%

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

   7. Taylor expanded in uy around 0 96.9%

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

      1. associate-*l* 97.0%

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

      2. fma-def 97.0%

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

      3. sub-neg 97.0%

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

      4. metadata-eval 97.0%

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

      5. unpow2 97.0%

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

      6. associate-*r* 97.0%

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

      7. associate--l+ 97.0%

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

      8. neg-mul-1 97.0%

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

      9. sub-neg 97.0%

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

      10. metadata-eval 97.0%

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

      11. associate-+r- 97.0%

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

   9. Simplified 97.0%

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

   10. Taylor expanded in maxCos around 0 93.3%

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

       1. +-commutative 93.3%

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

       2. neg-mul-1 93.3%

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

       3. unsub-neg 93.3%

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

       4. unpow2 93.3%

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

   12. Simplified 93.3%

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

   if 0.00100000005 < uy

   1. Initial program 53.9%

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

      1. associate-*l* 53.9%

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

      2. +-commutative 53.9%

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

      3. associate-+r- 53.9%

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

      4. fma-def 53.9%

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

      5. +-commutative 53.9%

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

      6. associate-+r- 53.8%

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

      7. fma-def 53.8%

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

   3. Simplified 53.8%

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

   4. Taylor expanded in ux around 0 77.2%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.0010000000474974513:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{2 \cdot ux - ux \cdot ux}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \]

Alternative 10: 92.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.1%

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

   1. associate-*l* 56.1%

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

   2. sub-neg 56.1%

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

   3. +-commutative 56.1%

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

   4. distribute-rgt-neg-in 56.1%

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

   5. fma-def 56.2%

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

   6. +-commutative 56.2%

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

   7. associate-+r- 56.2%

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

   8. fma-def 56.2%

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

   9. neg-sub0 56.2%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{0 - \left(\left(1 - ux\right) + ux \cdot maxCos\right)}, 1\right)} \]

   10. +-commutative 56.2%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}, 1\right)} \]

   11. associate-+r- 56.1%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)}, 1\right)} \]

   12. associate--r- 56.1%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(ux \cdot maxCos + 1\right)\right) + ux}, 1\right)} \]

   13. neg-sub0 56.1%

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

   14. +-commutative 56.1%

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

   15. sub-neg 56.1%

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

   16. fma-def 56.1%

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

3. Simplified 56.1%

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

4. Taylor expanded in ux around 0 98.3%

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

5. Taylor expanded in maxCos around 0 94.2%

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

   1. associate-*r* 94.2%

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

   2. mul-1-neg 94.2%

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

   3. +-commutative 94.2%

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

   4. sub-neg 94.2%

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

   5. unpow2 94.2%

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

7. Simplified 94.2%

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

8. Final simplification 94.2%

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

Alternative 11: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.0010000000474974513:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{2 \cdot ux - ux \cdot ux}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.0010000000474974513)
   (* 2.0 (* uy (* PI (sqrt (- (* 2.0 ux) (* ux ux))))))
   (* (sin (* uy (* 2.0 PI))) (sqrt (* 2.0 ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0010000000474974513f) {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf(((2.0f * ux) - (ux * ux)))));
	} 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 (uy <= Float32(0.0010000000474974513))
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(Float32(Float32(2.0) * ux) - Float32(ux * ux))))));
	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(0.0010000000474974513))
		tmp = single(2.0) * (uy * (single(pi) * sqrt(((single(2.0) * ux) - (ux * ux)))));
	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 uy < 0.00100000005

   1. Initial program 57.0%

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

      1. associate-*l* 57.0%

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

      2. sub-neg 57.0%

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

      3. +-commutative 57.0%

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

      4. distribute-rgt-neg-in 57.0%

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

      5. fma-def 57.0%

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

      6. +-commutative 57.0%

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

      7. associate-+r- 57.0%

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

      8. fma-def 57.0%

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

      9. neg-sub0 57.0%

         \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{0 - \left(\left(1 - ux\right) + ux \cdot maxCos\right)}, 1\right)} \]

      10. +-commutative 57.0%

          \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}, 1\right)} \]

      11. associate-+r- 56.9%

          \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)}, 1\right)} \]

      12. associate--r- 56.9%

          \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(ux \cdot maxCos + 1\right)\right) + ux}, 1\right)} \]

      13. neg-sub0 56.9%

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

      14. +-commutative 56.9%

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

      15. sub-neg 56.9%

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

      16. fma-def 56.9%

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

   3. Simplified 56.9%

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

   4. Taylor expanded in ux around 0 98.5%

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

      1. add-sqr-sqrt 98.0%

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

      2. pow2 98.0%

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

   6. Applied egg-rr 98.0%

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

   7. Taylor expanded in uy around 0 96.9%

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

      1. associate-*l* 97.0%

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

      2. fma-def 97.0%

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

      3. sub-neg 97.0%

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

      4. metadata-eval 97.0%

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

      5. unpow2 97.0%

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

      6. associate-*r* 97.0%

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

      7. associate--l+ 97.0%

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

      8. neg-mul-1 97.0%

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

      9. sub-neg 97.0%

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

      10. metadata-eval 97.0%

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

      11. associate-+r- 97.0%

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

   9. Simplified 97.0%

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

   10. Taylor expanded in maxCos around 0 93.3%

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

       1. +-commutative 93.3%

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

       2. neg-mul-1 93.3%

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

       3. unsub-neg 93.3%

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

       4. unpow2 93.3%

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

   12. Simplified 93.3%

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

   if 0.00100000005 < uy

   1. Initial program 53.9%

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

      1. associate-*l* 53.9%

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

      2. +-commutative 53.9%

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

      3. associate-+r- 53.9%

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

      4. fma-def 53.9%

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

      5. +-commutative 53.9%

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

      6. associate-+r- 53.8%

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

      7. fma-def 53.8%

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

   3. Simplified 53.8%

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

   4. Taylor expanded in ux around 0 42.6%

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

   5. Taylor expanded in maxCos around 0 74.0%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.0010000000474974513:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{2 \cdot ux - ux \cdot ux}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]

Alternative 12: 65.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.1%

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

   1. associate-*l* 56.1%

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

   2. sub-neg 56.1%

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

   3. +-commutative 56.1%

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

   4. distribute-rgt-neg-in 56.1%

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

   5. fma-def 56.2%

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

   6. +-commutative 56.2%

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

   7. associate-+r- 56.2%

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

   8. fma-def 56.2%

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

   9. neg-sub0 56.2%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{0 - \left(\left(1 - ux\right) + ux \cdot maxCos\right)}, 1\right)} \]

   10. +-commutative 56.2%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}, 1\right)} \]

   11. associate-+r- 56.1%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)}, 1\right)} \]

   12. associate--r- 56.1%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(ux \cdot maxCos + 1\right)\right) + ux}, 1\right)} \]

   13. neg-sub0 56.1%

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

   14. +-commutative 56.1%

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

   15. sub-neg 56.1%

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

   16. fma-def 56.1%

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

3. Simplified 56.1%

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

4. Taylor expanded in uy around 0 49.6%

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

5. Taylor expanded in ux around 0 67.2%

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

   1. neg-mul-1 67.2%

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

   2. associate--l+ 67.2%

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

   3. sub-neg 67.2%

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

   4. metadata-eval 67.2%

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

   5. distribute-neg-in 67.2%

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

   6. metadata-eval 67.2%

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

   7. +-commutative 67.2%

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

   8. sub-neg 67.2%

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

7. Simplified 67.2%

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

8. Taylor expanded in uy around 0 67.2%

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

   1. associate-*l* 67.2%

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

   2. *-commutative 67.2%

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

   3. cancel-sign-sub-inv 67.2%

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

   4. metadata-eval 67.2%

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

   5. *-commutative 67.2%

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

10. Simplified 67.2%

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

11. Final simplification 67.2%

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

Alternative 13: 76.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.1%

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

   1. associate-*l* 56.1%

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

   2. sub-neg 56.1%

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

   3. +-commutative 56.1%

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

   4. distribute-rgt-neg-in 56.1%

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

   5. fma-def 56.2%

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

   6. +-commutative 56.2%

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

   7. associate-+r- 56.2%

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

   8. fma-def 56.2%

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

   9. neg-sub0 56.2%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{0 - \left(\left(1 - ux\right) + ux \cdot maxCos\right)}, 1\right)} \]

   10. +-commutative 56.2%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}, 1\right)} \]

   11. associate-+r- 56.1%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)}, 1\right)} \]

   12. associate--r- 56.1%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(ux \cdot maxCos + 1\right)\right) + ux}, 1\right)} \]

   13. neg-sub0 56.1%

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

   14. +-commutative 56.1%

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

   15. sub-neg 56.1%

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

   16. fma-def 56.1%

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

3. Simplified 56.1%

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

4. Taylor expanded in ux around 0 98.3%

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

   1. add-sqr-sqrt 94.7%

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

   2. pow2 94.7%

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

6. Applied egg-rr 94.7%

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

7. Taylor expanded in uy around 0 81.6%

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

   1. associate-*l* 81.6%

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

   2. fma-def 81.6%

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

   3. sub-neg 81.6%

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

   4. metadata-eval 81.6%

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

   5. unpow2 81.6%

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

   6. associate-*r* 81.6%

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

   7. associate--l+ 81.6%

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

   8. neg-mul-1 81.6%

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

   9. sub-neg 81.6%

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

   10. metadata-eval 81.6%

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

   11. associate-+r- 81.6%

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

9. Simplified 81.6%

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

10. Taylor expanded in maxCos around 0 78.9%

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

    1. +-commutative 78.9%

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

    2. neg-mul-1 78.9%

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

    3. unsub-neg 78.9%

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

    4. unpow2 78.9%

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

12. Simplified 78.9%

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

13. Final simplification 78.9%

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

Alternative 14: 62.6% 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 56.1%

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

   1. associate-*l* 56.1%

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

   2. sub-neg 56.1%

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

   3. +-commutative 56.1%

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

   4. distribute-rgt-neg-in 56.1%

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

   5. fma-def 56.2%

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

   6. +-commutative 56.2%

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

   7. associate-+r- 56.2%

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

   8. fma-def 56.2%

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

   9. neg-sub0 56.2%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{0 - \left(\left(1 - ux\right) + ux \cdot maxCos\right)}, 1\right)} \]

   10. +-commutative 56.2%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}, 1\right)} \]

   11. associate-+r- 56.1%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)}, 1\right)} \]

   12. associate--r- 56.1%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(ux \cdot maxCos + 1\right)\right) + ux}, 1\right)} \]

   13. neg-sub0 56.1%

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

   14. +-commutative 56.1%

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

   15. sub-neg 56.1%

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

   16. fma-def 56.1%

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

3. Simplified 56.1%

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

4. Taylor expanded in uy around 0 49.6%

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

5. Taylor expanded in ux around 0 67.2%

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

   1. neg-mul-1 67.2%

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

   2. associate--l+ 67.2%

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

   3. sub-neg 67.2%

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

   4. metadata-eval 67.2%

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

   5. distribute-neg-in 67.2%

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

   6. metadata-eval 67.2%

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

   7. +-commutative 67.2%

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

   8. sub-neg 67.2%

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

7. Simplified 67.2%

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

8. Taylor expanded in maxCos around 0 65.3%

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

9. Final simplification 65.3%

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

Alternative 15: 7.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.1%

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

   1. associate-*l* 56.1%

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

   2. sub-neg 56.1%

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

   3. +-commutative 56.1%

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

   4. distribute-rgt-neg-in 56.1%

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

   5. fma-def 56.2%

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

   6. +-commutative 56.2%

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

   7. associate-+r- 56.2%

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

   8. fma-def 56.2%

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

   9. neg-sub0 56.2%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{0 - \left(\left(1 - ux\right) + ux \cdot maxCos\right)}, 1\right)} \]

   10. +-commutative 56.2%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}, 1\right)} \]

   11. associate-+r- 56.1%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)}, 1\right)} \]

   12. associate--r- 56.1%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(ux \cdot maxCos + 1\right)\right) + ux}, 1\right)} \]

   13. neg-sub0 56.1%

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

   14. +-commutative 56.1%

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

   15. sub-neg 56.1%

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

   16. fma-def 56.1%

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

3. Simplified 56.1%

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

4. Taylor expanded in uy around 0 49.6%

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

5. Taylor expanded in ux around 0 7.1%

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

6. Final simplification 7.1%

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

Reproduce

herbie shell --seed 2023238 
(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)))))))