UniformSampleCone, y

Percentage Accurate: 58.1% → 98.4%

Time: 15.4s

Alternatives: 16

Speedup: 2.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 58.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in ux around 0 98.2%

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

   1. associate--l+ 98.2%

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

   2. associate-*r* 98.2%

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

   3. mul-1-neg 98.2%

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

   4. sub-neg 98.2%

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

   5. metadata-eval 98.2%

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

   6. +-commutative 98.2%

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

5. Simplified 98.2%

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

   1. *-commutative 98.2%

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

   2. add-cbrt-cube 98.2%

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

   3. *-commutative 98.2%

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

   4. associate-*r* 98.2%

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

   5. add-cbrt-cube 98.2%

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

   6. cbrt-unprod 98.1%

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

7. Applied egg-rr 98.3%

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

   1. *-commutative 98.3%

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

9. Simplified 98.3%

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

Alternative 2: 98.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in ux around 0 98.2%

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

   1. associate--l+ 98.2%

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

   2. associate-*r* 98.2%

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

   3. mul-1-neg 98.2%

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

   4. sub-neg 98.2%

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

   5. metadata-eval 98.2%

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

   6. +-commutative 98.2%

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

5. Simplified 98.2%

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

   1. add-sqr-sqrt 98.3%

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

   2. pow2 98.3%

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

   3. cancel-sign-sub-inv 98.3%

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

   4. fma-define 98.3%

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

   5. metadata-eval 98.3%

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

7. Applied egg-rr 98.3%

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

8. Final simplification 98.3%

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

Alternative 3: 98.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in ux around 0 98.2%

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

   1. associate--l+ 98.2%

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

   2. associate-*r* 98.2%

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

   3. mul-1-neg 98.2%

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

   4. sub-neg 98.2%

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

   5. metadata-eval 98.2%

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

   6. +-commutative 98.2%

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

5. Simplified 98.2%

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

6. Final simplification 98.2%

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

Alternative 4: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in ux around 0 98.2%

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

   1. associate--l+ 98.2%

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

   2. associate-*r* 98.2%

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

   3. mul-1-neg 98.2%

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

   4. sub-neg 98.2%

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

   5. metadata-eval 98.2%

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

   6. +-commutative 98.2%

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

5. Simplified 98.2%

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

6. Taylor expanded in maxCos around 0 98.2%

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

7. Final simplification 98.2%

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

Alternative 5: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in ux around 0 98.2%

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

   1. associate--l+ 98.2%

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

   2. associate-*r* 98.2%

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

   3. mul-1-neg 98.2%

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

   4. sub-neg 98.2%

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

   5. metadata-eval 98.2%

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

   6. +-commutative 98.2%

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

5. Simplified 98.2%

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

6. Taylor expanded in maxCos around 0 98.2%

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

7. Taylor expanded in uy around inf 98.2%

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

   1. *-commutative 98.2%

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

   2. neg-mul-1 98.2%

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

   3. associate-+r+ 98.2%

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

   4. sub-neg 98.2%

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

   5. sub-neg 98.2%

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

   6. neg-mul-1 98.2%

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

   7. +-commutative 98.2%

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

   8. distribute-lft-neg-in 98.2%

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

   9. mul-1-neg 98.2%

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

   10. distribute-rgt-in 98.2%

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

   11. mul-1-neg 98.2%

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

   12. metadata-eval 98.2%

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

9. Simplified 98.2%

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

10. Final simplification 98.2%

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

Alternative 6: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in ux around 0 98.2%

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

   1. associate--l+ 98.2%

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

   2. associate-*r* 98.2%

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

   3. mul-1-neg 98.2%

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

   4. sub-neg 98.2%

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

   5. metadata-eval 98.2%

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

   6. +-commutative 98.2%

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

5. Simplified 98.2%

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

6. Taylor expanded in maxCos around -inf 48.1%

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

   1. mul-1-neg 48.1%

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

   2. unsub-neg 48.1%

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

   3. neg-mul-1 48.1%

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

   4. mul-1-neg 48.1%

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

   5. unsub-neg 48.1%

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

   6. *-commutative 48.1%

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

   7. neg-mul-1 48.1%

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

   8. unsub-neg 48.1%

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

8. Simplified 48.1%

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

9. Taylor expanded in maxCos around 0 97.5%

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

    1. associate--l+ 97.5%

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

    2. associate-*r* 97.5%

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

    3. mul-1-neg 97.5%

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

    4. *-commutative 97.5%

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

11. Simplified 97.5%

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

12. Final simplification 97.5%

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

Alternative 7: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in ux around 0 98.2%

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

   1. associate--l+ 98.2%

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

   2. associate-*r* 98.2%

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

   3. mul-1-neg 98.2%

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

   4. sub-neg 98.2%

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

   5. metadata-eval 98.2%

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

   6. +-commutative 98.2%

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

5. Simplified 98.2%

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

6. Taylor expanded in maxCos around -inf 48.1%

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

   1. mul-1-neg 48.1%

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

   2. unsub-neg 48.1%

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

   3. neg-mul-1 48.1%

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

   4. mul-1-neg 48.1%

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

   5. unsub-neg 48.1%

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

   6. *-commutative 48.1%

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

   7. neg-mul-1 48.1%

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

   8. unsub-neg 48.1%

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

8. Simplified 48.1%

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

9. Taylor expanded in maxCos around 0 97.5%

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

    1. mul-1-neg 97.5%

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

    2. unsub-neg 97.5%

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

    3. *-commutative 97.5%

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

11. Simplified 97.5%

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

12. Final simplification 97.5%

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

Alternative 8: 92.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in ux around 0 98.2%

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

   1. associate--l+ 98.2%

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

   2. associate-*r* 98.2%

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

   3. mul-1-neg 98.2%

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

   4. sub-neg 98.2%

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

   5. metadata-eval 98.2%

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

   6. +-commutative 98.2%

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

5. Simplified 98.2%

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

6. Taylor expanded in maxCos around 0 93.5%

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

   1. neg-mul-1 93.5%

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

   2. unsub-neg 93.5%

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

8. Simplified 93.5%

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

9. Final simplification 93.5%

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

Alternative 9: 81.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in ux around 0 98.2%

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

   1. associate--l+ 98.2%

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

   2. associate-*r* 98.2%

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

   3. mul-1-neg 98.2%

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

   4. sub-neg 98.2%

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

   5. metadata-eval 98.2%

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

   6. +-commutative 98.2%

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

5. Simplified 98.2%

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

6. Taylor expanded in maxCos around 0 98.2%

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

7. Taylor expanded in uy around 0 81.7%

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

8. Final simplification 81.7%

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

Alternative 10: 81.8% accurate, 1.8× speedup

Math

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

FPCore

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

Julia

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

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in ux around 0 98.2%

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

   1. associate--l+ 98.2%

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

   2. associate-*r* 98.2%

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

   3. mul-1-neg 98.2%

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

   4. sub-neg 98.2%

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

   5. metadata-eval 98.2%

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

   6. +-commutative 98.2%

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

5. Simplified 98.2%

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

   1. *-commutative 98.2%

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

   2. add-cbrt-cube 98.2%

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

   3. *-commutative 98.2%

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

   4. associate-*r* 98.2%

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

   5. add-cbrt-cube 98.2%

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

   6. cbrt-unprod 98.1%

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

7. Applied egg-rr 98.3%

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

   1. *-commutative 98.3%

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

9. Simplified 98.3%

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

10. Taylor expanded in uy around 0 81.7%

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

    1. associate-+r+ 81.7%

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

    2. mul-1-neg 81.7%

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

    3. unsub-neg 81.7%

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

    4. +-commutative 81.7%

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

    5. sub-neg 81.7%

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

    6. metadata-eval 81.7%

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

    7. +-commutative 81.7%

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

    8. +-commutative 81.7%

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

12. Simplified 81.7%

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

13. Taylor expanded in maxCos around 0 81.7%

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

    1. +-commutative 81.7%

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

    2. *-commutative 81.7%

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

    3. *-commutative 81.7%

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

    4. distribute-lft-in 81.7%

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

15. Simplified 81.7%

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

16. Final simplification 81.7%

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

Alternative 11: 81.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in ux around 0 98.2%

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

   1. associate--l+ 98.2%

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

   2. associate-*r* 98.2%

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

   3. mul-1-neg 98.2%

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

   4. sub-neg 98.2%

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

   5. metadata-eval 98.2%

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

   6. +-commutative 98.2%

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

5. Simplified 98.2%

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

6. Taylor expanded in maxCos around 0 98.2%

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

7. Taylor expanded in uy around 0 81.7%

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

   1. *-commutative 81.7%

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

   2. neg-mul-1 81.7%

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

   3. associate-+r+ 81.7%

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

   4. sub-neg 81.7%

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

   5. sub-neg 81.7%

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

9. Simplified 81.7%

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

10. Final simplification 81.7%

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

Alternative 12: 81.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in ux around 0 98.2%

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

   1. associate--l+ 98.2%

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

   2. associate-*r* 98.2%

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

   3. mul-1-neg 98.2%

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

   4. sub-neg 98.2%

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

   5. metadata-eval 98.2%

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

   6. +-commutative 98.2%

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

5. Simplified 98.2%

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

   1. *-commutative 98.2%

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

   2. add-cbrt-cube 98.2%

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

   3. *-commutative 98.2%

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

   4. associate-*r* 98.2%

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

   5. add-cbrt-cube 98.2%

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

   6. cbrt-unprod 98.1%

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

7. Applied egg-rr 98.3%

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

   1. *-commutative 98.3%

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

9. Simplified 98.3%

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

10. Taylor expanded in uy around 0 81.7%

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

    1. associate-+r+ 81.7%

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

    2. mul-1-neg 81.7%

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

    3. unsub-neg 81.7%

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

    4. +-commutative 81.7%

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

    5. sub-neg 81.7%

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

    6. metadata-eval 81.7%

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

    7. +-commutative 81.7%

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

    8. +-commutative 81.7%

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

12. Simplified 81.7%

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

13. Taylor expanded in maxCos around 0 81.1%

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

    1. +-commutative 81.1%

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

    2. mul-1-neg 81.1%

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

    3. unsub-neg 81.1%

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

    4. associate-*r* 81.1%

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

    5. *-commutative 81.1%

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

    6. *-commutative 81.1%

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

15. Simplified 81.1%

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

16. Final simplification 81.1%

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

Alternative 13: 81.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in ux around 0 98.2%

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

   1. associate--l+ 98.2%

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

   2. associate-*r* 98.2%

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

   3. mul-1-neg 98.2%

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

   4. sub-neg 98.2%

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

   5. metadata-eval 98.2%

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

   6. +-commutative 98.2%

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

5. Simplified 98.2%

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

6. Taylor expanded in maxCos around 0 97.5%

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

7. Taylor expanded in uy around 0 81.2%

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

8. Final simplification 81.2%

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

Alternative 14: 80.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in ux around 0 98.2%

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

   1. associate--l+ 98.2%

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

   2. associate-*r* 98.2%

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

   3. mul-1-neg 98.2%

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

   4. sub-neg 98.2%

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

   5. metadata-eval 98.2%

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

   6. +-commutative 98.2%

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

5. Simplified 98.2%

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

   1. *-commutative 98.2%

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

   2. add-cbrt-cube 98.2%

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

   3. *-commutative 98.2%

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

   4. associate-*r* 98.2%

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

   5. add-cbrt-cube 98.2%

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

   6. cbrt-unprod 98.1%

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

7. Applied egg-rr 98.3%

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

   1. *-commutative 98.3%

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

9. Simplified 98.3%

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

10. Taylor expanded in uy around 0 81.7%

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

    1. associate-+r+ 81.7%

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

    2. mul-1-neg 81.7%

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

    3. unsub-neg 81.7%

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

    4. +-commutative 81.7%

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

    5. sub-neg 81.7%

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

    6. metadata-eval 81.7%

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

    7. +-commutative 81.7%

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

    8. +-commutative 81.7%

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

12. Simplified 81.7%

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

13. Taylor expanded in maxCos around 0 80.8%

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

14. Final simplification 80.8%

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

Alternative 15: 77.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in ux around 0 98.2%

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

   1. associate--l+ 98.2%

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

   2. associate-*r* 98.2%

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

   3. mul-1-neg 98.2%

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

   4. sub-neg 98.2%

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

   5. metadata-eval 98.2%

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

   6. +-commutative 98.2%

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

5. Simplified 98.2%

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

   1. *-commutative 98.2%

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

   2. add-cbrt-cube 98.2%

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

   3. *-commutative 98.2%

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

   4. associate-*r* 98.2%

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

   5. add-cbrt-cube 98.2%

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

   6. cbrt-unprod 98.1%

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

7. Applied egg-rr 98.3%

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

   1. *-commutative 98.3%

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

9. Simplified 98.3%

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

10. Taylor expanded in uy around 0 81.7%

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

    1. associate-+r+ 81.7%

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

    2. mul-1-neg 81.7%

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

    3. unsub-neg 81.7%

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

    4. +-commutative 81.7%

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

    5. sub-neg 81.7%

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

    6. metadata-eval 81.7%

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

    7. +-commutative 81.7%

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

    8. +-commutative 81.7%

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

12. Simplified 81.7%

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

13. Taylor expanded in maxCos around 0 78.4%

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

    1. *-commutative 78.4%

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

15. Simplified 78.4%

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

Alternative 16: 7.1% accurate, 223.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 0.0)

C

float code(float ux, float uy, float maxCos) {
	return 0.0f;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = 0.0e0
end function

Julia

function code(ux, uy, maxCos)
	return Float32(0.0)
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = single(0.0);
end

Derivation

1. Initial program 55.6%

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

   1. associate-*l* 55.6%

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

   2. sub-neg 55.6%

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

   3. +-commutative 55.6%

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

   4. distribute-rgt-neg-in 55.6%

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

   5. fma-define 55.9%

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

3. Simplified 55.9%

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

5. Taylor expanded in uy around 0 49.6%

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

7. Simplified 49.6%

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

8. Taylor expanded in ux around 0 7.1%

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

9. Taylor expanded in uy around 0 7.1%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024165 
(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)))))))