UniformSampleCone, y

Percentage Accurate: 57.4% → 98.2%

Time: 15.3s

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: 57.4% 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: 98.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in ux around inf 98.5%

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

   1. add-cbrt-cube 98.4%

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

   2. pow1/3 96.2%

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

5. Applied egg-rr 96.3%

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

   1. unpow1/3 98.5%

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

7. Simplified 98.5%

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

Alternative 2: 98.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in ux around inf 98.5%

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

   1. add-cbrt-cube 98.4%

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

   2. pow1/3 96.2%

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

5. Applied egg-rr 96.3%

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

   1. unpow1/3 98.5%

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

7. Simplified 98.5%

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

8. Taylor expanded in uy around inf 98.4%

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

9. Taylor expanded in ux around 0 98.5%

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

10. Final simplification 98.5%

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

Alternative 3: 98.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in ux around 0 98.4%

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

   1. cancel-sign-sub-inv 98.4%

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

   2. metadata-eval 98.4%

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

   3. associate-*r* 98.4%

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

   4. mul-1-neg 98.4%

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

   5. sub-neg 98.4%

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

   6. metadata-eval 98.4%

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

   7. +-commutative 98.4%

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

   8. *-commutative 98.4%

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

5. Simplified 98.4%

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

6. Final simplification 98.4%

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

Alternative 4: 95.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.00011200000153621659:\\ \;\;\;\;\left(2 \cdot \left(ux \cdot \left(uy \cdot \pi\right)\right)\right) \cdot \sqrt{\frac{2 + maxCos \cdot -2}{ux} - {\left(maxCos + -1\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \left(ux \cdot \sqrt{-1 + \frac{2}{ux}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.00011200000153621659)
   (*
    (* 2.0 (* ux (* uy PI)))
    (sqrt (- (/ (+ 2.0 (* maxCos -2.0)) ux) (pow (+ maxCos -1.0) 2.0))))
   (* (sin (* (* uy 2.0) PI)) (* ux (sqrt (+ -1.0 (/ 2.0 ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.00011200000153621659f) {
		tmp = (2.0f * (ux * (uy * ((float) M_PI)))) * sqrtf((((2.0f + (maxCos * -2.0f)) / ux) - powf((maxCos + -1.0f), 2.0f)));
	} else {
		tmp = sinf(((uy * 2.0f) * ((float) M_PI))) * (ux * sqrtf((-1.0f + (2.0f / ux))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.00011200000153621659))
		tmp = Float32(Float32(Float32(2.0) * Float32(ux * Float32(uy * Float32(pi)))) * sqrt(Float32(Float32(Float32(Float32(2.0) + Float32(maxCos * Float32(-2.0))) / ux) - (Float32(maxCos + Float32(-1.0)) ^ Float32(2.0)))));
	else
		tmp = Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * Float32(ux * sqrt(Float32(Float32(-1.0) + Float32(Float32(2.0) / ux)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if ((uy * single(2.0)) <= single(0.00011200000153621659))
		tmp = (single(2.0) * (ux * (uy * single(pi)))) * sqrt((((single(2.0) + (maxCos * single(-2.0))) / ux) - ((maxCos + single(-1.0)) ^ single(2.0))));
	else
		tmp = sin(((uy * single(2.0)) * single(pi))) * (ux * sqrt((single(-1.0) + (single(2.0) / ux))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 1.12000002e-4

   1. Initial program 59.9%

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

   3. Taylor expanded in ux around inf 98.6%

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

   4. Taylor expanded in uy around 0 98.5%

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

      1. associate-*r* 98.5%

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

      2. associate--r+ 98.6%

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

      3. associate-*r/ 98.6%

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

      4. metadata-eval 98.6%

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

      5. associate-*r/ 98.6%

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

      6. div-sub 98.6%

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

      7. cancel-sign-sub-inv 98.6%

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

      8. metadata-eval 98.6%

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

      9. *-commutative 98.6%

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

      10. sub-neg 98.6%

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

      11. metadata-eval 98.6%

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

   6. Simplified 98.6%

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

   if 1.12000002e-4 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 62.4%

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

   3. Taylor expanded in ux around inf 98.3%

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

   4. Taylor expanded in maxCos around 0 93.5%

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

      1. sub-neg 93.5%

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

      2. associate-*r/ 93.5%

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

      3. metadata-eval 93.5%

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

      4. metadata-eval 93.5%

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

   6. Simplified 93.5%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.00011200000153621659:\\ \;\;\;\;\left(2 \cdot \left(ux \cdot \left(uy \cdot \pi\right)\right)\right) \cdot \sqrt{\frac{2 + maxCos \cdot -2}{ux} - {\left(maxCos + -1\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \left(ux \cdot \sqrt{-1 + \frac{2}{ux}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in ux around inf 98.5%

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

   1. add-cbrt-cube 98.4%

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

   2. pow1/3 96.2%

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

5. Applied egg-rr 96.3%

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

   1. unpow1/3 98.5%

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

7. Simplified 98.5%

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

8. Taylor expanded in uy around inf 98.4%

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

9. Taylor expanded in maxCos around 0 98.4%

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

    1. associate--l+ 98.4%

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

    2. associate-*r/ 98.4%

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

    3. metadata-eval 98.4%

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

    4. associate--l+ 98.4%

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

    5. mul-1-neg 98.4%

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

    6. associate-*r/ 98.4%

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

    7. metadata-eval 98.4%

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

11. Simplified 98.4%

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

12. Final simplification 98.4%

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

Alternative 6: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in ux around inf 98.5%

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

4. Taylor expanded in maxCos around 0 97.6%

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

   1. associate--l+ 97.6%

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

   2. associate-*r/ 97.6%

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

   3. metadata-eval 97.6%

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

   4. associate-*r/ 97.6%

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

   5. metadata-eval 97.6%

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

6. Simplified 97.6%

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

7. Taylor expanded in ux around 0 97.5%

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

8. Final simplification 97.5%

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

Alternative 7: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in ux around inf 98.5%

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

4. Taylor expanded in maxCos around 0 97.6%

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

   1. associate--l+ 97.6%

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

   2. associate-*r/ 97.6%

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

   3. metadata-eval 97.6%

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

   4. associate-*r/ 97.6%

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

   5. metadata-eval 97.6%

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

6. Simplified 97.6%

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

7. Taylor expanded in uy around inf 97.5%

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

   1. associate-*l* 97.3%

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

   2. associate--l+ 97.3%

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

   3. associate-*r/ 97.3%

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

   4. metadata-eval 97.3%

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

   5. associate-*r/ 97.3%

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

   6. metadata-eval 97.3%

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

9. Simplified 97.3%

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

10. Final simplification 97.3%

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

Alternative 8: 89.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.00249999994

   1. Initial program 61.3%

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

   3. Taylor expanded in ux around inf 98.6%

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

   4. Taylor expanded in maxCos around 0 97.5%

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

      1. associate--l+ 97.5%

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

      2. associate-*r/ 97.5%

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

      3. metadata-eval 97.5%

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

      4. associate-*r/ 97.5%

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

      5. metadata-eval 97.5%

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

   6. Simplified 97.5%

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

   7. Taylor expanded in uy around 0 95.3%

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

      1. associate--l+ 95.3%

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

      2. associate-*r/ 95.3%

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

      3. metadata-eval 95.3%

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

      4. associate-*r/ 95.3%

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

      5. metadata-eval 95.3%

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

   9. Simplified 95.3%

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

   if 0.00249999994 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 60.3%

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

   3. Taylor expanded in ux around 0 73.4%

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

   4. Taylor expanded in maxCos around 0 70.2%

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

4. Final simplification 88.7%

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

Alternative 9: 92.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in ux around inf 98.5%

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

   1. add-cbrt-cube 98.4%

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

   2. pow1/3 96.2%

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

5. Applied egg-rr 96.3%

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

   1. unpow1/3 98.5%

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

7. Simplified 98.5%

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

8. Taylor expanded in maxCos around 0 92.8%

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

   1. sub-neg 92.8%

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

   2. associate-*r/ 92.8%

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

   3. metadata-eval 92.8%

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

   4. metadata-eval 92.8%

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

10. Simplified 92.8%

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

11. Final simplification 92.8%

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

Alternative 10: 92.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in ux around inf 98.5%

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

4. Taylor expanded in maxCos around 0 92.7%

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

   1. sub-neg 92.7%

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

   2. associate-*r/ 92.7%

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

   3. metadata-eval 92.7%

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

   4. metadata-eval 92.7%

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

6. Simplified 92.7%

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

7. Final simplification 92.7%

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

Alternative 11: 92.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in ux around inf 98.5%

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

4. Taylor expanded in maxCos around 0 92.8%

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

   1. associate-*l* 92.6%

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

   2. sub-neg 92.6%

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

   3. associate-*r/ 92.6%

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

   4. metadata-eval 92.6%

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

   5. metadata-eval 92.6%

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

6. Simplified 92.6%

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

7. Final simplification 92.6%

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

Alternative 12: 75.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00011300000187475234:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{1 + \left(1 + \left(ux \cdot maxCos - ux\right)\right) \cdot \left(ux + -1\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00011300000187475234)
   (* 2.0 (* uy (* PI (sqrt (* ux (- 2.0 (* 2.0 maxCos)))))))
   (*
    2.0
    (*
     uy
     (* PI (sqrt (+ 1.0 (* (+ 1.0 (- (* ux maxCos) ux)) (+ ux -1.0)))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00011300000187475234f) {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf((ux * (2.0f - (2.0f * maxCos))))));
	} else {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf((1.0f + ((1.0f + ((ux * maxCos) - ux)) * (ux + -1.0f))))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00011300000187475234))
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))))));
	else
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) + Float32(Float32(ux * maxCos) - ux)) * Float32(ux + Float32(-1.0))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00011300000187475234))
		tmp = single(2.0) * (uy * (single(pi) * sqrt((ux * (single(2.0) - (single(2.0) * maxCos))))));
	else
		tmp = single(2.0) * (uy * (single(pi) * sqrt((single(1.0) + ((single(1.0) + ((ux * maxCos) - ux)) * (ux + single(-1.0)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.13000002e-4

   1. Initial program 34.9%

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

      1. associate-*l* 34.9%

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

      2. sub-neg 34.9%

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

      3. +-commutative 34.9%

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

      4. distribute-rgt-neg-in 34.9%

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

      5. fma-define 34.9%

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

   3. Simplified 35.0%

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

   5. Taylor expanded in uy around 0 32.5%

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

   7. Simplified 32.5%

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

   8. Taylor expanded in ux around 0 80.2%

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

   if 1.13000002e-4 < ux

   1. Initial program 89.7%

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

      1. associate-*l* 89.7%

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

      2. sub-neg 89.7%

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

      3. +-commutative 89.7%

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

      4. distribute-rgt-neg-in 89.7%

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

      5. fma-define 90.1%

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

   3. Simplified 90.2%

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

   5. Taylor expanded in uy around 0 78.2%

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

   7. Simplified 78.4%

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

   8. Taylor expanded in maxCos around 0 74.9%

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

      1. neg-mul-1 74.9%

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

   10. Simplified 74.9%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00011300000187475234:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{1 + \left(1 + \left(ux \cdot maxCos - ux\right)\right) \cdot \left(ux + -1\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 80.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in ux around inf 98.5%

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

4. Taylor expanded in maxCos around 0 97.6%

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

   1. associate--l+ 97.6%

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

   2. associate-*r/ 97.6%

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

   3. metadata-eval 97.6%

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

   4. associate-*r/ 97.6%

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

   5. metadata-eval 97.6%

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

6. Simplified 97.6%

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

7. Taylor expanded in uy around 0 82.9%

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

   1. associate--l+ 82.9%

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

   2. associate-*r/ 82.9%

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

   3. metadata-eval 82.9%

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

   4. associate-*r/ 82.9%

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

   5. metadata-eval 82.9%

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

9. Simplified 82.9%

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

10. Final simplification 82.9%

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

Alternative 14: 75.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00011300000187475234:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{1 + \left(1 - ux\right) \cdot \left(ux + -1\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00011300000187475234)
   (* 2.0 (* uy (* PI (sqrt (* ux (- 2.0 (* 2.0 maxCos)))))))
   (* 2.0 (* uy (* PI (sqrt (+ 1.0 (* (- 1.0 ux) (+ ux -1.0)))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00011300000187475234f) {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf((ux * (2.0f - (2.0f * maxCos))))));
	} else {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf((1.0f + ((1.0f - ux) * (ux + -1.0f))))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00011300000187475234))
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))))));
	else
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00011300000187475234))
		tmp = single(2.0) * (uy * (single(pi) * sqrt((ux * (single(2.0) - (single(2.0) * maxCos))))));
	else
		tmp = single(2.0) * (uy * (single(pi) * sqrt((single(1.0) + ((single(1.0) - ux) * (ux + single(-1.0)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.13000002e-4

   1. Initial program 34.9%

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

      1. associate-*l* 34.9%

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

      2. sub-neg 34.9%

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

      3. +-commutative 34.9%

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

      4. distribute-rgt-neg-in 34.9%

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

      5. fma-define 34.9%

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

   3. Simplified 35.0%

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

   5. Taylor expanded in uy around 0 32.5%

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

   7. Simplified 32.5%

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

   8. Taylor expanded in ux around 0 80.2%

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

   if 1.13000002e-4 < ux

   1. Initial program 89.7%

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

      1. associate-*l* 89.7%

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

      2. sub-neg 89.7%

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

      3. +-commutative 89.7%

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

      4. distribute-rgt-neg-in 89.7%

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

      5. fma-define 90.1%

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

   3. Simplified 90.2%

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

   5. Taylor expanded in uy around 0 78.2%

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

   7. Simplified 78.4%

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

   8. Taylor expanded in maxCos around 0 74.6%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00011300000187475234:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{1 + \left(1 - ux\right) \cdot \left(ux + -1\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 65.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.0%

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

   1. associate-*l* 61.0%

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

   2. sub-neg 61.0%

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

   3. +-commutative 61.0%

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

   4. distribute-rgt-neg-in 61.0%

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

   5. fma-define 61.2%

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

3. Simplified 61.3%

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

5. Taylor expanded in uy around 0 54.3%

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

7. Simplified 54.4%

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

8. Taylor expanded in ux around 0 65.3%

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

9. Final simplification 65.3%

   \[\leadsto 2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right)\right) \]
10. 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 61.0%

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

   1. associate-*l* 61.0%

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

   2. sub-neg 61.0%

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

   3. +-commutative 61.0%

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

   4. distribute-rgt-neg-in 61.0%

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

   5. fma-define 61.2%

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

3. Simplified 61.3%

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

5. Taylor expanded in uy around 0 54.3%

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

7. Simplified 54.4%

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

8. Taylor expanded in ux around 0 7.1%

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

9. Taylor expanded in uy around 0 7.1%

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

Reproduce

herbie shell --seed 2024114 
(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)))))))