UniformSampleCone, y

Percentage Accurate: 57.4% → 98.3%

Time: 15.7s

Alternatives: 20

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: 98.3% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (cbrt
  (*
   (pow (sin (* 2.0 (* uy PI))) 3.0)
   (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 cbrtf((powf(sinf((2.0f * (uy * ((float) M_PI)))), 3.0f) * 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 cbrt(Float32((sin(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) ^ Float32(3.0)) * (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 58.6%

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

   1. add-cbrt-cube 98.3%

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

   2. add-cbrt-cube 98.2%

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

   3. cbrt-unprod 98.2%

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

5. Applied egg-rr 98.3%

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

7. Simplified 98.3%

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

Alternative 2: 98.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.6%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. 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 ux around 0 98.3%

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

5. Final simplification 98.3%

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

Alternative 3: 98.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.6%

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

   1. associate-*l* 58.6%

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

   2. sub-neg 58.6%

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

   3. +-commutative 58.6%

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

   4. distribute-rgt-neg-in 58.6%

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

   5. fma-define 58.7%

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

3. Simplified 58.8%

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

5. Taylor expanded in ux around inf 98.2%

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

6. Final simplification 98.2%

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

Alternative 4: 95.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0006000000284984708:\\ \;\;\;\;\sqrt{\frac{2 + maxCos \cdot -2}{ux} - {\left(maxCos + -1\right)}^{2}} \cdot \left(2 \cdot \left(\left(uy \cdot \pi\right) \cdot ux\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{{ux}^{2} \cdot \left(-1 + \frac{2}{ux}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.0006000000284984708)
   (*
    (sqrt (- (/ (+ 2.0 (* maxCos -2.0)) ux) (pow (+ maxCos -1.0) 2.0)))
    (* 2.0 (* (* uy PI) ux)))
   (* (sin (* PI (* 2.0 uy))) (sqrt (* (pow ux 2.0) (+ -1.0 (/ 2.0 ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((2.0f * uy) <= 0.0006000000284984708f) {
		tmp = sqrtf((((2.0f + (maxCos * -2.0f)) / ux) - powf((maxCos + -1.0f), 2.0f))) * (2.0f * ((uy * ((float) M_PI)) * ux));
	} else {
		tmp = sinf((((float) M_PI) * (2.0f * uy))) * sqrtf((powf(ux, 2.0f) * (-1.0f + (2.0f / ux))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.0006000000284984708))
		tmp = Float32(sqrt(Float32(Float32(Float32(Float32(2.0) + Float32(maxCos * Float32(-2.0))) / ux) - (Float32(maxCos + Float32(-1.0)) ^ Float32(2.0)))) * Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * ux)));
	else
		tmp = Float32(sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy))) * sqrt(Float32((ux ^ Float32(2.0)) * 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 ((single(2.0) * uy) <= single(0.0006000000284984708))
		tmp = sqrt((((single(2.0) + (maxCos * single(-2.0))) / ux) - ((maxCos + single(-1.0)) ^ single(2.0)))) * (single(2.0) * ((uy * single(pi)) * ux));
	else
		tmp = sin((single(pi) * (single(2.0) * uy))) * sqrt(((ux ^ single(2.0)) * (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)) < 6.00000028e-4

   1. Initial program 57.8%

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

   3. Taylor expanded in ux around inf 98.4%

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

   4. Taylor expanded in uy around 0 98.0%

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

      1. associate-*r* 98.0%

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

      2. associate--r+ 98.0%

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

      3. associate-*r/ 98.0%

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

      4. metadata-eval 98.0%

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

      5. associate-*r/ 98.0%

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

      6. div-sub 98.0%

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

      7. cancel-sign-sub-inv 98.0%

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

      8. metadata-eval 98.0%

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

      9. *-commutative 98.0%

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

      10. sub-neg 98.0%

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

      11. metadata-eval 98.0%

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

   6. Simplified 98.0%

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

   if 6.00000028e-4 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 59.8%

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

   3. Taylor expanded in ux around inf 98.1%

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

   4. Taylor expanded in maxCos around 0 91.5%

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

      1. sub-neg 91.5%

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

      2. associate-*r/ 91.5%

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

      3. metadata-eval 91.5%

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

      4. metadata-eval 91.5%

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

   6. Simplified 91.5%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0006000000284984708:\\ \;\;\;\;\sqrt{\frac{2 + maxCos \cdot -2}{ux} - {\left(maxCos + -1\right)}^{2}} \cdot \left(2 \cdot \left(\left(uy \cdot \pi\right) \cdot ux\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{{ux}^{2} \cdot \left(-1 + \frac{2}{ux}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.6%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. 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 uy around inf 98.2%

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

   1. *-commutative 98.2%

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

   2. associate--r+ 98.2%

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

   3. associate-*r/ 98.2%

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

   4. metadata-eval 98.2%

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

   5. associate-*r/ 98.2%

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

   6. div-sub 98.2%

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

   7. cancel-sign-sub-inv 98.2%

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

   8. metadata-eval 98.2%

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

   9. *-commutative 98.2%

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

   10. sub-neg 98.2%

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

   11. metadata-eval 98.2%

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

6. Simplified 98.2%

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

7. Final simplification 98.2%

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

Alternative 6: 98.3% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sin (* PI (* 2.0 uy)))
  (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((((float) M_PI) * (2.0f * uy))) * 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(pi) * Float32(Float32(2.0) * uy))) * 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((single(pi) * (single(2.0) * uy))) * sqrt((ux * ((maxCos * single(-2.0)) + (single(2.0) - (ux * ((maxCos + single(-1.0)) ^ single(2.0)))))));
end

Derivation

1. Initial program 58.6%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. 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 ux around 0 98.2%

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

   1. cancel-sign-sub-inv 98.2%

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

   2. mul-1-neg 98.2%

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

   3. unsub-neg 98.2%

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

   4. sub-neg 98.2%

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

   5. metadata-eval 98.2%

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

   6. metadata-eval 98.2%

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

   7. *-commutative 98.2%

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

6. Simplified 98.2%

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

7. Final simplification 98.2%

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

Alternative 7: 95.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0006000000284984708:\\ \;\;\;\;\sqrt{\frac{2 + maxCos \cdot -2}{ux} - {\left(maxCos + -1\right)}^{2}} \cdot \left(2 \cdot \left(\left(uy \cdot \pi\right) \cdot ux\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \left(ux \cdot \sqrt{-1 + \frac{2}{ux}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.0006000000284984708)
   (*
    (sqrt (- (/ (+ 2.0 (* maxCos -2.0)) ux) (pow (+ maxCos -1.0) 2.0)))
    (* 2.0 (* (* uy PI) ux)))
   (* (sin (* PI (* 2.0 uy))) (* ux (sqrt (+ -1.0 (/ 2.0 ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((2.0f * uy) <= 0.0006000000284984708f) {
		tmp = sqrtf((((2.0f + (maxCos * -2.0f)) / ux) - powf((maxCos + -1.0f), 2.0f))) * (2.0f * ((uy * ((float) M_PI)) * ux));
	} else {
		tmp = sinf((((float) M_PI) * (2.0f * uy))) * (ux * sqrtf((-1.0f + (2.0f / ux))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.0006000000284984708))
		tmp = Float32(sqrt(Float32(Float32(Float32(Float32(2.0) + Float32(maxCos * Float32(-2.0))) / ux) - (Float32(maxCos + Float32(-1.0)) ^ Float32(2.0)))) * Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * ux)));
	else
		tmp = Float32(sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy))) * 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 ((single(2.0) * uy) <= single(0.0006000000284984708))
		tmp = sqrt((((single(2.0) + (maxCos * single(-2.0))) / ux) - ((maxCos + single(-1.0)) ^ single(2.0)))) * (single(2.0) * ((uy * single(pi)) * ux));
	else
		tmp = sin((single(pi) * (single(2.0) * uy))) * (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)) < 6.00000028e-4

   1. Initial program 57.8%

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

   3. Taylor expanded in ux around inf 98.4%

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

   4. Taylor expanded in uy around 0 98.0%

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

      1. associate-*r* 98.0%

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

      2. associate--r+ 98.0%

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

      3. associate-*r/ 98.0%

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

      4. metadata-eval 98.0%

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

      5. associate-*r/ 98.0%

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

      6. div-sub 98.0%

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

      7. cancel-sign-sub-inv 98.0%

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

      8. metadata-eval 98.0%

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

      9. *-commutative 98.0%

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

      10. sub-neg 98.0%

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

      11. metadata-eval 98.0%

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

   6. Simplified 98.0%

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

   if 6.00000028e-4 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 59.8%

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

   3. Taylor expanded in ux around inf 98.1%

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

   4. Taylor expanded in maxCos around 0 91.2%

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

      1. sub-neg 91.2%

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

      2. associate-*r/ 91.2%

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

      3. metadata-eval 91.2%

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

      4. metadata-eval 91.2%

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

   6. Simplified 91.2%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0006000000284984708:\\ \;\;\;\;\sqrt{\frac{2 + maxCos \cdot -2}{ux} - {\left(maxCos + -1\right)}^{2}} \cdot \left(2 \cdot \left(\left(uy \cdot \pi\right) \cdot ux\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \left(ux \cdot \sqrt{-1 + \frac{2}{ux}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{if}\;maxCos \leq 4.999999873689376 \cdot 10^{-6}:\\ \;\;\;\;\left(t\_0 \cdot ux\right) \cdot \sqrt{-1 + \frac{2}{ux}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{maxCos \cdot \left(ux \cdot -2 + 2 \cdot \frac{ux}{maxCos}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (sin (* 2.0 (* uy PI)))))
   (if (<= maxCos 4.999999873689376e-6)
     (* (* t_0 ux) (sqrt (+ -1.0 (/ 2.0 ux))))
     (* t_0 (sqrt (* maxCos (+ (* ux -2.0) (* 2.0 (/ ux maxCos)))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = sinf((2.0f * (uy * ((float) M_PI))));
	float tmp;
	if (maxCos <= 4.999999873689376e-6f) {
		tmp = (t_0 * ux) * sqrtf((-1.0f + (2.0f / ux)));
	} else {
		tmp = t_0 * sqrtf((maxCos * ((ux * -2.0f) + (2.0f * (ux / maxCos)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = sin(Float32(Float32(2.0) * Float32(uy * Float32(pi))))
	tmp = Float32(0.0)
	if (maxCos <= Float32(4.999999873689376e-6))
		tmp = Float32(Float32(t_0 * ux) * sqrt(Float32(Float32(-1.0) + Float32(Float32(2.0) / ux))));
	else
		tmp = Float32(t_0 * sqrt(Float32(maxCos * Float32(Float32(ux * Float32(-2.0)) + Float32(Float32(2.0) * Float32(ux / maxCos))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = sin((single(2.0) * (uy * single(pi))));
	tmp = single(0.0);
	if (maxCos <= single(4.999999873689376e-6))
		tmp = (t_0 * ux) * sqrt((single(-1.0) + (single(2.0) / ux)));
	else
		tmp = t_0 * sqrt((maxCos * ((ux * single(-2.0)) + (single(2.0) * (ux / maxCos)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 4.99999987e-6

   1. Initial program 59.7%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. 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 97.9%

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

      1. sub-neg 97.9%

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

      2. associate-*r/ 97.9%

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

      3. metadata-eval 97.9%

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

      4. metadata-eval 97.9%

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

   6. Simplified 97.9%

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

   if 4.99999987e-6 < maxCos

   1. Initial program 53.5%

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

   3. Taylor expanded in ux around 0 81.9%

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

   4. Taylor expanded in maxCos around inf 82.0%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 4.999999873689376 \cdot 10^{-6}:\\ \;\;\;\;\left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot ux\right) \cdot \sqrt{-1 + \frac{2}{ux}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{maxCos \cdot \left(ux \cdot -2 + 2 \cdot \frac{ux}{maxCos}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{if}\;maxCos \leq 4.999999873689376 \cdot 10^{-6}:\\ \;\;\;\;\left(t\_0 \cdot ux\right) \cdot \sqrt{-1 + \frac{2}{ux}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (sin (* 2.0 (* uy PI)))))
   (if (<= maxCos 4.999999873689376e-6)
     (* (* t_0 ux) (sqrt (+ -1.0 (/ 2.0 ux))))
     (* t_0 (sqrt (* ux (- 2.0 (* 2.0 maxCos))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = sinf((2.0f * (uy * ((float) M_PI))));
	float tmp;
	if (maxCos <= 4.999999873689376e-6f) {
		tmp = (t_0 * ux) * sqrtf((-1.0f + (2.0f / ux)));
	} else {
		tmp = t_0 * sqrtf((ux * (2.0f - (2.0f * maxCos))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = sin(Float32(Float32(2.0) * Float32(uy * Float32(pi))))
	tmp = Float32(0.0)
	if (maxCos <= Float32(4.999999873689376e-6))
		tmp = Float32(Float32(t_0 * ux) * sqrt(Float32(Float32(-1.0) + Float32(Float32(2.0) / ux))));
	else
		tmp = Float32(t_0 * sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = sin((single(2.0) * (uy * single(pi))));
	tmp = single(0.0);
	if (maxCos <= single(4.999999873689376e-6))
		tmp = (t_0 * ux) * sqrt((single(-1.0) + (single(2.0) / ux)));
	else
		tmp = t_0 * sqrt((ux * (single(2.0) - (single(2.0) * maxCos))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 4.99999987e-6

   1. Initial program 59.7%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. 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 97.9%

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

      1. sub-neg 97.9%

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

      2. associate-*r/ 97.9%

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

      3. metadata-eval 97.9%

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

      4. metadata-eval 97.9%

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

   6. Simplified 97.9%

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

   if 4.99999987e-6 < maxCos

   1. Initial program 53.5%

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

   3. Taylor expanded in ux around 0 81.9%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 4.999999873689376 \cdot 10^{-6}:\\ \;\;\;\;\left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot ux\right) \cdot \sqrt{-1 + \frac{2}{ux}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 94.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 4.999999873689376 \cdot 10^{-6}:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \left(ux \cdot \sqrt{-1 + \frac{2}{ux}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 4.999999873689376e-6)
   (* (sin (* PI (* 2.0 uy))) (* ux (sqrt (+ -1.0 (/ 2.0 ux)))))
   (* (sin (* 2.0 (* uy PI))) (sqrt (* ux (- 2.0 (* 2.0 maxCos)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 4.999999873689376e-6f) {
		tmp = sinf((((float) M_PI) * (2.0f * uy))) * (ux * sqrtf((-1.0f + (2.0f / ux))));
	} else {
		tmp = sinf((2.0f * (uy * ((float) M_PI)))) * sqrtf((ux * (2.0f - (2.0f * maxCos))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(4.999999873689376e-6))
		tmp = Float32(sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy))) * Float32(ux * sqrt(Float32(Float32(-1.0) + Float32(Float32(2.0) / ux)))));
	else
		tmp = Float32(sin(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (maxCos <= single(4.999999873689376e-6))
		tmp = sin((single(pi) * (single(2.0) * uy))) * (ux * sqrt((single(-1.0) + (single(2.0) / ux))));
	else
		tmp = sin((single(2.0) * (uy * single(pi)))) * sqrt((ux * (single(2.0) - (single(2.0) * maxCos))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 4.99999987e-6

   1. Initial program 59.7%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. 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 97.8%

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

      1. sub-neg 97.8%

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

      2. associate-*r/ 97.8%

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

      3. metadata-eval 97.8%

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

      4. metadata-eval 97.8%

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

   6. Simplified 97.8%

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

   if 4.99999987e-6 < maxCos

   1. Initial program 53.5%

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

   3. Taylor expanded in ux around 0 81.9%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 4.999999873689376 \cdot 10^{-6}:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \left(ux \cdot \sqrt{-1 + \frac{2}{ux}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 82.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00019999999494757503:\\ \;\;\;\;\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 + \left(1 + ux \cdot \left(maxCos \cdot \left(1 + \frac{-1}{maxCos}\right)\right)\right) \cdot \left(-1 + \left(ux - ux \cdot maxCos\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00019999999494757503)
   (* (sin (* 2.0 (* uy PI))) (sqrt (* 2.0 ux)))
   (*
    2.0
    (*
     (* uy PI)
     (sqrt
      (+
       1.0
       (*
        (+ 1.0 (* ux (* maxCos (+ 1.0 (/ -1.0 maxCos)))))
        (+ -1.0 (- ux (* ux maxCos))))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00019999999494757503f) {
		tmp = sinf((2.0f * (uy * ((float) M_PI)))) * sqrtf((2.0f * ux));
	} else {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf((1.0f + ((1.0f + (ux * (maxCos * (1.0f + (-1.0f / maxCos))))) * (-1.0f + (ux - (ux * maxCos)))))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00019999999494757503))
		tmp = Float32(sin(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) * sqrt(Float32(Float32(2.0) * ux)));
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) + Float32(ux * Float32(maxCos * Float32(Float32(1.0) + Float32(Float32(-1.0) / maxCos))))) * Float32(Float32(-1.0) + Float32(ux - Float32(ux * maxCos))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00019999999494757503))
		tmp = sin((single(2.0) * (uy * single(pi)))) * sqrt((single(2.0) * ux));
	else
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((single(1.0) + ((single(1.0) + (ux * (maxCos * (single(1.0) + (single(-1.0) / maxCos))))) * (single(-1.0) + (ux - (ux * maxCos)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.99999995e-4

   1. Initial program 39.0%

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

   3. Taylor expanded in ux around 0 91.7%

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

   4. Taylor expanded in maxCos around 0 84.8%

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

   if 1.99999995e-4 < ux

   1. Initial program 89.1%

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

      1. associate-*l* 89.1%

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

      2. sub-neg 89.1%

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

      3. +-commutative 89.1%

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

      4. distribute-rgt-neg-in 89.1%

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

      5. fma-define 89.2%

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

   3. Simplified 89.2%

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

   5. Taylor expanded in uy around 0 74.9%

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

   7. Simplified 75.0%

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

   8. Taylor expanded in maxCos around inf 75.0%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00019999999494757503:\\ \;\;\;\;\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 + \left(1 + ux \cdot \left(maxCos \cdot \left(1 + \frac{-1}{maxCos}\right)\right)\right) \cdot \left(-1 + \left(ux - ux \cdot maxCos\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 92.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.6%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. 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 90.6%

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

   1. sub-neg 90.6%

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

   2. associate-*r/ 90.6%

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

   3. metadata-eval 90.6%

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

   4. metadata-eval 90.6%

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

6. Simplified 90.6%

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

7. Final simplification 90.6%

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

Alternative 13: 92.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.6%

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

   1. add-cbrt-cube 98.3%

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

   2. add-cbrt-cube 98.2%

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

   3. cbrt-unprod 98.2%

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

5. Applied egg-rr 98.3%

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

7. Simplified 98.3%

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

   1. *-commutative 98.3%

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

   2. cbrt-prod 98.2%

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

9. Applied egg-rr 98.1%

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

    1. associate-*l* 98.1%

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

11. Simplified 98.1%

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

12. Taylor expanded in maxCos around 0 90.6%

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

    1. sub-neg 90.6%

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

    2. associate-*r/ 90.6%

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

    3. metadata-eval 90.6%

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

    4. metadata-eval 90.6%

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

14. Simplified 90.6%

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

15. Final simplification 90.6%

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

Alternative 14: 76.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00019999999494757503:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 + maxCos \cdot -2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 + \left(1 + ux \cdot \left(maxCos \cdot \left(1 + \frac{-1}{maxCos}\right)\right)\right) \cdot \left(-1 + \left(ux - ux \cdot maxCos\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00019999999494757503)
   (* 2.0 (* (* uy PI) (sqrt (* ux (+ 2.0 (* maxCos -2.0))))))
   (*
    2.0
    (*
     (* uy PI)
     (sqrt
      (+
       1.0
       (*
        (+ 1.0 (* ux (* maxCos (+ 1.0 (/ -1.0 maxCos)))))
        (+ -1.0 (- ux (* ux maxCos))))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00019999999494757503f) {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf((ux * (2.0f + (maxCos * -2.0f)))));
	} else {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf((1.0f + ((1.0f + (ux * (maxCos * (1.0f + (-1.0f / maxCos))))) * (-1.0f + (ux - (ux * maxCos)))))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00019999999494757503))
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(ux * Float32(Float32(2.0) + Float32(maxCos * Float32(-2.0)))))));
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) + Float32(ux * Float32(maxCos * Float32(Float32(1.0) + Float32(Float32(-1.0) / maxCos))))) * Float32(Float32(-1.0) + Float32(ux - Float32(ux * maxCos))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00019999999494757503))
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((ux * (single(2.0) + (maxCos * single(-2.0))))));
	else
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((single(1.0) + ((single(1.0) + (ux * (maxCos * (single(1.0) + (single(-1.0) / maxCos))))) * (single(-1.0) + (ux - (ux * maxCos)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.99999995e-4

   1. Initial program 39.0%

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

      1. associate-*l* 39.0%

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

      2. sub-neg 39.0%

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

      3. +-commutative 39.0%

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

      4. distribute-rgt-neg-in 39.0%

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

      5. fma-define 39.1%

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

   3. Simplified 39.3%

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

   5. Taylor expanded in uy around 0 35.9%

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

   7. Simplified 35.8%

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

   8. Taylor expanded in ux around 0 75.9%

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

      1. *-commutative 75.9%

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

      2. cancel-sign-sub-inv 75.9%

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

      3. metadata-eval 75.9%

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

      4. *-commutative 75.9%

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

   10. Simplified 75.9%

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

   if 1.99999995e-4 < ux

   1. Initial program 89.1%

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

      1. associate-*l* 89.1%

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

      2. sub-neg 89.1%

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

      3. +-commutative 89.1%

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

      4. distribute-rgt-neg-in 89.1%

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

      5. fma-define 89.2%

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

   3. Simplified 89.2%

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

   5. Taylor expanded in uy around 0 74.9%

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

   7. Simplified 75.0%

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

   8. Taylor expanded in maxCos around inf 75.0%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00019999999494757503:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 + maxCos \cdot -2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 + \left(1 + ux \cdot \left(maxCos \cdot \left(1 + \frac{-1}{maxCos}\right)\right)\right) \cdot \left(-1 + \left(ux - ux \cdot maxCos\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 76.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00019999999494757503:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 + maxCos \cdot -2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 + \left(1 + \left(ux \cdot maxCos - ux\right)\right) \cdot \left(-1 - ux \cdot \left(maxCos + -1\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00019999999494757503)
   (* 2.0 (* (* uy PI) (sqrt (* ux (+ 2.0 (* maxCos -2.0))))))
   (*
    2.0
    (*
     (* uy PI)
     (sqrt
      (+
       1.0
       (* (+ 1.0 (- (* ux maxCos) ux)) (- -1.0 (* ux (+ maxCos -1.0))))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00019999999494757503f) {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf((ux * (2.0f + (maxCos * -2.0f)))));
	} else {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf((1.0f + ((1.0f + ((ux * maxCos) - ux)) * (-1.0f - (ux * (maxCos + -1.0f)))))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00019999999494757503))
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(ux * Float32(Float32(2.0) + Float32(maxCos * Float32(-2.0)))))));
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) + Float32(Float32(ux * maxCos) - ux)) * Float32(Float32(-1.0) - Float32(ux * Float32(maxCos + Float32(-1.0)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00019999999494757503))
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((ux * (single(2.0) + (maxCos * single(-2.0))))));
	else
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((single(1.0) + ((single(1.0) + ((ux * maxCos) - ux)) * (single(-1.0) - (ux * (maxCos + single(-1.0))))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.99999995e-4

   1. Initial program 39.0%

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

      1. associate-*l* 39.0%

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

      2. sub-neg 39.0%

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

      3. +-commutative 39.0%

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

      4. distribute-rgt-neg-in 39.0%

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

      5. fma-define 39.1%

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

   3. Simplified 39.3%

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

   5. Taylor expanded in uy around 0 35.9%

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

   7. Simplified 35.8%

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

   8. Taylor expanded in ux around 0 75.9%

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

      1. *-commutative 75.9%

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

      2. cancel-sign-sub-inv 75.9%

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

      3. metadata-eval 75.9%

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

      4. *-commutative 75.9%

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

   10. Simplified 75.9%

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

   if 1.99999995e-4 < ux

   1. Initial program 89.1%

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

      1. associate-*l* 89.1%

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

      2. sub-neg 89.1%

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

      3. +-commutative 89.1%

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

      4. distribute-rgt-neg-in 89.1%

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

      5. fma-define 89.2%

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

   3. Simplified 89.2%

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

   5. Taylor expanded in uy around 0 74.9%

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

   7. Simplified 75.0%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00019999999494757503:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 + maxCos \cdot -2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 + \left(1 + \left(ux \cdot maxCos - ux\right)\right) \cdot \left(-1 - ux \cdot \left(maxCos + -1\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 76.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00019999999494757503:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 + maxCos \cdot -2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(\left(1 + ux \cdot maxCos\right) - ux\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00019999999494757503)
   (* 2.0 (* (* uy PI) (sqrt (* ux (+ 2.0 (* maxCos -2.0))))))
   (*
    2.0
    (*
     (* uy PI)
     (sqrt
      (-
       1.0
       (* (+ 1.0 (* ux (+ maxCos -1.0))) (- (+ 1.0 (* ux maxCos)) ux))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00019999999494757503f) {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf((ux * (2.0f + (maxCos * -2.0f)))));
	} else {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf((1.0f - ((1.0f + (ux * (maxCos + -1.0f))) * ((1.0f + (ux * maxCos)) - ux)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00019999999494757503))
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(ux * Float32(Float32(2.0) + Float32(maxCos * Float32(-2.0)))))));
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(1.0) + Float32(ux * Float32(maxCos + Float32(-1.0)))) * Float32(Float32(Float32(1.0) + Float32(ux * maxCos)) - ux))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00019999999494757503))
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((ux * (single(2.0) + (maxCos * single(-2.0))))));
	else
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((single(1.0) - ((single(1.0) + (ux * (maxCos + single(-1.0)))) * ((single(1.0) + (ux * maxCos)) - ux)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.99999995e-4

   1. Initial program 39.0%

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

      1. associate-*l* 39.0%

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

      2. sub-neg 39.0%

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

      3. +-commutative 39.0%

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

      4. distribute-rgt-neg-in 39.0%

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

      5. fma-define 39.1%

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

   3. Simplified 39.3%

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

   5. Taylor expanded in uy around 0 35.9%

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

   7. Simplified 35.8%

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

   8. Taylor expanded in ux around 0 75.9%

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

      1. *-commutative 75.9%

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

      2. cancel-sign-sub-inv 75.9%

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

      3. metadata-eval 75.9%

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

      4. *-commutative 75.9%

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

   10. Simplified 75.9%

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

   if 1.99999995e-4 < ux

   1. Initial program 89.1%

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

      1. associate-*l* 89.1%

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

      2. sub-neg 89.1%

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

      3. +-commutative 89.1%

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

      4. distribute-rgt-neg-in 89.1%

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

      5. fma-define 89.2%

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

   3. Simplified 89.2%

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

   5. Taylor expanded in uy around 0 74.9%

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

   7. Simplified 75.0%

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

   8. Taylor expanded in uy around 0 74.9%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00019999999494757503:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 + maxCos \cdot -2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(\left(1 + ux \cdot maxCos\right) - ux\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 74.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00031999999191612005:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 + maxCos \cdot -2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 + \left(1 + \left(ux \cdot maxCos - ux\right)\right) \cdot \left(ux + -1\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00031999999191612005)
   (* 2.0 (* (* uy PI) (sqrt (* ux (+ 2.0 (* maxCos -2.0))))))
   (*
    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 <= 0.00031999999191612005f) {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf((ux * (2.0f + (maxCos * -2.0f)))));
	} 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(0.00031999999191612005))
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(ux * Float32(Float32(2.0) + Float32(maxCos * Float32(-2.0)))))));
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) + Float32(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(0.00031999999191612005))
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((ux * (single(2.0) + (maxCos * single(-2.0))))));
	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 < 3.19999992e-4

   1. Initial program 41.1%

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

      1. associate-*l* 41.1%

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

      2. sub-neg 41.1%

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

      3. +-commutative 41.1%

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

      4. distribute-rgt-neg-in 41.1%

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

      5. fma-define 41.3%

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

   3. Simplified 41.4%

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

   5. Taylor expanded in uy around 0 38.1%

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

   7. Simplified 37.9%

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

   8. Taylor expanded in ux around 0 75.4%

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

      1. *-commutative 75.4%

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

      2. cancel-sign-sub-inv 75.4%

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

      3. metadata-eval 75.4%

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

      4. *-commutative 75.4%

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

   10. Simplified 75.4%

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

   if 3.19999992e-4 < ux

   1. Initial program 90.8%

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

      1. associate-*l* 90.8%

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

      2. sub-neg 90.8%

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

      3. +-commutative 90.8%

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

      4. distribute-rgt-neg-in 90.8%

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

      5. fma-define 90.8%

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

   3. Simplified 90.8%

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

   5. Taylor expanded in uy around 0 75.2%

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

   7. Simplified 75.4%

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

   8. Taylor expanded in maxCos around 0 72.8%

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

      1. neg-mul-1 72.8%

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

   10. Simplified 72.8%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00031999999191612005:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 + maxCos \cdot -2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 + \left(1 + \left(ux \cdot maxCos - ux\right)\right) \cdot \left(ux + -1\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 74.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00031999999191612005:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 + maxCos \cdot -2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{1 + \left(1 - ux\right) \cdot \left(ux + -1\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00031999999191612005)
   (* 2.0 (* (* uy PI) (sqrt (* ux (+ 2.0 (* maxCos -2.0))))))
   (* 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 <= 0.00031999999191612005f) {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf((ux * (2.0f + (maxCos * -2.0f)))));
	} 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(0.00031999999191612005))
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(ux * Float32(Float32(2.0) + Float32(maxCos * Float32(-2.0)))))));
	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(0.00031999999191612005))
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((ux * (single(2.0) + (maxCos * single(-2.0))))));
	else
		tmp = single(2.0) * (uy * (single(pi) * sqrt((single(1.0) + ((single(1.0) - ux) * (ux + single(-1.0)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 3.19999992e-4

   1. Initial program 41.1%

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

      1. associate-*l* 41.1%

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

      2. sub-neg 41.1%

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

      3. +-commutative 41.1%

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

      4. distribute-rgt-neg-in 41.1%

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

      5. fma-define 41.3%

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

   3. Simplified 41.4%

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

   5. Taylor expanded in uy around 0 38.1%

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

   7. Simplified 37.9%

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

   8. Taylor expanded in ux around 0 75.4%

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

      1. *-commutative 75.4%

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

      2. cancel-sign-sub-inv 75.4%

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

      3. metadata-eval 75.4%

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

      4. *-commutative 75.4%

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

   10. Simplified 75.4%

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

   if 3.19999992e-4 < ux

   1. Initial program 90.8%

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

      1. associate-*l* 90.8%

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

      2. sub-neg 90.8%

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

      3. +-commutative 90.8%

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

      4. distribute-rgt-neg-in 90.8%

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

      5. fma-define 90.8%

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

   3. Simplified 90.8%

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

   5. Taylor expanded in uy around 0 75.2%

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

   7. Simplified 75.4%

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

      1. add-cube-cbrt 75.4%

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

      2. pow3 75.4%

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

   9. Applied egg-rr 75.4%

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

   10. Taylor expanded in maxCos around 0 72.5%

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

       1. associate-*l* 72.6%

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

       2. mul-1-neg 72.6%

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

       3. sub-neg 72.6%

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

   12. Simplified 72.6%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00031999999191612005:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 + maxCos \cdot -2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{1 + \left(1 - ux\right) \cdot \left(ux + -1\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 65.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.6%

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

   1. associate-*l* 58.6%

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

   2. sub-neg 58.6%

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

   3. +-commutative 58.6%

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

   4. distribute-rgt-neg-in 58.6%

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

   5. fma-define 58.7%

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

3. Simplified 58.8%

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

5. Taylor expanded in uy around 0 51.2%

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

7. Simplified 51.1%

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

8. Taylor expanded in ux around 0 65.3%

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

   1. *-commutative 65.3%

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

   2. cancel-sign-sub-inv 65.3%

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

   3. metadata-eval 65.3%

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

   4. *-commutative 65.3%

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

10. Simplified 65.3%

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

Alternative 20: 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 58.6%

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

   1. associate-*l* 58.6%

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

   2. sub-neg 58.6%

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

   3. +-commutative 58.6%

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

   4. distribute-rgt-neg-in 58.6%

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

   5. fma-define 58.7%

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

3. Simplified 58.8%

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

5. Taylor expanded in uy around 0 51.2%

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

7. Simplified 51.1%

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

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 2024141 
(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)))))))