UniformSampleCone, y

Percentage Accurate: 57.4% → 98.3%

Time: 21.5s

Alternatives: 10

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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.9%

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

3. Taylor expanded in ux around 0 98.4%

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

   1. cancel-sign-sub-inv 98.4%

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

   2. metadata-eval 98.4%

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

   3. associate-*r* 98.4%

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

   4. mul-1-neg 98.4%

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

   5. sub-neg 98.4%

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

   6. metadata-eval 98.4%

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

   7. +-commutative 98.4%

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

   8. *-commutative 98.4%

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

5. Simplified 98.4%

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

6. Final simplification 98.4%

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

Alternative 2: 95.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0012000000569969416:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{ux \cdot \left(\left(2 - ux \cdot {\left(-1 + maxCos\right)}^{2}\right) - 2 \cdot maxCos\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(2 - ux\right) - maxCos\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.0012000000569969416)
   (*
    2.0
    (*
     uy
     (*
      PI
      (sqrt
       (* ux (- (- 2.0 (* ux (pow (+ -1.0 maxCos) 2.0))) (* 2.0 maxCos)))))))
   (* (sin (* uy (* 2.0 PI))) (sqrt (* ux (- (- 2.0 ux) maxCos))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.0012000000569969416f) {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf((ux * ((2.0f - (ux * powf((-1.0f + maxCos), 2.0f))) - (2.0f * maxCos))))));
	} else {
		tmp = sinf((uy * (2.0f * ((float) M_PI)))) * sqrtf((ux * ((2.0f - ux) - maxCos)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.0012000000569969416))
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(ux * Float32(Float32(Float32(2.0) - Float32(ux * (Float32(Float32(-1.0) + maxCos) ^ Float32(2.0)))) - Float32(Float32(2.0) * maxCos)))))));
	else
		tmp = Float32(sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(Float32(2.0) - ux) - maxCos))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if ((uy * single(2.0)) <= single(0.0012000000569969416))
		tmp = single(2.0) * (uy * (single(pi) * sqrt((ux * ((single(2.0) - (ux * ((single(-1.0) + maxCos) ^ single(2.0)))) - (single(2.0) * maxCos))))));
	else
		tmp = sin((uy * (single(2.0) * single(pi)))) * sqrt((ux * ((single(2.0) - ux) - maxCos)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.00120000006

   1. Initial program 56.6%

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

      1. associate-*l* 56.6%

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

      2. sub-neg 56.6%

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

      3. +-commutative 56.6%

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

      4. distribute-rgt-neg-in 56.6%

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

      5. fma-define 56.6%

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

   3. Simplified 56.9%

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

   5. Taylor expanded in uy around 0 56.5%

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

   7. Simplified 56.5%

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

   8. Taylor expanded in ux around 0 98.1%

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

   if 0.00120000006 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 60.8%

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

      1. associate-*l* 60.8%

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

      2. sub-neg 60.8%

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

      3. +-commutative 60.8%

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

      4. distribute-rgt-neg-in 60.8%

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

      5. fma-define 61.1%

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

   3. Simplified 61.1%

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

   5. Taylor expanded in ux around inf 97.7%

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

   6. Taylor expanded in ux around 0 98.0%

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

   7. Taylor expanded in maxCos around 0 94.8%

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

      1. neg-mul-1 94.8%

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

      2. unsub-neg 94.8%

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

   9. Simplified 94.8%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0012000000569969416:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{ux \cdot \left(\left(2 - ux \cdot {\left(-1 + maxCos\right)}^{2}\right) - 2 \cdot maxCos\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(2 - ux\right) - maxCos\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.9%

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

   1. associate-*l* 57.9%

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

   2. sub-neg 57.9%

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

   3. +-commutative 57.9%

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

   4. distribute-rgt-neg-in 57.9%

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

   5. fma-define 58.0%

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

3. Simplified 58.2%

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

5. Taylor expanded in 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. Taylor expanded in ux around 0 98.4%

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

7. Final simplification 98.4%

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

Alternative 4: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.9%

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

3. Taylor expanded in ux around 0 98.4%

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

   1. cancel-sign-sub-inv 98.4%

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

   2. metadata-eval 98.4%

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

   3. associate-*r* 98.4%

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

   4. mul-1-neg 98.4%

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

   5. sub-neg 98.4%

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

   6. metadata-eval 98.4%

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

   7. +-commutative 98.4%

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

   8. *-commutative 98.4%

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

5. Simplified 98.4%

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

6. Taylor expanded in maxCos around 0 97.7%

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

7. Final simplification 97.7%

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

Alternative 5: 95.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0012000000569969416:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{ux \cdot \left(1 - \left(maxCos + \left(\left(-1 + maxCos\right) + ux \cdot \left(\left(-1 + maxCos\right) \cdot \left(-1 + maxCos\right)\right)\right)\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(2 - ux\right) - maxCos\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.0012000000569969416)
   (*
    2.0
    (*
     uy
     (*
      PI
      (sqrt
       (*
        ux
        (-
         1.0
         (+
          maxCos
          (+ (+ -1.0 maxCos) (* ux (* (+ -1.0 maxCos) (+ -1.0 maxCos)))))))))))
   (* (sin (* uy (* 2.0 PI))) (sqrt (* ux (- (- 2.0 ux) maxCos))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.0012000000569969416f) {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf((ux * (1.0f - (maxCos + ((-1.0f + maxCos) + (ux * ((-1.0f + maxCos) * (-1.0f + maxCos))))))))));
	} else {
		tmp = sinf((uy * (2.0f * ((float) M_PI)))) * sqrtf((ux * ((2.0f - ux) - maxCos)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.0012000000569969416))
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(ux * Float32(Float32(1.0) - Float32(maxCos + Float32(Float32(Float32(-1.0) + maxCos) + Float32(ux * Float32(Float32(Float32(-1.0) + maxCos) * Float32(Float32(-1.0) + maxCos)))))))))));
	else
		tmp = Float32(sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(Float32(2.0) - ux) - maxCos))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if ((uy * single(2.0)) <= single(0.0012000000569969416))
		tmp = single(2.0) * (uy * (single(pi) * sqrt((ux * (single(1.0) - (maxCos + ((single(-1.0) + maxCos) + (ux * ((single(-1.0) + maxCos) * (single(-1.0) + maxCos))))))))));
	else
		tmp = sin((uy * (single(2.0) * single(pi)))) * sqrt((ux * ((single(2.0) - ux) - maxCos)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.00120000006

   1. Initial program 56.6%

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

      1. associate-*l* 56.6%

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

      2. sub-neg 56.6%

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

      3. +-commutative 56.6%

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

      4. distribute-rgt-neg-in 56.6%

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

      5. fma-define 56.6%

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

   3. Simplified 56.9%

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

   5. Taylor expanded in uy around 0 56.5%

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

   7. Simplified 56.5%

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

   8. Taylor expanded in ux around -inf 97.9%

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

   9. Taylor expanded in ux around 0 98.0%

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

       1. mul-1-neg 98.0%

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

       2. +-commutative 98.0%

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

       3. unsub-neg 98.0%

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

       4. +-commutative 98.0%

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

       5. mul-1-neg 98.0%

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

       6. unsub-neg 98.0%

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

       7. mul-1-neg 98.0%

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

       8. sub-neg 98.0%

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

       9. sub-neg 98.0%

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

       10. metadata-eval 98.0%

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

       11. sub-neg 98.0%

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

       12. metadata-eval 98.0%

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

   11. Simplified 98.0%

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

   if 0.00120000006 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 60.8%

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

      1. associate-*l* 60.8%

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

      2. sub-neg 60.8%

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

      3. +-commutative 60.8%

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

      4. distribute-rgt-neg-in 60.8%

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

      5. fma-define 61.1%

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

   3. Simplified 61.1%

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

   5. Taylor expanded in ux around inf 97.7%

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

   6. Taylor expanded in ux around 0 98.0%

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

   7. Taylor expanded in maxCos around 0 94.8%

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

      1. neg-mul-1 94.8%

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

      2. unsub-neg 94.8%

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

   9. Simplified 94.8%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0012000000569969416:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{ux \cdot \left(1 - \left(maxCos + \left(\left(-1 + maxCos\right) + ux \cdot \left(\left(-1 + maxCos\right) \cdot \left(-1 + maxCos\right)\right)\right)\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(2 - ux\right) - maxCos\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0012000000569969416:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{ux \cdot \left(1 - \left(maxCos + \left(\left(-1 + maxCos\right) + ux \cdot \left(\left(-1 + maxCos\right) \cdot \left(-1 + maxCos\right)\right)\right)\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.0012000000569969416)
   (*
    2.0
    (*
     uy
     (*
      PI
      (sqrt
       (*
        ux
        (-
         1.0
         (+
          maxCos
          (+ (+ -1.0 maxCos) (* ux (* (+ -1.0 maxCos) (+ -1.0 maxCos)))))))))))
   (* (sin (* uy (* 2.0 PI))) (sqrt (* ux (- 2.0 ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.0012000000569969416f) {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf((ux * (1.0f - (maxCos + ((-1.0f + maxCos) + (ux * ((-1.0f + maxCos) * (-1.0f + maxCos))))))))));
	} else {
		tmp = sinf((uy * (2.0f * ((float) M_PI)))) * sqrtf((ux * (2.0f - ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.0012000000569969416))
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(ux * Float32(Float32(1.0) - Float32(maxCos + Float32(Float32(Float32(-1.0) + maxCos) + Float32(ux * Float32(Float32(Float32(-1.0) + maxCos) * Float32(Float32(-1.0) + maxCos)))))))))));
	else
		tmp = Float32(sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if ((uy * single(2.0)) <= single(0.0012000000569969416))
		tmp = single(2.0) * (uy * (single(pi) * sqrt((ux * (single(1.0) - (maxCos + ((single(-1.0) + maxCos) + (ux * ((single(-1.0) + maxCos) * (single(-1.0) + maxCos))))))))));
	else
		tmp = sin((uy * (single(2.0) * single(pi)))) * sqrt((ux * (single(2.0) - ux)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.00120000006

   1. Initial program 56.6%

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

      1. associate-*l* 56.6%

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

      2. sub-neg 56.6%

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

      3. +-commutative 56.6%

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

      4. distribute-rgt-neg-in 56.6%

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

      5. fma-define 56.6%

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

   3. Simplified 56.9%

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

   5. Taylor expanded in uy around 0 56.5%

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

   7. Simplified 56.5%

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

   8. Taylor expanded in ux around -inf 97.9%

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

   9. Taylor expanded in ux around 0 98.0%

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

       1. mul-1-neg 98.0%

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

       2. +-commutative 98.0%

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

       3. unsub-neg 98.0%

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

       4. +-commutative 98.0%

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

       5. mul-1-neg 98.0%

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

       6. unsub-neg 98.0%

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

       7. mul-1-neg 98.0%

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

       8. sub-neg 98.0%

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

       9. sub-neg 98.0%

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

       10. metadata-eval 98.0%

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

       11. sub-neg 98.0%

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

       12. metadata-eval 98.0%

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

   11. Simplified 98.0%

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

   if 0.00120000006 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 60.8%

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

      1. associate-*l* 60.8%

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

      2. sub-neg 60.8%

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

      3. +-commutative 60.8%

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

      4. distribute-rgt-neg-in 60.8%

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

      5. fma-define 61.1%

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

   3. Simplified 61.1%

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

   5. Taylor expanded in ux around inf 97.7%

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

   6. Taylor expanded in ux around 0 98.0%

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

   7. Taylor expanded in maxCos around 0 94.3%

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

      1. neg-mul-1 94.3%

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

      2. unsub-neg 94.3%

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

   9. Simplified 94.3%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0012000000569969416:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{ux \cdot \left(1 - \left(maxCos + \left(\left(-1 + maxCos\right) + ux \cdot \left(\left(-1 + maxCos\right) \cdot \left(-1 + maxCos\right)\right)\right)\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0019000000320374966:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{ux \cdot \left(1 - \left(maxCos + \left(\left(-1 + maxCos\right) + ux \cdot \left(\left(-1 + maxCos\right) \cdot \left(-1 + maxCos\right)\right)\right)\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.0019000000320374966)
   (*
    2.0
    (*
     uy
     (*
      PI
      (sqrt
       (*
        ux
        (-
         1.0
         (+
          maxCos
          (+ (+ -1.0 maxCos) (* ux (* (+ -1.0 maxCos) (+ -1.0 maxCos)))))))))))
   (* (sin (* uy (* 2.0 PI))) (sqrt (* 2.0 ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.0019000000320374966f) {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf((ux * (1.0f - (maxCos + ((-1.0f + maxCos) + (ux * ((-1.0f + maxCos) * (-1.0f + maxCos))))))))));
	} else {
		tmp = sinf((uy * (2.0f * ((float) M_PI)))) * sqrtf((2.0f * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.0019000000320374966))
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(ux * Float32(Float32(1.0) - Float32(maxCos + Float32(Float32(Float32(-1.0) + maxCos) + Float32(ux * Float32(Float32(Float32(-1.0) + maxCos) * Float32(Float32(-1.0) + maxCos)))))))))));
	else
		tmp = Float32(sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(2.0) * ux)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if ((uy * single(2.0)) <= single(0.0019000000320374966))
		tmp = single(2.0) * (uy * (single(pi) * sqrt((ux * (single(1.0) - (maxCos + ((single(-1.0) + maxCos) + (ux * ((single(-1.0) + maxCos) * (single(-1.0) + maxCos))))))))));
	else
		tmp = sin((uy * (single(2.0) * single(pi)))) * sqrt((single(2.0) * ux));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.00190000003

   1. Initial program 57.0%

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

      1. associate-*l* 57.0%

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

      2. sub-neg 57.0%

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

      3. +-commutative 57.0%

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

      4. distribute-rgt-neg-in 57.0%

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

      5. fma-define 57.1%

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

   3. Simplified 57.3%

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

   5. Taylor expanded in uy around 0 56.8%

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

   7. Simplified 56.8%

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

   8. Taylor expanded in ux around -inf 97.6%

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

   9. Taylor expanded in ux around 0 97.7%

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

       1. mul-1-neg 97.7%

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

       2. +-commutative 97.7%

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

       3. unsub-neg 97.7%

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

       4. +-commutative 97.7%

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

       5. mul-1-neg 97.7%

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

       6. unsub-neg 97.7%

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

       7. mul-1-neg 97.7%

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

       8. sub-neg 97.7%

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

       9. sub-neg 97.7%

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

       10. metadata-eval 97.7%

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

       11. sub-neg 97.7%

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

       12. metadata-eval 97.7%

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

   11. Simplified 97.7%

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

   if 0.00190000003 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 60.0%

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

      1. associate-*l* 60.0%

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

      2. sub-neg 60.0%

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

      3. +-commutative 60.0%

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

      4. distribute-rgt-neg-in 60.0%

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

      5. fma-define 60.3%

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

   3. Simplified 60.3%

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

   5. Taylor expanded in maxCos around 0 58.5%

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

   6. Taylor expanded in ux around 0 74.6%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0019000000320374966:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{ux \cdot \left(1 - \left(maxCos + \left(\left(-1 + maxCos\right) + ux \cdot \left(\left(-1 + maxCos\right) \cdot \left(-1 + maxCos\right)\right)\right)\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.9%

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

   1. associate-*l* 57.9%

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

   2. sub-neg 57.9%

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

   3. +-commutative 57.9%

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

   4. distribute-rgt-neg-in 57.9%

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

   5. fma-define 58.0%

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

3. Simplified 58.2%

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

5. Taylor expanded in uy around 0 50.5%

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

7. Simplified 50.5%

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

8. Taylor expanded in ux around -inf 82.4%

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

9. Taylor expanded in ux around 0 82.5%

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

    1. mul-1-neg 82.5%

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

    2. +-commutative 82.5%

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

    3. unsub-neg 82.5%

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

    4. +-commutative 82.5%

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

    5. mul-1-neg 82.5%

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

    6. unsub-neg 82.5%

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

    7. mul-1-neg 82.5%

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

    8. sub-neg 82.5%

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

    9. sub-neg 82.5%

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

    10. metadata-eval 82.5%

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

    11. sub-neg 82.5%

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

    12. metadata-eval 82.5%

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

11. Simplified 82.5%

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

12. Final simplification 82.5%

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

Alternative 9: 77.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.9%

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

   1. associate-*l* 57.9%

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

   2. sub-neg 57.9%

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

   3. +-commutative 57.9%

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

   4. distribute-rgt-neg-in 57.9%

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

   5. fma-define 58.0%

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

3. Simplified 58.2%

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

5. Taylor expanded in uy around 0 50.5%

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

7. Simplified 50.5%

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

8. Taylor expanded in ux around -inf 82.4%

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

9. Taylor expanded in maxCos around 0 78.3%

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

    1. sub-neg 78.3%

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

    2. metadata-eval 78.3%

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

    3. +-commutative 78.3%

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

    4. associate-*r/ 78.3%

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

    5. metadata-eval 78.3%

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

11. Simplified 78.3%

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

12. Final simplification 78.3%

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

Alternative 10: 63.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.9%

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

   1. associate-*l* 57.9%

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

   2. sub-neg 57.9%

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

   3. +-commutative 57.9%

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

   4. distribute-rgt-neg-in 57.9%

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

   5. fma-define 58.0%

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

3. Simplified 58.2%

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

5. Taylor expanded in uy around 0 50.5%

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

7. Simplified 50.5%

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

8. Taylor expanded in ux around 0 67.3%

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

9. Taylor expanded in maxCos around 0 65.2%

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

10. Final simplification 65.2%

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

Reproduce

herbie shell --seed 2024079 
(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)))))))