Beckmann Sample, near normal, slope_y

Percentage Accurate: 58.4% → 98.4%

Time: 16.6s

Alternatives: 10

Speedup: 2.9×

Specification

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\begin{array}{l} \\ \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-logf((1.0f - u1))) * sinf(((2.0f * ((float) M_PI)) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(-log((single(1.0) - u1))) * sin(((single(2.0) * single(pi)) * u2));
end

Initial Program: 58.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-logf((1.0f - u1))) * sinf(((2.0f * ((float) M_PI)) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(-log((single(1.0) - u1))) * sin(((single(2.0) * single(pi)) * u2));
end

Alternative 1: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \sqrt[3]{\left(t\_0 \cdot {t\_0}^{2}\right) \cdot {\left(-\mathsf{log1p}\left(-u1\right)\right)}^{1.5}} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* 2.0 (* PI u2)))))
   (cbrt (* (* t_0 (pow t_0 2.0)) (pow (- (log1p (- u1))) 1.5)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sinf((2.0f * (((float) M_PI) * u2)));
	return cbrtf(((t_0 * powf(t_0, 2.0f)) * powf(-log1pf(-u1), 1.5f)));
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sin(Float32(Float32(2.0) * Float32(Float32(pi) * u2)))
	return cbrt(Float32(Float32(t_0 * (t_0 ^ Float32(2.0))) * (Float32(-log1p(Float32(-u1))) ^ Float32(1.5))))
end

Derivation

1. Initial program 58.6%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Step-by-step derivation

   1. sub-neg 58.6%

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. log1p-define 98.5%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

3. Simplified 98.5%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*l* 98.5%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(2 \cdot \left(\pi \cdot u2\right)\right)} \]

   2. sin-2 98.3%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right)} \]

6. Applied egg-rr 98.3%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right)} \]
7. Step-by-step derivation

   1. sin-2 98.5%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)} \]

   2. *-commutative 98.5%

      \[\leadsto \color{blue}{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}} \]

   3. add-cbrt-cube 98.5%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\sin \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot u2\right)\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}} \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)} \]

   4. add-cbrt-cube 98.5%

      \[\leadsto \sqrt[3]{\left(\sin \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot u2\right)\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot u2\right)\right)} \cdot \color{blue}{\sqrt[3]{\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}}} \]

   5. cbrt-unprod 98.3%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sin \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot u2\right)\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot u2\right)\right)\right) \cdot \left(\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right)}} \]

   6. pow3 98.2%

      \[\leadsto \sqrt[3]{\color{blue}{{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3}} \cdot \left(\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right)} \]

8. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{\sqrt[3]{{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3} \cdot {\left(-\mathsf{log1p}\left(-u1\right)\right)}^{1.5}}} \]
9. Step-by-step derivation

   1. unpow3 98.5%

      \[\leadsto \sqrt[3]{\color{blue}{\left(\left(\sin \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot u2\right)\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)} \cdot {\left(-\mathsf{log1p}\left(-u1\right)\right)}^{1.5}} \]

   2. pow2 98.5%

      \[\leadsto \sqrt[3]{\left(\color{blue}{{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}^{2}} \cdot \sin \left(2 \cdot \left(\pi \cdot u2\right)\right)\right) \cdot {\left(-\mathsf{log1p}\left(-u1\right)\right)}^{1.5}} \]

10. Applied egg-rr 98.5%

    \[\leadsto \sqrt[3]{\color{blue}{\left({\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}^{2} \cdot \sin \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)} \cdot {\left(-\mathsf{log1p}\left(-u1\right)\right)}^{1.5}} \]

11. Final simplification 98.5%

    \[\leadsto \sqrt[3]{\left(\sin \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot {\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}^{2}\right) \cdot {\left(-\mathsf{log1p}\left(-u1\right)\right)}^{1.5}} \]
12. Add Preprocessing

Alternative 2: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(u2 \cdot \left(2 \cdot \pi\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log1p (- u1)))) (sin (* u2 (* 2.0 PI)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * sinf((u2 * (2.0f * ((float) M_PI))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * sin(Float32(u2 * Float32(Float32(2.0) * Float32(pi)))))
end

Derivation

1. Initial program 58.6%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Step-by-step derivation

   1. sub-neg 58.6%

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. log1p-define 98.5%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

3. Simplified 98.5%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)} \]
4. Add Preprocessing

5. Final simplification 98.5%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(u2 \cdot \left(2 \cdot \pi\right)\right) \]
6. Add Preprocessing

Alternative 3: 95.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(2 \cdot \pi\right)\\ \mathbf{if}\;t\_0 \leq 0.002099999925121665:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 - u1 \cdot -0.3333333333333333\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* 2.0 PI))))
   (if (<= t_0 0.002099999925121665)
     (* (sqrt (- (log1p (- u1)))) (* 2.0 (* PI u2)))
     (*
      (sin t_0)
      (sqrt (* u1 (+ 1.0 (* u1 (- 0.5 (* u1 -0.3333333333333333))))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u2 * (2.0f * ((float) M_PI));
	float tmp;
	if (t_0 <= 0.002099999925121665f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (((float) M_PI) * u2));
	} else {
		tmp = sinf(t_0) * sqrtf((u1 * (1.0f + (u1 * (0.5f - (u1 * -0.3333333333333333f))))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u2 * Float32(Float32(2.0) * Float32(pi)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.002099999925121665))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	else
		tmp = Float32(sin(t_0) * sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(u1 * Float32(Float32(0.5) - Float32(u1 * Float32(-0.3333333333333333))))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00209999993

   1. Initial program 58.8%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Step-by-step derivation

      1. sub-neg 58.8%

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. log1p-define 98.7%

         \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. Simplified 98.7%

      \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l* 98.7%

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(2 \cdot \left(\pi \cdot u2\right)\right)} \]

      2. sin-2 98.6%

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right)} \]

   6. Applied egg-rr 98.6%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right)} \]

   7. Taylor expanded in u2 around 0 98.3%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \color{blue}{\left(u2 \cdot \pi\right)}\right) \]

   if 0.00209999993 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 58.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u1 around 0 89.7%

      \[\leadsto \sqrt{-\color{blue}{u1 \cdot \left(u1 \cdot \left(-0.3333333333333333 \cdot u1 - 0.5\right) - 1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.002099999925121665:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 - u1 \cdot -0.3333333333333333\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(2 \cdot \pi\right)\\ \mathbf{if}\;t\_0 \leq 0.002099999925121665:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{u1 \cdot \left(1 - u1 \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* 2.0 PI))))
   (if (<= t_0 0.002099999925121665)
     (* (sqrt (- (log1p (- u1)))) (* 2.0 (* PI u2)))
     (* (sin t_0) (sqrt (* u1 (- 1.0 (* u1 -0.5))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u2 * (2.0f * ((float) M_PI));
	float tmp;
	if (t_0 <= 0.002099999925121665f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (((float) M_PI) * u2));
	} else {
		tmp = sinf(t_0) * sqrtf((u1 * (1.0f - (u1 * -0.5f))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u2 * Float32(Float32(2.0) * Float32(pi)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.002099999925121665))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	else
		tmp = Float32(sin(t_0) * sqrt(Float32(u1 * Float32(Float32(1.0) - Float32(u1 * Float32(-0.5))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00209999993

   1. Initial program 58.8%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Step-by-step derivation

      1. sub-neg 58.8%

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. log1p-define 98.7%

         \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. Simplified 98.7%

      \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l* 98.7%

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(2 \cdot \left(\pi \cdot u2\right)\right)} \]

      2. sin-2 98.6%

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right)} \]

   6. Applied egg-rr 98.6%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right)} \]

   7. Taylor expanded in u2 around 0 98.3%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \color{blue}{\left(u2 \cdot \pi\right)}\right) \]

   if 0.00209999993 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 58.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u1 around 0 86.2%

      \[\leadsto \sqrt{-\color{blue}{u1 \cdot \left(-0.5 \cdot u1 - 1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.002099999925121665:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{u1 \cdot \left(1 - u1 \cdot -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \sin \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 - u1 \cdot \left(u1 \cdot -0.25 - 0.3333333333333333\right)\right)\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sin (* u2 (* 2.0 PI)))
  (sqrt
   (* u1 (+ 1.0 (* u1 (- 0.5 (* u1 (- (* u1 -0.25) 0.3333333333333333)))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf((u2 * (2.0f * ((float) M_PI)))) * sqrtf((u1 * (1.0f + (u1 * (0.5f - (u1 * ((u1 * -0.25f) - 0.3333333333333333f)))))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sin(Float32(u2 * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(u1 * Float32(Float32(0.5) - Float32(u1 * Float32(Float32(u1 * Float32(-0.25)) - Float32(0.3333333333333333)))))))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sin((u2 * (single(2.0) * single(pi)))) * sqrt((u1 * (single(1.0) + (u1 * (single(0.5) - (u1 * ((u1 * single(-0.25)) - single(0.3333333333333333))))))));
end

Derivation

1. Initial program 58.6%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u1 around 0 93.3%

   \[\leadsto \sqrt{-\color{blue}{u1 \cdot \left(u1 \cdot \left(u1 \cdot \left(-0.25 \cdot u1 - 0.3333333333333333\right) - 0.5\right) - 1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

4. Final simplification 93.3%

   \[\leadsto \sin \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 - u1 \cdot \left(u1 \cdot -0.25 - 0.3333333333333333\right)\right)\right)} \]
5. Add Preprocessing

Alternative 6: 90.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.04500000178813934:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 (* 2.0 PI)) 0.04500000178813934)
   (* (sqrt (- (log1p (- u1)))) (* 2.0 (* PI u2)))
   (* (sqrt u1) (sin (* PI (* 2.0 u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * (2.0f * ((float) M_PI))) <= 0.04500000178813934f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (((float) M_PI) * u2));
	} else {
		tmp = sqrtf(u1) * sinf((((float) M_PI) * (2.0f * u2)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(Float32(2.0) * Float32(pi))) <= Float32(0.04500000178813934))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	else
		tmp = Float32(sqrt(u1) * sin(Float32(Float32(pi) * Float32(Float32(2.0) * u2))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.0450000018

   1. Initial program 59.3%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Step-by-step derivation

      1. sub-neg 59.3%

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. log1p-define 98.6%

         \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. Simplified 98.6%

      \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l* 98.6%

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(2 \cdot \left(\pi \cdot u2\right)\right)} \]

      2. sin-2 98.5%

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right)} \]

   6. Applied egg-rr 98.5%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right)} \]

   7. Taylor expanded in u2 around 0 94.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \color{blue}{\left(u2 \cdot \pi\right)}\right) \]

   if 0.0450000018 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 55.8%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Step-by-step derivation

      1. sub-neg 55.8%

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. log1p-define 98.0%

         \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. Simplified 98.0%

      \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cbrt-cube 98.0%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. pow1/3 95.5%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right)}^{0.3333333333333333}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   6. Applied egg-rr 72.3%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   7. Step-by-step derivation

      1. unpow1/3 74.2%

         \[\leadsto \color{blue}{\sqrt[3]{{\left(\mathsf{log1p}\left(u1\right)\right)}^{1.5}}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   8. Simplified 74.2%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\mathsf{log1p}\left(u1\right)\right)}^{1.5}}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   9. Taylor expanded in u1 around 0 76.1%

      \[\leadsto \color{blue}{\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* 76.1%

          \[\leadsto \sqrt{u1} \cdot \sin \color{blue}{\left(\left(2 \cdot u2\right) \cdot \pi\right)} \]

       2. *-commutative 76.1%

          \[\leadsto \sqrt{u1} \cdot \sin \color{blue}{\left(\pi \cdot \left(2 \cdot u2\right)\right)} \]

       3. *-commutative 76.1%

          \[\leadsto \sqrt{u1} \cdot \sin \left(\pi \cdot \color{blue}{\left(u2 \cdot 2\right)}\right) \]

   11. Simplified 76.1%

       \[\leadsto \color{blue}{\sqrt{u1} \cdot \sin \left(\pi \cdot \left(u2 \cdot 2\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.04500000178813934:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt u1) (sin (* PI (* 2.0 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(u1) * sinf((((float) M_PI) * (2.0f * u2)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(u1) * sin(Float32(Float32(pi) * Float32(Float32(2.0) * u2))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(u1) * sin((single(pi) * (single(2.0) * u2)));
end

Derivation

1. Initial program 58.6%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Step-by-step derivation

   1. sub-neg 58.6%

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. log1p-define 98.5%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

3. Simplified 98.5%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-cbrt-cube 98.5%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. pow1/3 96.0%

      \[\leadsto \color{blue}{{\left(\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right)}^{0.3333333333333333}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

6. Applied egg-rr 72.1%

   \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
7. Step-by-step derivation

   1. unpow1/3 73.7%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\mathsf{log1p}\left(u1\right)\right)}^{1.5}}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

8. Simplified 73.7%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\mathsf{log1p}\left(u1\right)\right)}^{1.5}}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

9. Taylor expanded in u1 around 0 75.8%

   \[\leadsto \color{blue}{\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)} \]
10. Step-by-step derivation

    1. associate-*r* 75.8%

       \[\leadsto \sqrt{u1} \cdot \sin \color{blue}{\left(\left(2 \cdot u2\right) \cdot \pi\right)} \]

    2. *-commutative 75.8%

       \[\leadsto \sqrt{u1} \cdot \sin \color{blue}{\left(\pi \cdot \left(2 \cdot u2\right)\right)} \]

    3. *-commutative 75.8%

       \[\leadsto \sqrt{u1} \cdot \sin \left(\pi \cdot \color{blue}{\left(u2 \cdot 2\right)}\right) \]

11. Simplified 75.8%

    \[\leadsto \color{blue}{\sqrt{u1} \cdot \sin \left(\pi \cdot \left(u2 \cdot 2\right)\right)} \]

12. Final simplification 75.8%

    \[\leadsto \sqrt{u1} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right) \]
13. Add Preprocessing

Alternative 8: 66.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\pi \cdot \left(u2 \cdot \sqrt{u1}\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (* 2.0 (* PI (* u2 (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return 2.0f * (((float) M_PI) * (u2 * sqrtf(u1)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(2.0) * Float32(Float32(pi) * Float32(u2 * sqrt(u1))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(2.0) * (single(pi) * (u2 * sqrt(u1)));
end

Derivation

1. Initial program 58.6%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u1 around 0 -0.0%

   \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
4. Step-by-step derivation

   1. associate-*r* -0.0%

      \[\leadsto \color{blue}{\left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}} \]

   2. unpow2 -0.0%

      \[\leadsto \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right) \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \]

   3. rem-square-sqrt 4.0%

      \[\leadsto \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right) \cdot \color{blue}{-1} \]

   4. *-commutative 4.0%

      \[\leadsto \color{blue}{\left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \sqrt{u1}\right)} \cdot -1 \]

   5. associate-*l* 4.0%

      \[\leadsto \color{blue}{\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \left(\sqrt{u1} \cdot -1\right)} \]

   6. *-commutative 4.0%

      \[\leadsto \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \color{blue}{\left(-1 \cdot \sqrt{u1}\right)} \]

   7. mul-1-neg 4.0%

      \[\leadsto \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \color{blue}{\left(-\sqrt{u1}\right)} \]

5. Simplified 4.0%

   \[\leadsto \color{blue}{\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \left(-\sqrt{u1}\right)} \]
6. Step-by-step derivation

   1. add-cube-cbrt 4.0%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \left(-\sqrt{u1}\right)} \cdot \sqrt[3]{\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \left(-\sqrt{u1}\right)}\right) \cdot \sqrt[3]{\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \left(-\sqrt{u1}\right)}} \]

   2. pow3 4.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \left(-\sqrt{u1}\right)}\right)}^{3}} \]

   3. *-commutative 4.0%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(-\sqrt{u1}\right) \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)}}\right)}^{3} \]

   4. add-sqr-sqrt -0.0%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(\sqrt{-\sqrt{u1}} \cdot \sqrt{-\sqrt{u1}}\right)} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)}\right)}^{3} \]

   5. sqrt-unprod 75.5%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{\sqrt{\left(-\sqrt{u1}\right) \cdot \left(-\sqrt{u1}\right)}} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)}\right)}^{3} \]

   6. sqr-neg 75.5%

      \[\leadsto {\left(\sqrt[3]{\sqrt{\color{blue}{\sqrt{u1} \cdot \sqrt{u1}}} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)}\right)}^{3} \]

   7. add-sqr-sqrt 75.5%

      \[\leadsto {\left(\sqrt[3]{\sqrt{\color{blue}{u1}} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)}\right)}^{3} \]

   8. *-commutative 75.5%

      \[\leadsto {\left(\sqrt[3]{\sqrt{u1} \cdot \sin \left(2 \cdot \color{blue}{\left(\pi \cdot u2\right)}\right)}\right)}^{3} \]

   9. associate-*r* 75.5%

      \[\leadsto {\left(\sqrt[3]{\sqrt{u1} \cdot \sin \color{blue}{\left(\left(2 \cdot \pi\right) \cdot u2\right)}}\right)}^{3} \]

   10. *-commutative 75.5%

       \[\leadsto {\left(\sqrt[3]{\sqrt{u1} \cdot \sin \left(\color{blue}{\left(\pi \cdot 2\right)} \cdot u2\right)}\right)}^{3} \]

   11. associate-*l* 75.5%

       \[\leadsto {\left(\sqrt[3]{\sqrt{u1} \cdot \sin \color{blue}{\left(\pi \cdot \left(2 \cdot u2\right)\right)}}\right)}^{3} \]

7. Applied egg-rr 75.5%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{u1} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right)}\right)}^{3}} \]

8. Taylor expanded in u2 around 0 67.6%

   \[\leadsto \color{blue}{2 \cdot \left(\sqrt{u1} \cdot \left(u2 \cdot \pi\right)\right)} \]
9. Step-by-step derivation

   1. associate-*r* 67.7%

      \[\leadsto 2 \cdot \color{blue}{\left(\left(\sqrt{u1} \cdot u2\right) \cdot \pi\right)} \]

   2. *-commutative 67.7%

      \[\leadsto 2 \cdot \left(\color{blue}{\left(u2 \cdot \sqrt{u1}\right)} \cdot \pi\right) \]

10. Simplified 67.7%

    \[\leadsto \color{blue}{2 \cdot \left(\left(u2 \cdot \sqrt{u1}\right) \cdot \pi\right)} \]

11. Final simplification 67.7%

    \[\leadsto 2 \cdot \left(\pi \cdot \left(u2 \cdot \sqrt{u1}\right)\right) \]
12. Add Preprocessing

Alternative 9: 66.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(u2 \cdot \left(\pi \cdot \sqrt{u1}\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (* 2.0 (* u2 (* PI (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return 2.0f * (u2 * (((float) M_PI) * sqrtf(u1)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(2.0) * Float32(u2 * Float32(Float32(pi) * sqrt(u1))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(2.0) * (u2 * (single(pi) * sqrt(u1)));
end

Derivation

1. Initial program 58.6%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u1 around 0 -0.0%

   \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
4. Step-by-step derivation

   1. associate-*r* -0.0%

      \[\leadsto \color{blue}{\left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}} \]

   2. unpow2 -0.0%

      \[\leadsto \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right) \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \]

   3. rem-square-sqrt 4.0%

      \[\leadsto \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right) \cdot \color{blue}{-1} \]

   4. *-commutative 4.0%

      \[\leadsto \color{blue}{\left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \sqrt{u1}\right)} \cdot -1 \]

   5. associate-*l* 4.0%

      \[\leadsto \color{blue}{\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \left(\sqrt{u1} \cdot -1\right)} \]

   6. *-commutative 4.0%

      \[\leadsto \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \color{blue}{\left(-1 \cdot \sqrt{u1}\right)} \]

   7. mul-1-neg 4.0%

      \[\leadsto \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \color{blue}{\left(-\sqrt{u1}\right)} \]

5. Simplified 4.0%

   \[\leadsto \color{blue}{\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \left(-\sqrt{u1}\right)} \]
6. Step-by-step derivation

   1. add-cube-cbrt 4.0%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \left(-\sqrt{u1}\right)} \cdot \sqrt[3]{\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \left(-\sqrt{u1}\right)}\right) \cdot \sqrt[3]{\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \left(-\sqrt{u1}\right)}} \]

   2. pow3 4.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \left(-\sqrt{u1}\right)}\right)}^{3}} \]

   3. *-commutative 4.0%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(-\sqrt{u1}\right) \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)}}\right)}^{3} \]

   4. add-sqr-sqrt -0.0%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(\sqrt{-\sqrt{u1}} \cdot \sqrt{-\sqrt{u1}}\right)} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)}\right)}^{3} \]

   5. sqrt-unprod 75.5%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{\sqrt{\left(-\sqrt{u1}\right) \cdot \left(-\sqrt{u1}\right)}} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)}\right)}^{3} \]

   6. sqr-neg 75.5%

      \[\leadsto {\left(\sqrt[3]{\sqrt{\color{blue}{\sqrt{u1} \cdot \sqrt{u1}}} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)}\right)}^{3} \]

   7. add-sqr-sqrt 75.5%

      \[\leadsto {\left(\sqrt[3]{\sqrt{\color{blue}{u1}} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)}\right)}^{3} \]

   8. *-commutative 75.5%

      \[\leadsto {\left(\sqrt[3]{\sqrt{u1} \cdot \sin \left(2 \cdot \color{blue}{\left(\pi \cdot u2\right)}\right)}\right)}^{3} \]

   9. associate-*r* 75.5%

      \[\leadsto {\left(\sqrt[3]{\sqrt{u1} \cdot \sin \color{blue}{\left(\left(2 \cdot \pi\right) \cdot u2\right)}}\right)}^{3} \]

   10. *-commutative 75.5%

       \[\leadsto {\left(\sqrt[3]{\sqrt{u1} \cdot \sin \left(\color{blue}{\left(\pi \cdot 2\right)} \cdot u2\right)}\right)}^{3} \]

   11. associate-*l* 75.5%

       \[\leadsto {\left(\sqrt[3]{\sqrt{u1} \cdot \sin \color{blue}{\left(\pi \cdot \left(2 \cdot u2\right)\right)}}\right)}^{3} \]

7. Applied egg-rr 75.5%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{u1} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right)}\right)}^{3}} \]

8. Taylor expanded in u2 around 0 67.6%

   \[\leadsto \color{blue}{2 \cdot \left(\sqrt{u1} \cdot \left(u2 \cdot \pi\right)\right)} \]
9. Step-by-step derivation

   1. associate-*r* 67.7%

      \[\leadsto 2 \cdot \color{blue}{\left(\left(\sqrt{u1} \cdot u2\right) \cdot \pi\right)} \]

   2. *-commutative 67.7%

      \[\leadsto 2 \cdot \left(\color{blue}{\left(u2 \cdot \sqrt{u1}\right)} \cdot \pi\right) \]

   3. associate-*r* 67.7%

      \[\leadsto 2 \cdot \color{blue}{\left(u2 \cdot \left(\sqrt{u1} \cdot \pi\right)\right)} \]

   4. *-commutative 67.7%

      \[\leadsto 2 \cdot \left(u2 \cdot \color{blue}{\left(\pi \cdot \sqrt{u1}\right)}\right) \]

10. Simplified 67.7%

    \[\leadsto \color{blue}{2 \cdot \left(u2 \cdot \left(\pi \cdot \sqrt{u1}\right)\right)} \]
11. Add Preprocessing

Alternative 10: 4.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \left(\pi \cdot \left(u2 \cdot \sqrt{u1}\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* -2.0 (* PI (* u2 (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return -2.0f * (((float) M_PI) * (u2 * sqrtf(u1)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(-2.0) * Float32(Float32(pi) * Float32(u2 * sqrt(u1))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(-2.0) * (single(pi) * (u2 * sqrt(u1)));
end

Derivation

1. Initial program 58.6%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u1 around 0 -0.0%

   \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
4. Step-by-step derivation

   1. associate-*r* -0.0%

      \[\leadsto \color{blue}{\left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}} \]

   2. unpow2 -0.0%

      \[\leadsto \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right) \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \]

   3. rem-square-sqrt 4.0%

      \[\leadsto \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right) \cdot \color{blue}{-1} \]

   4. *-commutative 4.0%

      \[\leadsto \color{blue}{\left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \sqrt{u1}\right)} \cdot -1 \]

   5. associate-*l* 4.0%

      \[\leadsto \color{blue}{\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \left(\sqrt{u1} \cdot -1\right)} \]

   6. *-commutative 4.0%

      \[\leadsto \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \color{blue}{\left(-1 \cdot \sqrt{u1}\right)} \]

   7. mul-1-neg 4.0%

      \[\leadsto \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \color{blue}{\left(-\sqrt{u1}\right)} \]

5. Simplified 4.0%

   \[\leadsto \color{blue}{\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \left(-\sqrt{u1}\right)} \]

6. Taylor expanded in u2 around 0 4.3%

   \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{u1} \cdot \left(u2 \cdot \pi\right)\right)} \]

7. Taylor expanded in u1 around -inf -0.0%

   \[\leadsto -2 \cdot \color{blue}{\left(-1 \cdot \left(\sqrt{u1} \cdot \left(u2 \cdot \left(\pi \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)\right)\right)} \]
8. Step-by-step derivation

   1. mul-1-neg -0.0%

      \[\leadsto -2 \cdot \color{blue}{\left(-\sqrt{u1} \cdot \left(u2 \cdot \left(\pi \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)\right)} \]

   2. distribute-rgt-neg-in -0.0%

      \[\leadsto -2 \cdot \color{blue}{\left(\sqrt{u1} \cdot \left(-u2 \cdot \left(\pi \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)\right)} \]

   3. *-commutative -0.0%

      \[\leadsto -2 \cdot \left(\sqrt{u1} \cdot \left(-u2 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \pi\right)}\right)\right) \]

   4. unpow2 -0.0%

      \[\leadsto -2 \cdot \left(\sqrt{u1} \cdot \left(-u2 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \pi\right)\right)\right) \]

   5. rem-square-sqrt 4.3%

      \[\leadsto -2 \cdot \left(\sqrt{u1} \cdot \left(-u2 \cdot \left(\color{blue}{-1} \cdot \pi\right)\right)\right) \]

   6. neg-mul-1 4.3%

      \[\leadsto -2 \cdot \left(\sqrt{u1} \cdot \left(-u2 \cdot \color{blue}{\left(-\pi\right)}\right)\right) \]

   7. distribute-rgt-neg-in 4.3%

      \[\leadsto -2 \cdot \left(\sqrt{u1} \cdot \left(-\color{blue}{\left(-u2 \cdot \pi\right)}\right)\right) \]

   8. *-commutative 4.3%

      \[\leadsto -2 \cdot \left(\sqrt{u1} \cdot \left(-\left(-\color{blue}{\pi \cdot u2}\right)\right)\right) \]

   9. remove-double-neg 4.3%

      \[\leadsto -2 \cdot \left(\sqrt{u1} \cdot \color{blue}{\left(\pi \cdot u2\right)}\right) \]

   10. *-commutative 4.3%

       \[\leadsto -2 \cdot \color{blue}{\left(\left(\pi \cdot u2\right) \cdot \sqrt{u1}\right)} \]

   11. associate-*l* 4.3%

       \[\leadsto -2 \cdot \color{blue}{\left(\pi \cdot \left(u2 \cdot \sqrt{u1}\right)\right)} \]

9. Simplified 4.3%

   \[\leadsto -2 \cdot \color{blue}{\left(\pi \cdot \left(u2 \cdot \sqrt{u1}\right)\right)} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024081 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_y"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 PI) u2))))