Beckmann Sample, near normal, slope_y

Percentage Accurate: 58.5% → 98.3%

Time: 12.0s

Alternatives: 5

Speedup: 2.0×

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: 58.5% 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: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.4%

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

   1. sub-neg 55.4%

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

   2. log1p-def 98.4%

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

3. Simplified 98.4%

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

4. Final simplification 98.4%

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

Alternative 2: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t_0 \leq 0.0024999999441206455:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\pi \cdot \left(2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t_0 \cdot \sqrt{u1 \cdot \left(u1 \cdot \left(--0.5\right) - -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.0024999999441206455)
     (* (sqrt (- (log1p (- u1)))) (* PI (* 2.0 u2)))
     (* (sin t_0) (sqrt (* u1 (- (* u1 (- -0.5)) -1.0)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.0024999999441206455f) {
		tmp = sqrtf(-log1pf(-u1)) * (((float) M_PI) * (2.0f * u2));
	} else {
		tmp = sinf(t_0) * sqrtf((u1 * ((u1 * -(-0.5f)) - -1.0f)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(2.0) * Float32(pi)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0024999999441206455))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(pi) * Float32(Float32(2.0) * u2)));
	else
		tmp = Float32(sin(t_0) * sqrt(Float32(u1 * Float32(Float32(u1 * Float32(-Float32(-0.5))) - Float32(-1.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 2 (PI.f32)) u2) < 0.00249999994

   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-def 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. 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%

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

   5. Applied egg-rr 98.5%

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

      1. associate-*r* 98.5%

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

      2. *-commutative 98.5%

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

      3. *-commutative 98.5%

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

      4. *-commutative 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in u2 around 0 97.7%

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

      1. associate-*r* 97.7%

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

      2. *-commutative 97.7%

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

      3. *-commutative 97.7%

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

      4. *-commutative 97.7%

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

   10. Simplified 97.7%

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

   if 0.00249999994 < (*.f32 (*.f32 2 (PI.f32)) u2)

   1. Initial program 54.7%

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

   2. Taylor expanded in u1 around 0 87.9%

      \[\leadsto \sqrt{-\color{blue}{\left(-1 \cdot u1 + -0.5 \cdot {u1}^{2}\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   3. Step-by-step derivation

      1. *-commutative 87.9%

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

      2. *-commutative 87.9%

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

      3. unpow2 87.9%

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

      4. associate-*l* 87.9%

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

      5. distribute-lft-out 87.9%

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

   4. Simplified 87.9%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.0024999999441206455:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\pi \cdot \left(2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 \cdot \left(--0.5\right) - -1\right)}\\ \end{array} \]

Alternative 3: 90.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.014999999664723873:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\pi \cdot \left(2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* (* 2.0 PI) u2) 0.014999999664723873)
   (* (sqrt (- (log1p (- u1)))) (* PI (* 2.0 u2)))
   (* (sqrt u1) (sin (* 2.0 (* PI u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.014999999664723873f) {
		tmp = sqrtf(-log1pf(-u1)) * (((float) M_PI) * (2.0f * u2));
	} else {
		tmp = sqrtf(u1) * sinf((2.0f * (((float) M_PI) * u2)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.014999999664723873))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(pi) * Float32(Float32(2.0) * u2)));
	else
		tmp = Float32(sqrt(u1) * sin(Float32(Float32(2.0) * Float32(Float32(pi) * u2))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 2 (PI.f32)) u2) < 0.0149999997

   1. Initial program 55.7%

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

      1. sub-neg 55.7%

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

      2. log1p-def 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. 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 98.5%

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

   5. Applied egg-rr 98.5%

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

      1. associate-*r* 98.5%

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

      2. *-commutative 98.5%

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

      3. *-commutative 98.5%

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

      4. *-commutative 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in u2 around 0 96.0%

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

      1. associate-*r* 96.0%

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

      2. *-commutative 96.0%

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

      3. *-commutative 96.0%

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

      4. *-commutative 96.0%

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

   10. Simplified 96.0%

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

   if 0.0149999997 < (*.f32 (*.f32 2 (PI.f32)) u2)

   1. Initial program 54.6%

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

   2. Taylor expanded in u1 around 0 77.5%

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

      1. mul-1-neg 77.5%

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

   4. Simplified 77.5%

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

   5. Taylor expanded in u2 around inf 77.5%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.014999999664723873:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\pi \cdot \left(2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \end{array} \]

Alternative 4: 76.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.4%

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

2. Taylor expanded in u1 around 0 77.6%

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

   1. mul-1-neg 77.6%

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

4. Simplified 77.6%

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

5. Taylor expanded in u2 around inf 77.6%

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

6. Final simplification 77.6%

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

Alternative 5: 66.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \pi \cdot \left(u2 \cdot \sqrt{u1 \cdot 4}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (* PI (* u2 (sqrt (* u1 4.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return ((float) M_PI) * (u2 * sqrtf((u1 * 4.0f)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(pi) * Float32(u2 * sqrt(Float32(u1 * Float32(4.0)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(pi) * (u2 * sqrt((u1 * single(4.0))));
end

Derivation

1. Initial program 55.4%

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

2. Taylor expanded in u1 around 0 77.6%

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

   1. mul-1-neg 77.6%

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

4. Simplified 77.6%

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

5. Taylor expanded in u2 around 0 67.0%

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

   1. associate-*r* 67.0%

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

   2. *-commutative 67.0%

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

7. Simplified 67.0%

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

   1. expm1-log1p-u 67.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(u2 \cdot \pi\right) \cdot \left(2 \cdot \sqrt{u1}\right)\right)\right)} \]

   2. expm1-udef 26.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(u2 \cdot \pi\right) \cdot \left(2 \cdot \sqrt{u1}\right)\right)} - 1} \]

   3. *-commutative 26.8%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\pi \cdot u2\right)} \cdot \left(2 \cdot \sqrt{u1}\right)\right)} - 1 \]

   4. associate-*l* 26.8%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\pi \cdot \left(u2 \cdot \left(2 \cdot \sqrt{u1}\right)\right)}\right)} - 1 \]

   5. add-sqr-sqrt 26.8%

      \[\leadsto e^{\mathsf{log1p}\left(\pi \cdot \left(u2 \cdot \color{blue}{\left(\sqrt{2 \cdot \sqrt{u1}} \cdot \sqrt{2 \cdot \sqrt{u1}}\right)}\right)\right)} - 1 \]

   6. sqrt-unprod 26.8%

      \[\leadsto e^{\mathsf{log1p}\left(\pi \cdot \left(u2 \cdot \color{blue}{\sqrt{\left(2 \cdot \sqrt{u1}\right) \cdot \left(2 \cdot \sqrt{u1}\right)}}\right)\right)} - 1 \]

   7. *-commutative 26.8%

      \[\leadsto e^{\mathsf{log1p}\left(\pi \cdot \left(u2 \cdot \sqrt{\color{blue}{\left(\sqrt{u1} \cdot 2\right)} \cdot \left(2 \cdot \sqrt{u1}\right)}\right)\right)} - 1 \]

   8. *-commutative 26.8%

      \[\leadsto e^{\mathsf{log1p}\left(\pi \cdot \left(u2 \cdot \sqrt{\left(\sqrt{u1} \cdot 2\right) \cdot \color{blue}{\left(\sqrt{u1} \cdot 2\right)}}\right)\right)} - 1 \]

   9. swap-sqr 26.8%

      \[\leadsto e^{\mathsf{log1p}\left(\pi \cdot \left(u2 \cdot \sqrt{\color{blue}{\left(\sqrt{u1} \cdot \sqrt{u1}\right) \cdot \left(2 \cdot 2\right)}}\right)\right)} - 1 \]

   10. add-sqr-sqrt 26.8%

       \[\leadsto e^{\mathsf{log1p}\left(\pi \cdot \left(u2 \cdot \sqrt{\color{blue}{u1} \cdot \left(2 \cdot 2\right)}\right)\right)} - 1 \]

   11. metadata-eval 26.8%

       \[\leadsto e^{\mathsf{log1p}\left(\pi \cdot \left(u2 \cdot \sqrt{u1 \cdot \color{blue}{4}}\right)\right)} - 1 \]

9. Applied egg-rr 26.8%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\pi \cdot \left(u2 \cdot \sqrt{u1 \cdot 4}\right)\right)} - 1} \]
10. Step-by-step derivation

    1. expm1-def 67.0%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(u2 \cdot \sqrt{u1 \cdot 4}\right)\right)\right)} \]

    2. expm1-log1p 67.0%

       \[\leadsto \color{blue}{\pi \cdot \left(u2 \cdot \sqrt{u1 \cdot 4}\right)} \]

11. Simplified 67.0%

    \[\leadsto \color{blue}{\pi \cdot \left(u2 \cdot \sqrt{u1 \cdot 4}\right)} \]

12. Final simplification 67.0%

    \[\leadsto \pi \cdot \left(u2 \cdot \sqrt{u1 \cdot 4}\right) \]

Reproduce

herbie shell --seed 2023306 
(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))))