Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.0% → 98.4%

Time: 12.9s

Alternatives: 8

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: 57.0% 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, 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 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-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. Final simplification 98.5%

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

Alternative 2: 94.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t_0 \leq 0.0026000000070780516:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t_0 \cdot \sqrt{u1 \cdot \left(\left(--1\right) - u1 \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.0026000000070780516)
     (* (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 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.0026000000070780516f) {
		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(Float32(Float32(2.0) * Float32(pi)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0026000000070780516))
		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(-Float32(-1.0)) - Float32(u1 * Float32(-0.5))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 2 (PI.f32)) u2) < 0.00260000001

   1. Initial program 58.9%

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

      1. sub-neg 58.9%

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

      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. expm1-log1p-u 98.6%

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

      2. *-commutative 98.6%

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

      3. associate-*r* 98.6%

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

   5. Applied egg-rr 98.6%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\left(u2 \cdot 2\right) \cdot \pi\right)\right)\right)} \]
   6. Step-by-step derivation

      1. expm1-udef 41.5%

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

      2. log1p-udef 41.5%

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

      3. rem-exp-log 41.5%

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

      4. +-commutative 41.5%

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

      5. *-commutative 41.5%

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

   7. Applied egg-rr 41.5%

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

   8. Taylor expanded in u2 around 0 98.1%

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

   if 0.00260000001 < (*.f32 (*.f32 2 (PI.f32)) u2)

   1. Initial program 58.8%

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

   2. Taylor expanded in u1 around 0 88.1%

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

      1. *-commutative 88.1%

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

      2. *-commutative 88.1%

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

      3. unpow2 88.1%

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

      4. associate-*l* 88.1%

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

      5. distribute-lft-out 88.0%

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

   4. Simplified 88.0%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.0026000000070780516:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1 \cdot \left(\left(--1\right) - u1 \cdot -0.5\right)}\\ \end{array} \]

Alternative 3: 90.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(\pi \cdot u2\right)\\ \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.0026000000070780516:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* 2.0 (* PI u2))))
   (if (<= (* (* 2.0 PI) u2) 0.0026000000070780516)
     (* (sqrt (- (log1p (- u1)))) t_0)
     (* (sqrt u1) (sin t_0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = 2.0f * (((float) M_PI) * u2);
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.0026000000070780516f) {
		tmp = sqrtf(-log1pf(-u1)) * t_0;
	} else {
		tmp = sqrtf(u1) * sinf(t_0);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(2.0) * Float32(Float32(pi) * u2))
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.0026000000070780516))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * t_0);
	else
		tmp = Float32(sqrt(u1) * sin(t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 2 (PI.f32)) u2) < 0.00260000001

   1. Initial program 58.9%

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

      1. sub-neg 58.9%

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

      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. expm1-log1p-u 98.6%

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

      2. *-commutative 98.6%

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

      3. associate-*r* 98.6%

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

   5. Applied egg-rr 98.6%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\left(u2 \cdot 2\right) \cdot \pi\right)\right)\right)} \]
   6. Step-by-step derivation

      1. expm1-udef 41.5%

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

      2. log1p-udef 41.5%

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

      3. rem-exp-log 41.5%

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

      4. +-commutative 41.5%

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

      5. *-commutative 41.5%

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

   7. Applied egg-rr 41.5%

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

   8. Taylor expanded in u2 around 0 98.1%

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

   if 0.00260000001 < (*.f32 (*.f32 2 (PI.f32)) u2)

   1. Initial program 58.8%

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

   2. Taylor expanded in u1 around 0 76.0%

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

      1. mul-1-neg 76.0%

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

   4. Simplified 76.0%

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

   5. Taylor expanded in u2 around inf 76.0%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.0026000000070780516:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \end{array} \]

Alternative 4: 77.0% 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 58.8%

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

2. Taylor expanded in u1 around 0 76.3%

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

   1. mul-1-neg 76.3%

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

4. Simplified 76.3%

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

5. Taylor expanded in u2 around inf 76.3%

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

6. Final simplification 76.3%

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

Alternative 5: 66.7% accurate, 2.0× 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 58.8%

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

2. Taylor expanded in u1 around 0 76.3%

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

   1. mul-1-neg 76.3%

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

4. Simplified 76.3%

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

5. Taylor expanded in u2 around 0 66.6%

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

   1. associate-*r* 66.6%

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

7. Simplified 66.6%

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

   1. expm1-log1p-u 66.6%

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

   2. expm1-udef 27.2%

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

   3. *-commutative 27.2%

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

   4. associate-*l* 27.2%

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

9. Applied egg-rr 27.2%

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

    1. expm1-def 66.6%

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

    2. expm1-log1p 66.6%

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

11. Simplified 66.6%

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

12. Taylor expanded in u2 around 0 66.6%

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

    1. associate-*r* 66.6%

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

14. Simplified 66.6%

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

15. Final simplification 66.6%

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

Alternative 6: 66.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.8%

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

2. Taylor expanded in u1 around 0 76.3%

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

   1. mul-1-neg 76.3%

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

4. Simplified 76.3%

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

5. Taylor expanded in u2 around 0 66.6%

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

   1. associate-*r* 66.6%

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

7. Simplified 66.6%

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

   1. expm1-log1p-u 66.6%

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

   2. expm1-udef 27.2%

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

   3. *-commutative 27.2%

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

   4. associate-*l* 27.2%

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

9. Applied egg-rr 27.2%

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

    1. expm1-def 66.6%

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

    2. expm1-log1p 66.6%

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

11. Simplified 66.6%

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

12. Final simplification 66.6%

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

Alternative 7: 66.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.8%

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

2. Taylor expanded in u1 around 0 76.3%

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

   1. mul-1-neg 76.3%

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

4. Simplified 76.3%

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

5. Taylor expanded in u2 around 0 66.6%

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

   1. associate-*r* 66.6%

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

7. Simplified 66.6%

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

8. Final simplification 66.6%

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

Alternative 8: 7.1% accurate, 409.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 0.0)

C

float code(float cosTheta_i, float u1, float u2) {
	return 0.0f;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = 0.0e0
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(0.0)
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
end

Derivation

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-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. add-cube-cbrt 97.1%

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

   2. pow3 97.1%

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

   3. *-commutative 97.1%

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

   4. associate-*r* 97.1%

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

5. Applied egg-rr 97.1%

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

6. Taylor expanded in u2 around 0 7.1%

   \[\leadsto \color{blue}{0} \]

7. Final simplification 7.1%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023322 
(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))))