Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.7% → 99.0%

Time: 39.0s

Alternatives: 13

Speedup: 3.1×

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{{\left(\cos \left(\left(\pi \cdot 2\right) \cdot u2\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right)}^{3}} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (cbrt (pow (* (cos (* (* PI 2.0) u2)) (sqrt (- (log1p (- u1))))) 3.0)))

C

float code(float cosTheta_i, float u1, float u2) {
	return cbrtf(powf((cosf(((((float) M_PI) * 2.0f) * u2)) * sqrtf(-log1pf(-u1))), 3.0f));
}

Julia

function code(cosTheta_i, u1, u2)
	return cbrt((Float32(cos(Float32(Float32(Float32(pi) * Float32(2.0)) * u2)) * sqrt(Float32(-log1p(Float32(-u1))))) ^ Float32(3.0)))
end

Derivation

1. Initial program 60.1%

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

   1. sub-neg 60.1%

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

   2. log1p-define 99.0%

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

3. Simplified 99.0%

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

   1. expm1-log1p-u 98.9%

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

   2. expm1-undefine 98.9%

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

   3. *-commutative 98.9%

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

   4. associate-*l* 98.9%

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

6. Applied egg-rr 98.9%

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

   1. log1p-undefine 98.9%

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

   2. rem-exp-log 99.0%

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

   3. +-commutative 99.0%

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

   4. *-commutative 99.0%

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

   5. associate-*r* 99.0%

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

8. Applied egg-rr 99.0%

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

   1. add-cbrt-cube 99.0%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(\left(\left(\pi \cdot u2\right) \cdot 2 + 1\right) - 1\right)\right) \cdot \left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(\left(\left(\pi \cdot u2\right) \cdot 2 + 1\right) - 1\right)\right)\right) \cdot \left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(\left(\left(\pi \cdot u2\right) \cdot 2 + 1\right) - 1\right)\right)}} \]

   2. pow3 99.0%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(\left(\left(\pi \cdot u2\right) \cdot 2 + 1\right) - 1\right)\right)}^{3}}} \]

10. Applied egg-rr 99.0%

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

11. Final simplification 99.0%

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

Alternative 2: 94.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(\pi \cdot 2\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq 0.9999989867210388:\\ \;\;\;\;t\_0 \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* PI 2.0) u2))))
   (if (<= t_0 0.9999989867210388)
     (* t_0 (sqrt (* u1 (+ 1.0 (* u1 0.5)))))
     (sqrt (- (log1p (- u1)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf(((((float) M_PI) * 2.0f) * u2));
	float tmp;
	if (t_0 <= 0.9999989867210388f) {
		tmp = t_0 * sqrtf((u1 * (1.0f + (u1 * 0.5f))));
	} else {
		tmp = sqrtf(-log1pf(-u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(Float32(pi) * Float32(2.0)) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9999989867210388))
		tmp = Float32(t_0 * sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(u1 * Float32(0.5))))));
	else
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)) < 0.999998987

   1. Initial program 59.1%

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

      1. sub-neg 59.1%

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

      2. log1p-define 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in u1 around 0 87.6%

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

      1. *-commutative 87.4%

         \[\leadsto \sqrt{u1 \cdot \left(1 + \color{blue}{u1 \cdot 0.5}\right)} \cdot \cos \left(e^{\mathsf{log1p}\left(\pi \cdot \left(2 \cdot u2\right)\right)} - 1\right) \]

   7. Simplified 87.6%

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

   if 0.999998987 < (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))

   1. Initial program 61.0%

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

      1. sub-neg 61.0%

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

      2. log1p-define 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in u2 around 0 98.9%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(\pi \cdot 2\right) \cdot u2\right) \leq 0.9999989867210388:\\ \;\;\;\;\cos \left(\left(\pi \cdot 2\right) \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\left(\pi \cdot 2\right) \cdot u2\right) \leq 0.9999780058860779:\\ \;\;\;\;\cos \left(\left(1 + 2 \cdot \left(\pi \cdot u2\right)\right) + -1\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (cos (* (* PI 2.0) u2)) 0.9999780058860779)
   (* (cos (+ (+ 1.0 (* 2.0 (* PI u2))) -1.0)) (sqrt u1))
   (sqrt (- (log1p (- u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (cosf(((((float) M_PI) * 2.0f) * u2)) <= 0.9999780058860779f) {
		tmp = cosf(((1.0f + (2.0f * (((float) M_PI) * u2))) + -1.0f)) * sqrtf(u1);
	} else {
		tmp = sqrtf(-log1pf(-u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(Float32(pi) * Float32(2.0)) * u2)) <= Float32(0.9999780058860779))
		tmp = Float32(cos(Float32(Float32(Float32(1.0) + Float32(Float32(2.0) * Float32(Float32(pi) * u2))) + Float32(-1.0))) * sqrt(u1));
	else
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)) < 0.999978006

   1. Initial program 59.1%

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

      1. sub-neg 59.1%

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

      2. log1p-define 98.2%

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

   3. Simplified 98.2%

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

      1. expm1-log1p-u 97.9%

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

      2. expm1-undefine 97.9%

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

      3. *-commutative 97.9%

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

      4. associate-*l* 97.9%

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

   6. Applied egg-rr 97.9%

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

      1. log1p-undefine 98.0%

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

      2. rem-exp-log 98.1%

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

      3. +-commutative 98.1%

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

      4. *-commutative 98.1%

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

      5. associate-*r* 98.1%

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

   8. Applied egg-rr 98.1%

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

   9. Taylor expanded in u1 around 0 75.2%

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

   if 0.999978006 < (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))

   1. Initial program 60.7%

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

      1. sub-neg 60.7%

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

      2. log1p-define 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in u2 around 0 95.9%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(\pi \cdot 2\right) \cdot u2\right) \leq 0.9999780058860779:\\ \;\;\;\;\cos \left(\left(1 + 2 \cdot \left(\pi \cdot u2\right)\right) + -1\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(\pi \cdot 2\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq 0.9999780058860779:\\ \;\;\;\;t\_0 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* PI 2.0) u2))))
   (if (<= t_0 0.9999780058860779)
     (* t_0 (sqrt u1))
     (sqrt (- (log1p (- u1)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf(((((float) M_PI) * 2.0f) * u2));
	float tmp;
	if (t_0 <= 0.9999780058860779f) {
		tmp = t_0 * sqrtf(u1);
	} else {
		tmp = sqrtf(-log1pf(-u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(Float32(pi) * Float32(2.0)) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9999780058860779))
		tmp = Float32(t_0 * sqrt(u1));
	else
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)) < 0.999978006

   1. Initial program 59.1%

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

      1. sub-neg 59.1%

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

      2. log1p-define 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in u1 around 0 75.2%

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

   if 0.999978006 < (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))

   1. Initial program 60.7%

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

      1. sub-neg 60.7%

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

      2. log1p-define 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in u2 around 0 95.9%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(\pi \cdot 2\right) \cdot u2\right) \leq 0.9999780058860779:\\ \;\;\;\;\cos \left(\left(\pi \cdot 2\right) \cdot u2\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.1%

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

   1. sub-neg 60.1%

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

   2. log1p-define 99.0%

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

3. Simplified 99.0%

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

5. Final simplification 99.0%

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

Alternative 6: 96.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi \cdot 2\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.001500000013038516:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos t\_0 \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* PI 2.0) u2)))
   (if (<= t_0 0.001500000013038516)
     (sqrt (- (log1p (- u1))))
     (*
      (cos 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 = (((float) M_PI) * 2.0f) * u2;
	float tmp;
	if (t_0 <= 0.001500000013038516f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = cosf(t_0) * sqrtf((u1 * (1.0f + (u1 * (0.5f + (u1 * 0.3333333333333333f))))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(pi) * Float32(2.0)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.001500000013038516))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(cos(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) < 0.00150000001

   1. Initial program 61.0%

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

      1. sub-neg 61.0%

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

      2. log1p-define 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in u2 around 0 98.9%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]

   if 0.00150000001 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 59.1%

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

      1. sub-neg 59.1%

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

      2. log1p-define 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in u1 around 0 91.3%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(0.5 + 0.3333333333333333 \cdot u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   6. Step-by-step derivation

      1. *-commutative 91.3%

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

   7. Simplified 91.3%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\pi \cdot 2\right) \cdot u2 \leq 0.001500000013038516:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(\pi \cdot 2\right) \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\pi \cdot 2\right) \cdot u2 \leq 0.001500000013038516:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + u1 \cdot 0.5\right)} \cdot \cos \left(\left(1 + 2 \cdot \left(\pi \cdot u2\right)\right) + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* (* PI 2.0) u2) 0.001500000013038516)
   (sqrt (- (log1p (- u1))))
   (*
    (sqrt (* u1 (+ 1.0 (* u1 0.5))))
    (cos (+ (+ 1.0 (* 2.0 (* PI u2))) -1.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((((float) M_PI) * 2.0f) * u2) <= 0.001500000013038516f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = sqrtf((u1 * (1.0f + (u1 * 0.5f)))) * cosf(((1.0f + (2.0f * (((float) M_PI) * u2))) + -1.0f));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(pi) * Float32(2.0)) * u2) <= Float32(0.001500000013038516))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(u1 * Float32(0.5))))) * cos(Float32(Float32(Float32(1.0) + Float32(Float32(2.0) * Float32(Float32(pi) * u2))) + Float32(-1.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00150000001

   1. Initial program 61.0%

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

      1. sub-neg 61.0%

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

      2. log1p-define 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in u2 around 0 98.9%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]

   if 0.00150000001 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 59.1%

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

      1. sub-neg 59.1%

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

      2. log1p-define 98.4%

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

   3. Simplified 98.4%

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

      1. expm1-log1p-u 98.2%

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

      2. expm1-undefine 98.2%

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

      3. *-commutative 98.2%

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

      4. associate-*l* 98.2%

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

   6. Applied egg-rr 98.2%

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

   7. Taylor expanded in u1 around 0 87.4%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + 0.5 \cdot u1\right)}} \cdot \cos \left(e^{\mathsf{log1p}\left(\pi \cdot \left(2 \cdot u2\right)\right)} - 1\right) \]
   8. Step-by-step derivation

      1. *-commutative 87.4%

         \[\leadsto \sqrt{u1 \cdot \left(1 + \color{blue}{u1 \cdot 0.5}\right)} \cdot \cos \left(e^{\mathsf{log1p}\left(\pi \cdot \left(2 \cdot u2\right)\right)} - 1\right) \]

   9. Simplified 87.4%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot 0.5\right)}} \cdot \cos \left(e^{\mathsf{log1p}\left(\pi \cdot \left(2 \cdot u2\right)\right)} - 1\right) \]
   10. Step-by-step derivation

       1. log1p-undefine 98.2%

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

       2. rem-exp-log 98.4%

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

       3. +-commutative 98.4%

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

       4. *-commutative 98.4%

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

       5. associate-*r* 98.4%

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

   11. Applied egg-rr 87.6%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\pi \cdot 2\right) \cdot u2 \leq 0.001500000013038516:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + u1 \cdot 0.5\right)} \cdot \cos \left(\left(1 + 2 \cdot \left(\pi \cdot u2\right)\right) + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.1%

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

   1. sub-neg 60.1%

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

   2. log1p-define 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in u1 around 0 93.6%

   \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 + 0.25 \cdot u1\right)\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
6. Step-by-step derivation

   1. *-commutative 93.6%

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

7. Simplified 93.6%

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

8. Final simplification 93.6%

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

Alternative 9: 79.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (sqrt (- (log1p (- u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1));
}

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(-log1p(Float32(-u1))))
end

Derivation

1. Initial program 60.1%

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

   1. sub-neg 60.1%

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

   2. log1p-define 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in u2 around 0 77.2%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]

6. Final simplification 77.2%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \]
7. Add Preprocessing

Alternative 10: 76.2% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

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 = sqrt((u1 * (1.0e0 + (u1 * (0.5e0 + (u1 * (0.3333333333333333e0 + (u1 * 0.25e0))))))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 60.1%

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

   1. sub-neg 60.1%

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

   2. log1p-define 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in u2 around 0 77.2%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]

6. Taylor expanded in u1 around 0 73.9%

   \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 + 0.25 \cdot u1\right)\right)\right)}} \cdot 1 \]
7. Step-by-step derivation

   1. *-commutative 93.6%

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

8. Simplified 73.9%

   \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 + u1 \cdot 0.25\right)\right)\right)}} \cdot 1 \]

9. Final simplification 73.9%

   \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 + u1 \cdot 0.25\right)\right)\right)} \]
10. Add Preprocessing

Alternative 11: 74.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right)\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (sqrt (* u1 (+ 1.0 (* u1 (+ 0.5 (* u1 0.3333333333333333)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 * (1.0f + (u1 * (0.5f + (u1 * 0.3333333333333333f))))));
}

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 = sqrt((u1 * (1.0e0 + (u1 * (0.5e0 + (u1 * 0.3333333333333333e0))))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(u1 * Float32(Float32(0.5) + Float32(u1 * Float32(0.3333333333333333)))))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 * (single(1.0) + (u1 * (single(0.5) + (u1 * single(0.3333333333333333)))))));
end

Derivation

1. Initial program 60.1%

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

   1. sub-neg 60.1%

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

   2. log1p-define 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in u2 around 0 77.2%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]

6. Taylor expanded in u1 around 0 72.3%

   \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(0.5 + 0.3333333333333333 \cdot u1\right)\right)}} \cdot 1 \]
7. Step-by-step derivation

   1. *-commutative 91.4%

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

8. Simplified 72.3%

   \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right)\right)}} \cdot 1 \]

9. Final simplification 72.3%

   \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right)\right)} \]
10. Add Preprocessing

Alternative 12: 72.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1 \cdot \left(1 + u1 \cdot 0.5\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (sqrt (* u1 (+ 1.0 (* u1 0.5)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 * (1.0f + (u1 * 0.5f))));
}

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 = sqrt((u1 * (1.0e0 + (u1 * 0.5e0))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(u1 * Float32(0.5)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 * (single(1.0) + (u1 * single(0.5)))));
end

Derivation

1. Initial program 60.1%

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

   1. sub-neg 60.1%

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

   2. log1p-define 99.0%

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

3. Simplified 99.0%

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

   1. expm1-log1p-u 98.9%

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

   2. expm1-undefine 98.9%

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

   3. *-commutative 98.9%

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

   4. associate-*l* 98.9%

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

6. Applied egg-rr 98.9%

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

7. Taylor expanded in u1 around 0 87.2%

   \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + 0.5 \cdot u1\right)}} \cdot \cos \left(e^{\mathsf{log1p}\left(\pi \cdot \left(2 \cdot u2\right)\right)} - 1\right) \]
8. Step-by-step derivation

   1. *-commutative 87.2%

      \[\leadsto \sqrt{u1 \cdot \left(1 + \color{blue}{u1 \cdot 0.5}\right)} \cdot \cos \left(e^{\mathsf{log1p}\left(\pi \cdot \left(2 \cdot u2\right)\right)} - 1\right) \]

9. Simplified 87.2%

   \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot 0.5\right)}} \cdot \cos \left(e^{\mathsf{log1p}\left(\pi \cdot \left(2 \cdot u2\right)\right)} - 1\right) \]

10. Taylor expanded in u2 around 0 69.9%

    \[\leadsto \color{blue}{\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)}} \]

11. Final simplification 69.9%

    \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot 0.5\right)} \]
12. Add Preprocessing

Alternative 13: 64.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1} \end{array} \]

FPCore

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

C

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

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 = sqrt(u1)
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(u1)
end

MATLAB

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

Derivation

1. Initial program 60.1%

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

   1. sub-neg 60.1%

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

   2. log1p-define 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in u2 around 0 77.2%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]

6. Taylor expanded in u1 around 0 63.0%

   \[\leadsto \sqrt{\color{blue}{u1}} \cdot 1 \]

7. Taylor expanded in u1 around 0 63.0%

   \[\leadsto \color{blue}{\sqrt{u1}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024116 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_x"
  :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)))) (cos (* (* 2.0 PI) u2))))