Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.2% → 99.1%

Time: 14.7s

Alternatives: 6

Speedup: 4.0×

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

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

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: 57.2% 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: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.9%

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

   1. sub-neg 56.9%

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

   2. log1p-def 98.9%

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

   3. associate-*l* 98.9%

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

3. Simplified 98.9%

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

4. Final simplification 98.9%

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

Alternative 2: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.0007999999797903001:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{u1 + 0.5 \cdot \left(u1 \cdot u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 (* 2.0 PI)) 0.0007999999797903001)
   (sqrt (- (log1p (- u1))))
   (* (cos (* PI (+ u2 u2))) (sqrt (+ u1 (* 0.5 (* u1 u1)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * (2.0f * ((float) M_PI))) <= 0.0007999999797903001f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = cosf((((float) M_PI) * (u2 + u2))) * sqrtf((u1 + (0.5f * (u1 * u1))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(Float32(2.0) * Float32(pi))) <= Float32(0.0007999999797903001))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(cos(Float32(Float32(pi) * Float32(u2 + u2))) * sqrt(Float32(u1 + Float32(Float32(0.5) * Float32(u1 * u1)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 2 (PI.f32)) u2) < 7.9999998e-4

   1. Initial program 59.2%

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

      1. sub-neg 59.2%

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

      2. log1p-def 99.6%

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

      3. associate-*l* 99.6%

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

   3. Simplified 99.6%

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

      1. add-log-exp 99.6%

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

      2. log-pow 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. pow-exp 99.6%

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

      2. *-commutative 99.6%

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

   7. Applied egg-rr 99.6%

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

   8. Taylor expanded in u2 around 0 99.5%

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

   if 7.9999998e-4 < (*.f32 (*.f32 2 (PI.f32)) u2)

   1. Initial program 54.1%

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

      1. associate-*r* 54.1%

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

      2. expm1-log1p-u 54.2%

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right)} \]

   3. Applied egg-rr 54.2%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right)} \]

   4. Taylor expanded in u1 around 0 89.8%

      \[\leadsto \sqrt{-\color{blue}{\left(-1 \cdot u1 + -0.5 \cdot {u1}^{2}\right)}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right) \]
   5. Step-by-step derivation

      1. +-commutative 89.8%

         \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} + -1 \cdot u1\right)}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right) \]

      2. mul-1-neg 89.8%

         \[\leadsto \sqrt{-\left(-0.5 \cdot {u1}^{2} + \color{blue}{\left(-u1\right)}\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right) \]

      3. unsub-neg 89.8%

         \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} - u1\right)}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right) \]

      4. *-commutative 89.8%

         \[\leadsto \sqrt{-\left(\color{blue}{{u1}^{2} \cdot -0.5} - u1\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right) \]

      5. unpow2 89.8%

         \[\leadsto \sqrt{-\left(\color{blue}{\left(u1 \cdot u1\right)} \cdot -0.5 - u1\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right) \]

      6. associate-*l* 89.8%

         \[\leadsto \sqrt{-\left(\color{blue}{u1 \cdot \left(u1 \cdot -0.5\right)} - u1\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right) \]

   6. Simplified 89.8%

      \[\leadsto \sqrt{-\color{blue}{\left(u1 \cdot \left(u1 \cdot -0.5\right) - u1\right)}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right) \]

   7. Taylor expanded in u2 around inf 90.1%

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

      1. *-commutative 90.1%

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

      2. associate-*r* 90.1%

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

      3. *-commutative 90.1%

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

      4. associate-*l* 90.1%

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

      5. count-2 90.1%

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

      6. cancel-sign-sub-inv 90.1%

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

      7. metadata-eval 90.1%

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

      8. unpow2 90.1%

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

   9. Simplified 90.1%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.0007999999797903001:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{u1 + 0.5 \cdot \left(u1 \cdot u1\right)}\\ \end{array} \]

Alternative 3: 90.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.004600000102072954:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 (* 2.0 PI)) 0.004600000102072954)
   (sqrt (- (log1p (- u1))))
   (* (cos (* 2.0 (* PI u2))) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * (2.0f * ((float) M_PI))) <= 0.004600000102072954f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = cosf((2.0f * (((float) M_PI) * u2))) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(Float32(2.0) * Float32(pi))) <= Float32(0.004600000102072954))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(cos(Float32(Float32(2.0) * Float32(Float32(pi) * u2))) * sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 2 (PI.f32)) u2) < 0.0046000001

   1. Initial program 57.4%

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

      1. sub-neg 57.4%

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

      2. log1p-def 99.4%

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

      3. associate-*l* 99.4%

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

   3. Simplified 99.4%

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

      1. add-log-exp 99.4%

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

      2. log-pow 99.4%

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

   5. Applied egg-rr 99.4%

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

      1. pow-exp 99.4%

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

      2. *-commutative 99.4%

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

   7. Applied egg-rr 99.4%

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

   8. Taylor expanded in u2 around 0 97.0%

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

   if 0.0046000001 < (*.f32 (*.f32 2 (PI.f32)) u2)

   1. Initial program 55.7%

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

      1. sub-neg 55.7%

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

      2. log1p-def 97.9%

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

      3. associate-*l* 97.9%

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

   3. Simplified 97.9%

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

      1. neg-mul-1 97.9%

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

      2. log1p-udef 55.7%

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

      3. sub-neg 55.7%

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

      4. neg-mul-1 55.7%

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

      5. add-sqr-sqrt 55.8%

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

      6. pow2 55.8%

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

   5. Applied egg-rr 76.0%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{0.25}\right)}^{2}} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \]

   6. Taylor expanded in u1 around 0 78.4%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.004600000102072954:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 4: 80.3% accurate, 2.0× 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 56.9%

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

   1. sub-neg 56.9%

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

   2. log1p-def 98.9%

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

   3. associate-*l* 98.9%

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

3. Simplified 98.9%

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

   1. add-log-exp 98.8%

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

   2. log-pow 98.8%

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

5. Applied egg-rr 98.8%

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

   1. pow-exp 98.9%

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

   2. *-commutative 98.9%

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

7. Applied egg-rr 98.9%

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

8. Taylor expanded in u2 around 0 78.7%

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

9. Final simplification 78.7%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \]

Alternative 5: 73.0% accurate, 3.8× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 + (0.5f * (u1 * 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 + (0.5e0 * (u1 * u1))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 56.9%

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

   1. associate-*r* 56.9%

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

   2. expm1-log1p-u 56.9%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right)} \]

3. Applied egg-rr 56.9%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right)} \]

4. Taylor expanded in u1 around 0 88.4%

   \[\leadsto \sqrt{-\color{blue}{\left(-1 \cdot u1 + -0.5 \cdot {u1}^{2}\right)}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right) \]
5. Step-by-step derivation

   1. +-commutative 88.4%

      \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} + -1 \cdot u1\right)}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right) \]

   2. mul-1-neg 88.4%

      \[\leadsto \sqrt{-\left(-0.5 \cdot {u1}^{2} + \color{blue}{\left(-u1\right)}\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right) \]

   3. unsub-neg 88.4%

      \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} - u1\right)}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right) \]

   4. *-commutative 88.4%

      \[\leadsto \sqrt{-\left(\color{blue}{{u1}^{2} \cdot -0.5} - u1\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right) \]

   5. unpow2 88.4%

      \[\leadsto \sqrt{-\left(\color{blue}{\left(u1 \cdot u1\right)} \cdot -0.5 - u1\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right) \]

   6. associate-*l* 88.4%

      \[\leadsto \sqrt{-\left(\color{blue}{u1 \cdot \left(u1 \cdot -0.5\right)} - u1\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right) \]

6. Simplified 88.4%

   \[\leadsto \sqrt{-\color{blue}{\left(u1 \cdot \left(u1 \cdot -0.5\right) - u1\right)}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right) \]

7. Taylor expanded in u2 around 0 71.8%

   \[\leadsto \color{blue}{\sqrt{u1 - -0.5 \cdot {u1}^{2}}} \]
8. Step-by-step derivation

   1. cancel-sign-sub-inv 71.8%

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

   2. metadata-eval 71.8%

      \[\leadsto \sqrt{u1 + \color{blue}{0.5} \cdot {u1}^{2}} \]

   3. unpow2 71.8%

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

9. Simplified 71.8%

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

10. Final simplification 71.8%

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

Alternative 6: 65.1% accurate, 4.0× 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 56.9%

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

   1. associate-*r* 56.9%

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

   2. expm1-log1p-u 56.9%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right)} \]

3. Applied egg-rr 56.9%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right)} \]

4. Taylor expanded in u1 around 0 88.4%

   \[\leadsto \sqrt{-\color{blue}{\left(-1 \cdot u1 + -0.5 \cdot {u1}^{2}\right)}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right) \]
5. Step-by-step derivation

   1. +-commutative 88.4%

      \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} + -1 \cdot u1\right)}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right) \]

   2. mul-1-neg 88.4%

      \[\leadsto \sqrt{-\left(-0.5 \cdot {u1}^{2} + \color{blue}{\left(-u1\right)}\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right) \]

   3. unsub-neg 88.4%

      \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} - u1\right)}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right) \]

   4. *-commutative 88.4%

      \[\leadsto \sqrt{-\left(\color{blue}{{u1}^{2} \cdot -0.5} - u1\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right) \]

   5. unpow2 88.4%

      \[\leadsto \sqrt{-\left(\color{blue}{\left(u1 \cdot u1\right)} \cdot -0.5 - u1\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right) \]

   6. associate-*l* 88.4%

      \[\leadsto \sqrt{-\left(\color{blue}{u1 \cdot \left(u1 \cdot -0.5\right)} - u1\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right) \]

6. Simplified 88.4%

   \[\leadsto \sqrt{-\color{blue}{\left(u1 \cdot \left(u1 \cdot -0.5\right) - u1\right)}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right) \]

7. Taylor expanded in u2 around 0 71.8%

   \[\leadsto \color{blue}{\sqrt{u1 - -0.5 \cdot {u1}^{2}}} \]
8. Step-by-step derivation

   1. cancel-sign-sub-inv 71.8%

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

   2. metadata-eval 71.8%

      \[\leadsto \sqrt{u1 + \color{blue}{0.5} \cdot {u1}^{2}} \]

   3. unpow2 71.8%

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

9. Simplified 71.8%

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

10. Taylor expanded in u1 around 0 64.3%

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

11. Final simplification 64.3%

    \[\leadsto \sqrt{u1} \]

Reproduce

herbie shell --seed 2023293 
(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))))