Beckmann Sample, near normal, slope_x

Percentage Accurate: 58.1% → 99.0%

Time: 10.0s

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: 58.1% 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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.0%

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

   1. sub-neg 58.0%

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

   2. log1p-def 99.0%

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

   3. associate-*l* 99.0%

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

3. Simplified 99.0%

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

   1. add-cbrt-cube 99.0%

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

   2. add-cbrt-cube 99.0%

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

   3. cbrt-unprod 98.9%

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

   4. pow3 98.9%

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

   5. pow3 99.0%

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

5. Applied egg-rr 99.0%

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

6. Final simplification 99.0%

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

Alternative 2: 90.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(2 \cdot \pi\right)\\ \mathbf{if}\;t_0 \leq 0.004000000189989805:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* 2.0 PI))))
   (if (<= t_0 0.004000000189989805)
     (sqrt (- (log1p (- u1))))
     (* (sqrt u1) (cos t_0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u2 * (2.0f * ((float) M_PI));
	float tmp;
	if (t_0 <= 0.004000000189989805f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = sqrtf(u1) * cosf(t_0);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u2 * Float32(Float32(2.0) * Float32(pi)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.004000000189989805))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(sqrt(u1) * cos(t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 2 (PI.f32)) u2) < 0.00400000019

   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 99.5%

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

      3. associate-*l* 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in u2 around 0 97.8%

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

   if 0.00400000019 < (*.f32 (*.f32 2 (PI.f32)) u2)

   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-udef 97.9%

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

      3. add-cbrt-cube 97.9%

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

      4. pow1/3 95.2%

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

   3. Applied egg-rr 72.0%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. Taylor expanded in u1 around 0 75.5%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.004000000189989805:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\\ \end{array} \]

Alternative 3: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.0%

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

   1. sub-neg 58.0%

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

   2. log1p-def 99.0%

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

   3. associate-*l* 99.0%

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

3. Simplified 99.0%

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

4. Final simplification 99.0%

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

Alternative 4: 80.6% 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 58.0%

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

   1. sub-neg 58.0%

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

   2. log1p-def 99.0%

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

   3. associate-*l* 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in u2 around 0 80.2%

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

5. Final simplification 80.2%

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

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 58.0%

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

   1. sub-neg 58.0%

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

   2. log1p-def 99.0%

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

   3. associate-*l* 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in u2 around 0 80.2%

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

5. Taylor expanded in u1 around 0 71.8%

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

   1. +-commutative 71.8%

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

   2. mul-1-neg 71.8%

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

   3. unsub-neg 71.8%

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

   4. unpow2 71.8%

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

7. Simplified 71.8%

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

8. Final simplification 71.8%

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

Alternative 6: 64.9% 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 58.0%

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

   1. sub-neg 58.0%

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

   2. log1p-udef 99.0%

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

   3. add-cbrt-cube 98.9%

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

   4. pow1/3 95.4%

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

3. Applied egg-rr 72.4%

   \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

4. Taylor expanded in u1 around 0 76.3%

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

5. Taylor expanded in u2 around 0 65.1%

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

6. Final simplification 65.1%

   \[\leadsto \sqrt{u1} \]

Reproduce

herbie shell --seed 2023252 
(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))))