Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 55.7% → 99.0%

Time: 10.2s

Alternatives: 4

Speedup: 21.6×

Specification

\[\left(0.0001 \leq \alpha \land \alpha \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\]

Math

\[\begin{array}{l} \\ \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (* (* (- alpha) alpha) (log (- 1.0 u0))))

C

float code(float alpha, float u0) {
	return (-alpha * alpha) * logf((1.0f - u0));
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = (-alpha * alpha) * log((1.0e0 - u0))
end function

Julia

function code(alpha, u0)
	return Float32(Float32(Float32(-alpha) * alpha) * log(Float32(Float32(1.0) - u0)))
end

MATLAB

function tmp = code(alpha, u0)
	tmp = (-alpha * alpha) * log((single(1.0) - u0));
end

Initial Program: 55.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (* (* (- alpha) alpha) (log (- 1.0 u0))))

C

float code(float alpha, float u0) {
	return (-alpha * alpha) * logf((1.0f - u0));
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = (-alpha * alpha) * log((1.0e0 - u0))
end function

Julia

function code(alpha, u0)
	return Float32(Float32(Float32(-alpha) * alpha) * log(Float32(Float32(1.0) - u0)))
end

MATLAB

function tmp = code(alpha, u0)
	tmp = (-alpha * alpha) * log((single(1.0) - u0));
end

Alternative 1: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\alpha \cdot \alpha\right) \cdot \left(-\mathsf{log1p}\left(-u0\right)\right) \end{array} \]

FPCore

(FPCore (alpha u0) :precision binary32 (* (* alpha alpha) (- (log1p (- u0)))))

C

float code(float alpha, float u0) {
	return (alpha * alpha) * -log1pf(-u0);
}

Julia

function code(alpha, u0)
	return Float32(Float32(alpha * alpha) * Float32(-log1p(Float32(-u0))))
end

Derivation

1. Initial program 53.9%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
2. Step-by-step derivation

   1. distribute-lft-neg-out 53.9%

      \[\leadsto \color{blue}{\left(-\alpha \cdot \alpha\right)} \cdot \log \left(1 - u0\right) \]

   2. sub-neg 53.9%

      \[\leadsto \left(-\alpha \cdot \alpha\right) \cdot \log \color{blue}{\left(1 + \left(-u0\right)\right)} \]

   3. log1p-def 99.1%

      \[\leadsto \left(-\alpha \cdot \alpha\right) \cdot \color{blue}{\mathsf{log1p}\left(-u0\right)} \]

3. Simplified 99.1%

   \[\leadsto \color{blue}{\left(-\alpha \cdot \alpha\right) \cdot \mathsf{log1p}\left(-u0\right)} \]
4. Add Preprocessing

5. Final simplification 99.1%

   \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(-\mathsf{log1p}\left(-u0\right)\right) \]
6. Add Preprocessing

Alternative 2: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(-\alpha\right) \cdot \left(\alpha \cdot \mathsf{log1p}\left(-u0\right)\right) \end{array} \]

FPCore

(FPCore (alpha u0) :precision binary32 (* (- alpha) (* alpha (log1p (- u0)))))

C

float code(float alpha, float u0) {
	return -alpha * (alpha * log1pf(-u0));
}

Julia

function code(alpha, u0)
	return Float32(Float32(-alpha) * Float32(alpha * log1p(Float32(-u0))))
end

Derivation

1. Initial program 53.9%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
2. Step-by-step derivation

   1. associate-*l* 53.9%

      \[\leadsto \color{blue}{\left(-\alpha\right) \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)} \]

   2. distribute-lft-neg-out 53.9%

      \[\leadsto \color{blue}{-\alpha \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)} \]

   3. distribute-rgt-neg-in 53.9%

      \[\leadsto \color{blue}{\alpha \cdot \left(-\alpha \cdot \log \left(1 - u0\right)\right)} \]

   4. distribute-rgt-neg-in 53.9%

      \[\leadsto \alpha \cdot \color{blue}{\left(\alpha \cdot \left(-\log \left(1 - u0\right)\right)\right)} \]

   5. distribute-rgt-neg-in 53.9%

      \[\leadsto \alpha \cdot \color{blue}{\left(-\alpha \cdot \log \left(1 - u0\right)\right)} \]

   6. distribute-lft-neg-out 53.9%

      \[\leadsto \alpha \cdot \color{blue}{\left(\left(-\alpha\right) \cdot \log \left(1 - u0\right)\right)} \]

   7. sub-neg 53.9%

      \[\leadsto \alpha \cdot \left(\left(-\alpha\right) \cdot \log \color{blue}{\left(1 + \left(-u0\right)\right)}\right) \]

   8. log1p-def 99.0%

      \[\leadsto \alpha \cdot \left(\left(-\alpha\right) \cdot \color{blue}{\mathsf{log1p}\left(-u0\right)}\right) \]

3. Simplified 99.0%

   \[\leadsto \color{blue}{\alpha \cdot \left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(-u0\right)\right)} \]
4. Add Preprocessing

5. Final simplification 99.0%

   \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \mathsf{log1p}\left(-u0\right)\right) \]
6. Add Preprocessing

Alternative 3: 89.0% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \alpha \cdot \frac{\alpha}{\frac{1}{u0} - 0.5} \end{array} \]

FPCore

(FPCore (alpha u0) :precision binary32 (* alpha (/ alpha (- (/ 1.0 u0) 0.5))))

C

float code(float alpha, float u0) {
	return alpha * (alpha / ((1.0f / u0) - 0.5f));
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = alpha * (alpha / ((1.0e0 / u0) - 0.5e0))
end function

Julia

function code(alpha, u0)
	return Float32(alpha * Float32(alpha / Float32(Float32(Float32(1.0) / u0) - Float32(0.5))))
end

MATLAB

function tmp = code(alpha, u0)
	tmp = alpha * (alpha / ((single(1.0) / u0) - single(0.5)));
end

Derivation

1. Initial program 53.9%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
2. Step-by-step derivation

   1. associate-*l* 53.9%

      \[\leadsto \color{blue}{\left(-\alpha\right) \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)} \]

   2. distribute-lft-neg-out 53.9%

      \[\leadsto \color{blue}{-\alpha \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)} \]

   3. distribute-rgt-neg-in 53.9%

      \[\leadsto \color{blue}{\alpha \cdot \left(-\alpha \cdot \log \left(1 - u0\right)\right)} \]

   4. distribute-rgt-neg-in 53.9%

      \[\leadsto \alpha \cdot \color{blue}{\left(\alpha \cdot \left(-\log \left(1 - u0\right)\right)\right)} \]

   5. distribute-rgt-neg-in 53.9%

      \[\leadsto \alpha \cdot \color{blue}{\left(-\alpha \cdot \log \left(1 - u0\right)\right)} \]

   6. distribute-lft-neg-out 53.9%

      \[\leadsto \alpha \cdot \color{blue}{\left(\left(-\alpha\right) \cdot \log \left(1 - u0\right)\right)} \]

   7. sub-neg 53.9%

      \[\leadsto \alpha \cdot \left(\left(-\alpha\right) \cdot \log \color{blue}{\left(1 + \left(-u0\right)\right)}\right) \]

   8. log1p-def 99.0%

      \[\leadsto \alpha \cdot \left(\left(-\alpha\right) \cdot \color{blue}{\mathsf{log1p}\left(-u0\right)}\right) \]

3. Simplified 99.0%

   \[\leadsto \color{blue}{\alpha \cdot \left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(-u0\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in u0 around 0 87.3%

   \[\leadsto \alpha \cdot \color{blue}{\left(0.5 \cdot \left(\alpha \cdot {u0}^{2}\right) + \alpha \cdot u0\right)} \]
6. Step-by-step derivation

   1. +-commutative 87.3%

      \[\leadsto \alpha \cdot \color{blue}{\left(\alpha \cdot u0 + 0.5 \cdot \left(\alpha \cdot {u0}^{2}\right)\right)} \]

   2. *-commutative 87.3%

      \[\leadsto \alpha \cdot \left(\alpha \cdot u0 + \color{blue}{\left(\alpha \cdot {u0}^{2}\right) \cdot 0.5}\right) \]

   3. associate-*l* 87.3%

      \[\leadsto \alpha \cdot \left(\alpha \cdot u0 + \color{blue}{\alpha \cdot \left({u0}^{2} \cdot 0.5\right)}\right) \]

   4. distribute-lft-out 87.3%

      \[\leadsto \alpha \cdot \color{blue}{\left(\alpha \cdot \left(u0 + {u0}^{2} \cdot 0.5\right)\right)} \]

7. Simplified 87.3%

   \[\leadsto \alpha \cdot \color{blue}{\left(\alpha \cdot \left(u0 + {u0}^{2} \cdot 0.5\right)\right)} \]
8. Step-by-step derivation

   1. flip-+ 87.1%

      \[\leadsto \alpha \cdot \left(\alpha \cdot \color{blue}{\frac{u0 \cdot u0 - \left({u0}^{2} \cdot 0.5\right) \cdot \left({u0}^{2} \cdot 0.5\right)}{u0 - {u0}^{2} \cdot 0.5}}\right) \]

   2. associate-*r/ 86.9%

      \[\leadsto \alpha \cdot \color{blue}{\frac{\alpha \cdot \left(u0 \cdot u0 - \left({u0}^{2} \cdot 0.5\right) \cdot \left({u0}^{2} \cdot 0.5\right)\right)}{u0 - {u0}^{2} \cdot 0.5}} \]

   3. unpow2 86.9%

      \[\leadsto \alpha \cdot \frac{\alpha \cdot \left(\color{blue}{{u0}^{2}} - \left({u0}^{2} \cdot 0.5\right) \cdot \left({u0}^{2} \cdot 0.5\right)\right)}{u0 - {u0}^{2} \cdot 0.5} \]

   4. swap-sqr 86.9%

      \[\leadsto \alpha \cdot \frac{\alpha \cdot \left({u0}^{2} - \color{blue}{\left({u0}^{2} \cdot {u0}^{2}\right) \cdot \left(0.5 \cdot 0.5\right)}\right)}{u0 - {u0}^{2} \cdot 0.5} \]

   5. pow-sqr 86.9%

      \[\leadsto \alpha \cdot \frac{\alpha \cdot \left({u0}^{2} - \color{blue}{{u0}^{\left(2 \cdot 2\right)}} \cdot \left(0.5 \cdot 0.5\right)\right)}{u0 - {u0}^{2} \cdot 0.5} \]

   6. metadata-eval 86.9%

      \[\leadsto \alpha \cdot \frac{\alpha \cdot \left({u0}^{2} - {u0}^{\color{blue}{4}} \cdot \left(0.5 \cdot 0.5\right)\right)}{u0 - {u0}^{2} \cdot 0.5} \]

   7. metadata-eval 86.9%

      \[\leadsto \alpha \cdot \frac{\alpha \cdot \left({u0}^{2} - {u0}^{4} \cdot \color{blue}{0.25}\right)}{u0 - {u0}^{2} \cdot 0.5} \]

   8. *-commutative 86.9%

      \[\leadsto \alpha \cdot \frac{\alpha \cdot \left({u0}^{2} - {u0}^{4} \cdot 0.25\right)}{u0 - \color{blue}{0.5 \cdot {u0}^{2}}} \]

   9. cancel-sign-sub-inv 86.9%

      \[\leadsto \alpha \cdot \frac{\alpha \cdot \left({u0}^{2} - {u0}^{4} \cdot 0.25\right)}{\color{blue}{u0 + \left(-0.5\right) \cdot {u0}^{2}}} \]

   10. metadata-eval 86.9%

       \[\leadsto \alpha \cdot \frac{\alpha \cdot \left({u0}^{2} - {u0}^{4} \cdot 0.25\right)}{u0 + \color{blue}{-0.5} \cdot {u0}^{2}} \]

9. Applied egg-rr 86.9%

   \[\leadsto \alpha \cdot \color{blue}{\frac{\alpha \cdot \left({u0}^{2} - {u0}^{4} \cdot 0.25\right)}{u0 + -0.5 \cdot {u0}^{2}}} \]
10. Step-by-step derivation

    1. associate-/l* 87.0%

       \[\leadsto \alpha \cdot \color{blue}{\frac{\alpha}{\frac{u0 + -0.5 \cdot {u0}^{2}}{{u0}^{2} - {u0}^{4} \cdot 0.25}}} \]

    2. *-commutative 87.0%

       \[\leadsto \alpha \cdot \frac{\alpha}{\frac{u0 + \color{blue}{{u0}^{2} \cdot -0.5}}{{u0}^{2} - {u0}^{4} \cdot 0.25}} \]

    3. *-commutative 87.0%

       \[\leadsto \alpha \cdot \frac{\alpha}{\frac{u0 + {u0}^{2} \cdot -0.5}{{u0}^{2} - \color{blue}{0.25 \cdot {u0}^{4}}}} \]

11. Simplified 87.0%

    \[\leadsto \alpha \cdot \color{blue}{\frac{\alpha}{\frac{u0 + {u0}^{2} \cdot -0.5}{{u0}^{2} - 0.25 \cdot {u0}^{4}}}} \]

12. Taylor expanded in u0 around 0 88.9%

    \[\leadsto \alpha \cdot \frac{\alpha}{\color{blue}{\frac{1}{u0} - 0.5}} \]

13. Final simplification 88.9%

    \[\leadsto \alpha \cdot \frac{\alpha}{\frac{1}{u0} - 0.5} \]
14. Add Preprocessing

Alternative 4: 74.7% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \alpha \cdot \left(\alpha \cdot u0\right) \end{array} \]

FPCore

(FPCore (alpha u0) :precision binary32 (* alpha (* alpha u0)))

C

float code(float alpha, float u0) {
	return alpha * (alpha * u0);
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = alpha * (alpha * u0)
end function

Julia

function code(alpha, u0)
	return Float32(alpha * Float32(alpha * u0))
end

MATLAB

function tmp = code(alpha, u0)
	tmp = alpha * (alpha * u0);
end

Derivation

1. Initial program 53.9%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
2. Step-by-step derivation

   1. associate-*l* 53.9%

      \[\leadsto \color{blue}{\left(-\alpha\right) \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)} \]

   2. distribute-lft-neg-out 53.9%

      \[\leadsto \color{blue}{-\alpha \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)} \]

   3. distribute-rgt-neg-in 53.9%

      \[\leadsto \color{blue}{\alpha \cdot \left(-\alpha \cdot \log \left(1 - u0\right)\right)} \]

   4. distribute-rgt-neg-in 53.9%

      \[\leadsto \alpha \cdot \color{blue}{\left(\alpha \cdot \left(-\log \left(1 - u0\right)\right)\right)} \]

   5. distribute-rgt-neg-in 53.9%

      \[\leadsto \alpha \cdot \color{blue}{\left(-\alpha \cdot \log \left(1 - u0\right)\right)} \]

   6. distribute-lft-neg-out 53.9%

      \[\leadsto \alpha \cdot \color{blue}{\left(\left(-\alpha\right) \cdot \log \left(1 - u0\right)\right)} \]

   7. sub-neg 53.9%

      \[\leadsto \alpha \cdot \left(\left(-\alpha\right) \cdot \log \color{blue}{\left(1 + \left(-u0\right)\right)}\right) \]

   8. log1p-def 99.0%

      \[\leadsto \alpha \cdot \left(\left(-\alpha\right) \cdot \color{blue}{\mathsf{log1p}\left(-u0\right)}\right) \]

3. Simplified 99.0%

   \[\leadsto \color{blue}{\alpha \cdot \left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(-u0\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in u0 around 0 75.2%

   \[\leadsto \alpha \cdot \color{blue}{\left(\alpha \cdot u0\right)} \]

6. Final simplification 75.2%

   \[\leadsto \alpha \cdot \left(\alpha \cdot u0\right) \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024018 
(FPCore (alpha u0)
  :name "Beckmann Distribution sample, tan2theta, alphax == alphay"
  :precision binary32
  :pre (and (and (<= 0.0001 alpha) (<= alpha 1.0)) (and (<= 2.328306437e-10 u0) (<= u0 1.0)))
  (* (* (- alpha) alpha) (log (- 1.0 u0))))