Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 55.5% → 99.0%

Time: 10.7s

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: 55.5% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(-u0\right) \cdot \left(-{\alpha}^{2}\right) \end{array} \]

FPCore

(FPCore (alpha u0) :precision binary32 (* (log1p (- u0)) (- (pow alpha 2.0))))

C

float code(float alpha, float u0) {
	return log1pf(-u0) * -powf(alpha, 2.0f);
}

Julia

function code(alpha, u0)
	return Float32(log1p(Float32(-u0)) * Float32(-(alpha ^ Float32(2.0))))
end

Derivation

1. Initial program 57.5%

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

   1. associate-*l* 57.4%

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

   2. remove-double-neg 57.4%

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

   3. distribute-lft-neg-out 57.4%

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

   4. distribute-lft-neg-out 57.4%

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

   5. remove-double-neg 57.4%

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

   6. sub-neg 57.4%

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

   7. log1p-define 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in alpha around 0 57.5%

   \[\leadsto \color{blue}{-1 \cdot \left({\alpha}^{2} \cdot \log \left(1 - u0\right)\right)} \]
6. Step-by-step derivation

   1. mul-1-neg 57.5%

      \[\leadsto \color{blue}{-{\alpha}^{2} \cdot \log \left(1 - u0\right)} \]

   2. *-commutative 57.5%

      \[\leadsto -\color{blue}{\log \left(1 - u0\right) \cdot {\alpha}^{2}} \]

   3. distribute-rgt-neg-in 57.5%

      \[\leadsto \color{blue}{\log \left(1 - u0\right) \cdot \left(-{\alpha}^{2}\right)} \]

   4. sub-neg 57.5%

      \[\leadsto \log \color{blue}{\left(1 + \left(-u0\right)\right)} \cdot \left(-{\alpha}^{2}\right) \]

   5. mul-1-neg 57.5%

      \[\leadsto \log \left(1 + \color{blue}{-1 \cdot u0}\right) \cdot \left(-{\alpha}^{2}\right) \]

   6. log1p-define 99.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(-1 \cdot u0\right)} \cdot \left(-{\alpha}^{2}\right) \]

   7. mul-1-neg 99.0%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{-u0}\right) \cdot \left(-{\alpha}^{2}\right) \]

7. Simplified 99.0%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(-u0\right) \cdot \left(-{\alpha}^{2}\right)} \]

8. Final simplification 99.0%

   \[\leadsto \mathsf{log1p}\left(-u0\right) \cdot \left(-{\alpha}^{2}\right) \]
9. Add Preprocessing

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

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(alpha * Float32(alpha * Float32(-log1p(Float32(-u0)))))
end

Derivation

1. Initial program 57.5%

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

   1. associate-*l* 57.4%

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

   2. remove-double-neg 57.4%

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

   3. distribute-lft-neg-out 57.4%

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

   4. distribute-lft-neg-out 57.4%

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

   5. remove-double-neg 57.4%

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

   6. sub-neg 57.4%

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

   7. log1p-define 98.9%

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

3. Simplified 98.9%

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

5. Final simplification 98.9%

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

Alternative 3: 74.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ u0 \cdot {\alpha}^{2} \end{array} \]

FPCore

(FPCore (alpha u0) :precision binary32 (* u0 (pow alpha 2.0)))

C

float code(float alpha, float u0) {
	return u0 * powf(alpha, 2.0f);
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = u0 * (alpha ** 2.0e0)
end function

Julia

function code(alpha, u0)
	return Float32(u0 * (alpha ^ Float32(2.0)))
end

MATLAB

function tmp = code(alpha, u0)
	tmp = u0 * (alpha ^ single(2.0));
end

Derivation

1. Initial program 57.5%

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

   1. associate-*l* 57.4%

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

   2. remove-double-neg 57.4%

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

   3. distribute-lft-neg-out 57.4%

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

   4. distribute-lft-neg-out 57.4%

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

   5. remove-double-neg 57.4%

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

   6. sub-neg 57.4%

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

   7. log1p-define 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in u0 around 0 73.6%

   \[\leadsto \color{blue}{{\alpha}^{2} \cdot u0} \]

6. Final simplification 73.6%

   \[\leadsto u0 \cdot {\alpha}^{2} \]
7. Add Preprocessing

Alternative 4: 74.8% accurate, 21.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.5%

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

   1. associate-*l* 57.4%

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

   2. remove-double-neg 57.4%

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

   3. distribute-lft-neg-out 57.4%

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

   4. distribute-lft-neg-out 57.4%

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

   5. remove-double-neg 57.4%

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

   6. sub-neg 57.4%

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

   7. log1p-define 98.9%

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

3. Simplified 98.9%

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

   1. add0 98.9%

      \[\leadsto \color{blue}{\left(-\alpha\right) \cdot \left(\alpha \cdot \mathsf{log1p}\left(-u0\right)\right) + 0} \]

   2. fma-define 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-\alpha, \alpha \cdot \mathsf{log1p}\left(-u0\right), 0\right)} \]

   3. add-sqr-sqrt -0.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{-\alpha} \cdot \sqrt{-\alpha}}, \alpha \cdot \mathsf{log1p}\left(-u0\right), 0\right) \]

   4. sqrt-unprod 4.4%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\left(-\alpha\right) \cdot \left(-\alpha\right)}}, \alpha \cdot \mathsf{log1p}\left(-u0\right), 0\right) \]

   5. sqr-neg 4.4%

      \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\alpha \cdot \alpha}}, \alpha \cdot \mathsf{log1p}\left(-u0\right), 0\right) \]

   6. sqrt-unprod 4.4%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\alpha} \cdot \sqrt{\alpha}}, \alpha \cdot \mathsf{log1p}\left(-u0\right), 0\right) \]

   7. add-sqr-sqrt 4.4%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\alpha}, \alpha \cdot \mathsf{log1p}\left(-u0\right), 0\right) \]

   8. metadata-eval 4.4%

      \[\leadsto \mathsf{fma}\left(\alpha, \alpha \cdot \mathsf{log1p}\left(-u0\right), \color{blue}{-0}\right) \]

   9. fma-neg 4.4%

      \[\leadsto \color{blue}{\alpha \cdot \left(\alpha \cdot \mathsf{log1p}\left(-u0\right)\right) - 0} \]

   10. add-cube-cbrt 4.4%

       \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\alpha} \cdot \sqrt[3]{\alpha}\right) \cdot \sqrt[3]{\alpha}\right)} \cdot \left(\alpha \cdot \mathsf{log1p}\left(-u0\right)\right) - 0 \]

   11. associate-*l* 4.4%

       \[\leadsto \color{blue}{\left(\sqrt[3]{\alpha} \cdot \sqrt[3]{\alpha}\right) \cdot \left(\sqrt[3]{\alpha} \cdot \left(\alpha \cdot \mathsf{log1p}\left(-u0\right)\right)\right)} - 0 \]

   12. fma-neg 4.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\alpha} \cdot \sqrt[3]{\alpha}, \sqrt[3]{\alpha} \cdot \left(\alpha \cdot \mathsf{log1p}\left(-u0\right)\right), -0\right)} \]

   13. pow2 4.4%

       \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\alpha}\right)}^{2}}, \sqrt[3]{\alpha} \cdot \left(\alpha \cdot \mathsf{log1p}\left(-u0\right)\right), -0\right) \]

   14. add-sqr-sqrt -0.0%

       \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\alpha}\right)}^{2}, \sqrt[3]{\alpha} \cdot \left(\alpha \cdot \mathsf{log1p}\left(\color{blue}{\sqrt{-u0} \cdot \sqrt{-u0}}\right)\right), -0\right) \]

   15. sqrt-unprod 71.1%

       \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\alpha}\right)}^{2}, \sqrt[3]{\alpha} \cdot \left(\alpha \cdot \mathsf{log1p}\left(\color{blue}{\sqrt{\left(-u0\right) \cdot \left(-u0\right)}}\right)\right), -0\right) \]

   16. sqr-neg 71.1%

       \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\alpha}\right)}^{2}, \sqrt[3]{\alpha} \cdot \left(\alpha \cdot \mathsf{log1p}\left(\sqrt{\color{blue}{u0 \cdot u0}}\right)\right), -0\right) \]

   17. sqrt-unprod 71.0%

       \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\alpha}\right)}^{2}, \sqrt[3]{\alpha} \cdot \left(\alpha \cdot \mathsf{log1p}\left(\color{blue}{\sqrt{u0} \cdot \sqrt{u0}}\right)\right), -0\right) \]

   18. add-sqr-sqrt 71.1%

       \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\alpha}\right)}^{2}, \sqrt[3]{\alpha} \cdot \left(\alpha \cdot \mathsf{log1p}\left(\color{blue}{u0}\right)\right), -0\right) \]

   19. metadata-eval 71.1%

       \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\alpha}\right)}^{2}, \sqrt[3]{\alpha} \cdot \left(\alpha \cdot \mathsf{log1p}\left(u0\right)\right), \color{blue}{0}\right) \]

6. Applied egg-rr 71.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\alpha}\right)}^{2}, \sqrt[3]{\alpha} \cdot \left(\alpha \cdot \mathsf{log1p}\left(u0\right)\right), 0\right)} \]
7. Step-by-step derivation

   1. fma-undefine 71.1%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\alpha}\right)}^{2} \cdot \left(\sqrt[3]{\alpha} \cdot \left(\alpha \cdot \mathsf{log1p}\left(u0\right)\right)\right) + 0} \]

   2. add0 71.1%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\alpha}\right)}^{2} \cdot \left(\sqrt[3]{\alpha} \cdot \left(\alpha \cdot \mathsf{log1p}\left(u0\right)\right)\right)} \]

   3. *-commutative 71.1%

      \[\leadsto {\left(\sqrt[3]{\alpha}\right)}^{2} \cdot \color{blue}{\left(\left(\alpha \cdot \mathsf{log1p}\left(u0\right)\right) \cdot \sqrt[3]{\alpha}\right)} \]

   4. associate-*r* 71.0%

      \[\leadsto \color{blue}{\left({\left(\sqrt[3]{\alpha}\right)}^{2} \cdot \left(\alpha \cdot \mathsf{log1p}\left(u0\right)\right)\right) \cdot \sqrt[3]{\alpha}} \]

   5. *-commutative 71.0%

      \[\leadsto \color{blue}{\left(\left(\alpha \cdot \mathsf{log1p}\left(u0\right)\right) \cdot {\left(\sqrt[3]{\alpha}\right)}^{2}\right)} \cdot \sqrt[3]{\alpha} \]

   6. associate-*l* 71.0%

      \[\leadsto \color{blue}{\left(\alpha \cdot \left(\mathsf{log1p}\left(u0\right) \cdot {\left(\sqrt[3]{\alpha}\right)}^{2}\right)\right)} \cdot \sqrt[3]{\alpha} \]

   7. associate-*l* 71.0%

      \[\leadsto \color{blue}{\alpha \cdot \left(\left(\mathsf{log1p}\left(u0\right) \cdot {\left(\sqrt[3]{\alpha}\right)}^{2}\right) \cdot \sqrt[3]{\alpha}\right)} \]

   8. associate-*l* 71.0%

      \[\leadsto \alpha \cdot \color{blue}{\left(\mathsf{log1p}\left(u0\right) \cdot \left({\left(\sqrt[3]{\alpha}\right)}^{2} \cdot \sqrt[3]{\alpha}\right)\right)} \]

   9. pow-plus 71.1%

      \[\leadsto \alpha \cdot \left(\mathsf{log1p}\left(u0\right) \cdot \color{blue}{{\left(\sqrt[3]{\alpha}\right)}^{\left(2 + 1\right)}}\right) \]

   10. metadata-eval 71.1%

       \[\leadsto \alpha \cdot \left(\mathsf{log1p}\left(u0\right) \cdot {\left(\sqrt[3]{\alpha}\right)}^{\color{blue}{3}}\right) \]

8. Simplified 71.1%

   \[\leadsto \color{blue}{\alpha \cdot \left(\mathsf{log1p}\left(u0\right) \cdot {\left(\sqrt[3]{\alpha}\right)}^{3}\right)} \]

9. Taylor expanded in u0 around 0 73.6%

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

10. Final simplification 73.6%

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

Reproduce

herbie shell --seed 2024044 
(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))))