Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 55.6% → 89.7%

Time: 6.8s

Alternatives: 7

Speedup: 10.5×

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: 55.6% 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: 89.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u0 \leq 0.00018000000272877514:\\ \;\;\;\;\frac{\alpha \cdot u0}{\frac{1}{\alpha}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \frac{-1}{\frac{\alpha}{\alpha \cdot \alpha} \cdot \frac{1}{\alpha}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= u0 0.00018000000272877514)
   (/ (* alpha u0) (/ 1.0 alpha))
   (* (log (- 1.0 u0)) (/ -1.0 (* (/ alpha (* alpha alpha)) (/ 1.0 alpha))))))

C

float code(float alpha, float u0) {
	float tmp;
	if (u0 <= 0.00018000000272877514f) {
		tmp = (alpha * u0) / (1.0f / alpha);
	} else {
		tmp = logf((1.0f - u0)) * (-1.0f / ((alpha / (alpha * alpha)) * (1.0f / alpha)));
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: tmp
    if (u0 <= 0.00018000000272877514e0) then
        tmp = (alpha * u0) / (1.0e0 / alpha)
    else
        tmp = log((1.0e0 - u0)) * ((-1.0e0) / ((alpha / (alpha * alpha)) * (1.0e0 / alpha)))
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.00018000000272877514))
		tmp = Float32(Float32(alpha * u0) / Float32(Float32(1.0) / alpha));
	else
		tmp = Float32(log(Float32(Float32(1.0) - u0)) * Float32(Float32(-1.0) / Float32(Float32(alpha / Float32(alpha * alpha)) * Float32(Float32(1.0) / alpha))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if (u0 <= single(0.00018000000272877514))
		tmp = (alpha * u0) / (single(1.0) / alpha);
	else
		tmp = log((single(1.0) - u0)) * (single(-1.0) / ((alpha / (alpha * alpha)) * (single(1.0) / alpha)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u0 < 1.80000003e-4

   1. Initial program 34.2%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{{\alpha}^{2} \cdot u0} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f32 91.6

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

   5. Applied rewrites 91.6%

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

   7. Applied rewrites 91.7%

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

   if 1.80000003e-4 < u0

   1. Initial program 87.6%

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

      1. lift-*.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. distribute-lft-neg-out N/A

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

      4. neg-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - \alpha \cdot \alpha\right)} \cdot \log \left(1 - u0\right) \]

      5. flip-- N/A

         \[\leadsto \color{blue}{\frac{0 \cdot 0 - \left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right)}{0 + \alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

      6. +-lft-identity N/A

         \[\leadsto \frac{0 \cdot 0 - \left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right)}{\color{blue}{\alpha \cdot \alpha}} \cdot \log \left(1 - u0\right) \]

      7. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\alpha \cdot \alpha}{0 \cdot 0 - \left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right)}}} \cdot \log \left(1 - u0\right) \]

      8. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\alpha \cdot \alpha}{0 \cdot 0 - \left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right)}}} \cdot \log \left(1 - u0\right) \]

      9. lower-/.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\alpha \cdot \alpha}{0 \cdot 0 - \left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right)}}} \cdot \log \left(1 - u0\right) \]

      10. lower-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\alpha \cdot \alpha}}{0 \cdot 0 - \left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right)}} \cdot \log \left(1 - u0\right) \]

      11. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{\alpha \cdot \alpha}{\color{blue}{0} - \left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right)}} \cdot \log \left(1 - u0\right) \]

      12. sub0-neg N/A

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

      13. lower-neg.f32 N/A

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

      14. pow2 N/A

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

      15. pow2 N/A

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

      16. pow-prod-up N/A

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

      17. lower-pow.f32 N/A

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

      18. metadata-eval 87.3

          \[\leadsto \frac{1}{\frac{\alpha \cdot \alpha}{-{\alpha}^{\color{blue}{4}}}} \cdot \log \left(1 - u0\right) \]

   4. Applied rewrites 87.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\alpha \cdot \alpha}{-{\alpha}^{4}}}} \cdot \log \left(1 - u0\right) \]
   5. Step-by-step derivation

      1. lift-/.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\alpha \cdot \alpha}{-{\alpha}^{4}}}} \cdot \log \left(1 - u0\right) \]

      2. frac-2neg N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{neg}\left(\alpha \cdot \alpha\right)}{\mathsf{neg}\left(\left(-{\alpha}^{4}\right)\right)}}} \cdot \log \left(1 - u0\right) \]

      3. lift-*.f32 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{neg}\left(\color{blue}{\alpha \cdot \alpha}\right)}{\mathsf{neg}\left(\left(-{\alpha}^{4}\right)\right)}} \cdot \log \left(1 - u0\right) \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha}}{\mathsf{neg}\left(\left(-{\alpha}^{4}\right)\right)}} \cdot \log \left(1 - u0\right) \]

      5. lift-neg.f32 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(-\alpha\right)} \cdot \alpha}{\mathsf{neg}\left(\left(-{\alpha}^{4}\right)\right)}} \cdot \log \left(1 - u0\right) \]

      6. lift-neg.f32 N/A

         \[\leadsto \frac{1}{\frac{\left(-\alpha\right) \cdot \alpha}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left({\alpha}^{4}\right)\right)}\right)}} \cdot \log \left(1 - u0\right) \]

      7. remove-double-neg N/A

         \[\leadsto \frac{1}{\frac{\left(-\alpha\right) \cdot \alpha}{\color{blue}{{\alpha}^{4}}}} \cdot \log \left(1 - u0\right) \]

      8. lift-pow.f32 N/A

         \[\leadsto \frac{1}{\frac{\left(-\alpha\right) \cdot \alpha}{\color{blue}{{\alpha}^{4}}}} \cdot \log \left(1 - u0\right) \]

      9. metadata-eval N/A

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

      10. pow-pow N/A

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

      11. pow2 N/A

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

      12. lift-*.f32 N/A

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

      13. pow2 N/A

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

      14. times-frac N/A

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

      15. remove-double-neg N/A

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

      16. lift-neg.f32 N/A

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

      17. lift-neg.f32 N/A

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

      18. remove-double-neg N/A

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

      19. remove-double-div N/A

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

      20. associate-/l/ N/A

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

      21. inv-pow N/A

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

      22. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{-\alpha}{\alpha \cdot \alpha} \cdot \frac{1}{\left(\alpha \cdot \alpha\right) \cdot {\alpha}^{\color{blue}{\left(2 - 3\right)}}}} \cdot \log \left(1 - u0\right) \]

      23. pow-div N/A

          \[\leadsto \frac{1}{\frac{-\alpha}{\alpha \cdot \alpha} \cdot \frac{1}{\left(\alpha \cdot \alpha\right) \cdot \color{blue}{\frac{{\alpha}^{2}}{{\alpha}^{3}}}}} \cdot \log \left(1 - u0\right) \]

      24. pow2 N/A

          \[\leadsto \frac{1}{\frac{-\alpha}{\alpha \cdot \alpha} \cdot \frac{1}{\left(\alpha \cdot \alpha\right) \cdot \frac{\color{blue}{\alpha \cdot \alpha}}{{\alpha}^{3}}}} \cdot \log \left(1 - u0\right) \]

      25. lift-*.f32 N/A

          \[\leadsto \frac{1}{\frac{-\alpha}{\alpha \cdot \alpha} \cdot \frac{1}{\left(\alpha \cdot \alpha\right) \cdot \frac{\color{blue}{\alpha \cdot \alpha}}{{\alpha}^{3}}}} \cdot \log \left(1 - u0\right) \]

      26. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{-\alpha}{\alpha \cdot \alpha} \cdot \frac{1}{\left(\alpha \cdot \alpha\right) \cdot \frac{\alpha \cdot \alpha}{{\alpha}^{\color{blue}{\left(\frac{6}{2}\right)}}}}} \cdot \log \left(1 - u0\right) \]

   6. Applied rewrites 87.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{-\alpha}{\alpha \cdot \alpha} \cdot \frac{1}{\alpha}}} \cdot \log \left(1 - u0\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.00018000000272877514:\\ \;\;\;\;\frac{\alpha \cdot u0}{\frac{1}{\alpha}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \frac{-1}{\frac{\alpha}{\alpha \cdot \alpha} \cdot \frac{1}{\alpha}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u0 \leq 0.00018000000272877514:\\ \;\;\;\;\frac{\alpha \cdot u0}{\frac{1}{\alpha}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \alpha}{\alpha} \cdot \log \left(1 - u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= u0 0.00018000000272877514)
   (/ (* alpha u0) (/ 1.0 alpha))
   (* (/ (* (* (- alpha) alpha) alpha) alpha) (log (- 1.0 u0)))))

C

float code(float alpha, float u0) {
	float tmp;
	if (u0 <= 0.00018000000272877514f) {
		tmp = (alpha * u0) / (1.0f / alpha);
	} else {
		tmp = (((-alpha * alpha) * alpha) / alpha) * logf((1.0f - u0));
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: tmp
    if (u0 <= 0.00018000000272877514e0) then
        tmp = (alpha * u0) / (1.0e0 / alpha)
    else
        tmp = (((-alpha * alpha) * alpha) / alpha) * log((1.0e0 - u0))
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.00018000000272877514))
		tmp = Float32(Float32(alpha * u0) / Float32(Float32(1.0) / alpha));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(-alpha) * alpha) * alpha) / alpha) * log(Float32(Float32(1.0) - u0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if (u0 <= single(0.00018000000272877514))
		tmp = (alpha * u0) / (single(1.0) / alpha);
	else
		tmp = (((-alpha * alpha) * alpha) / alpha) * log((single(1.0) - u0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u0 < 1.80000003e-4

   1. Initial program 34.2%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{{\alpha}^{2} \cdot u0} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f32 91.6

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

   5. Applied rewrites 91.6%

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

   7. Applied rewrites 91.7%

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

   if 1.80000003e-4 < u0

   1. Initial program 87.6%

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

      1. lift-*.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. neg-sub0 N/A

         \[\leadsto \left(\color{blue}{\left(0 - \alpha\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      4. flip-- N/A

         \[\leadsto \left(\color{blue}{\frac{0 \cdot 0 - \alpha \cdot \alpha}{0 + \alpha}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{0} - \alpha \cdot \alpha}{0 + \alpha} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      6. neg-sub0 N/A

         \[\leadsto \left(\frac{\color{blue}{\mathsf{neg}\left(\alpha \cdot \alpha\right)}}{0 + \alpha} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      7. distribute-lft-neg-out N/A

         \[\leadsto \left(\frac{\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha}}{0 + \alpha} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      8. lift-neg.f32 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(-\alpha\right)} \cdot \alpha}{0 + \alpha} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      9. lift-*.f32 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(-\alpha\right) \cdot \alpha}}{0 + \alpha} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      10. +-lft-identity N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f32 N/A

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

      13. lower-*.f32 87.6

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

   4. Applied rewrites 87.6%

      \[\leadsto \color{blue}{\frac{\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \alpha}{\alpha}} \cdot \log \left(1 - u0\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.00018000000272877514:\\ \;\;\;\;\frac{\alpha \cdot u0}{\frac{1}{\alpha}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \alpha}{\alpha} \cdot \log \left(1 - u0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u0 \leq 0.00018000000272877514:\\ \;\;\;\;\frac{\alpha \cdot u0}{\frac{1}{\alpha}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= u0 0.00018000000272877514)
   (/ (* alpha u0) (/ 1.0 alpha))
   (* (* (- alpha) alpha) (log (- 1.0 u0)))))

C

float code(float alpha, float u0) {
	float tmp;
	if (u0 <= 0.00018000000272877514f) {
		tmp = (alpha * u0) / (1.0f / alpha);
	} else {
		tmp = (-alpha * alpha) * logf((1.0f - u0));
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: tmp
    if (u0 <= 0.00018000000272877514e0) then
        tmp = (alpha * u0) / (1.0e0 / alpha)
    else
        tmp = (-alpha * alpha) * log((1.0e0 - u0))
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.00018000000272877514))
		tmp = Float32(Float32(alpha * u0) / Float32(Float32(1.0) / alpha));
	else
		tmp = Float32(Float32(Float32(-alpha) * alpha) * log(Float32(Float32(1.0) - u0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if (u0 <= single(0.00018000000272877514))
		tmp = (alpha * u0) / (single(1.0) / alpha);
	else
		tmp = (-alpha * alpha) * log((single(1.0) - u0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u0 < 1.80000003e-4

   1. Initial program 34.2%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u0 around 0

      \[\leadsto \color{blue}{{\alpha}^{2} \cdot u0} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f32 91.6

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

   5. Applied rewrites 91.6%

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

   7. Applied rewrites 91.7%

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

   if 1.80000003e-4 < u0

   1. Initial program 87.6%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.00018000000272877514:\\ \;\;\;\;\frac{\alpha \cdot u0}{\frac{1}{\alpha}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.5% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 52.8%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
2. Add Preprocessing

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{{\alpha}^{2} \cdot u0} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 76.8

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

5. Applied rewrites 76.8%

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

7. Applied rewrites 76.8%

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

8. Final simplification 76.8%

   \[\leadsto \frac{\alpha \cdot u0}{\frac{1}{\alpha}} \]
9. Add Preprocessing

Alternative 5: 74.5% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 52.8%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
2. Add Preprocessing

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{{\alpha}^{2} \cdot u0} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 76.8

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

5. Applied rewrites 76.8%

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

7. Applied rewrites 76.8%

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

8. Final simplification 76.8%

   \[\leadsto \frac{\alpha}{\frac{1}{\alpha}} \cdot u0 \]
9. Add Preprocessing

Alternative 6: 74.6% accurate, 10.5× speedup

Math

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

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(Float32(alpha * u0) * alpha)
end

MATLAB

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

Derivation

1. Initial program 52.8%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
2. Add Preprocessing

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{{\alpha}^{2} \cdot u0} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 76.8

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

5. Applied rewrites 76.8%

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

7. Applied rewrites 76.8%

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

8. Final simplification 76.8%

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

Alternative 7: 74.6% accurate, 10.5× speedup

Math

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

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(Float32(alpha * alpha) * u0)
end

MATLAB

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

Derivation

1. Initial program 52.8%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
2. Add Preprocessing

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{{\alpha}^{2} \cdot u0} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 76.8

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

5. Applied rewrites 76.8%

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

6. Final simplification 76.8%

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

Reproduce

herbie shell --seed 2024244 
(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))))