Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 55.6% → 89.6%

Time: 6.7s

Alternatives: 5

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: 89.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u0 \leq 0.00011800000356743112:\\ \;\;\;\;\frac{u0}{{\alpha}^{-2}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \frac{1}{\frac{-1}{\alpha \cdot \alpha}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= u0 0.00011800000356743112)
   (/ u0 (pow alpha -2.0))
   (* (log (- 1.0 u0)) (/ 1.0 (/ -1.0 (* alpha alpha))))))

C

float code(float alpha, float u0) {
	float tmp;
	if (u0 <= 0.00011800000356743112f) {
		tmp = u0 / powf(alpha, -2.0f);
	} else {
		tmp = logf((1.0f - u0)) * (1.0f / (-1.0f / (alpha * 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 <= 0.00011800000356743112e0) then
        tmp = u0 / (alpha ** (-2.0e0))
    else
        tmp = log((1.0e0 - u0)) * (1.0e0 / ((-1.0e0) / (alpha * alpha)))
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.00011800000356743112))
		tmp = Float32(u0 / (alpha ^ Float32(-2.0)));
	else
		tmp = Float32(log(Float32(Float32(1.0) - u0)) * Float32(Float32(1.0) / Float32(Float32(-1.0) / Float32(alpha * alpha))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if (u0 <= single(0.00011800000356743112))
		tmp = u0 / (alpha ^ single(-2.0));
	else
		tmp = log((single(1.0) - u0)) * (single(1.0) / (single(-1.0) / (alpha * alpha)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u0 < 1.18000004e-4

   1. Initial program 35.7%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 91.8

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

   5. Applied rewrites 91.8%

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

   7. Applied rewrites 91.7%

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

   9. Applied rewrites 91.6%

      \[\leadsto \frac{u0}{\color{blue}{\frac{\frac{1}{\alpha}}{\alpha}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 91.8%

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

   if 1.18000004e-4 < u0

   1. Initial program 85.6%

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

      1. lift-neg.f32 N/A

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

      2. neg-sub0 N/A

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

      3. flip-- N/A

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

      4. metadata-eval N/A

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

      5. neg-sub0 N/A

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

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

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

      7. lift-neg.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. div-inv N/A

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

      10. lower-*.f32 N/A

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

      11. +-lft-identity N/A

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

      12. lower-/.f32 85.5

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

   4. Applied rewrites 85.5%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-/.f32 N/A

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

      7. rgt-mult-inverse N/A

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

      8. *-commutative N/A

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

      9. *-rgt-identity 85.6

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

      10. lift-*.f32 N/A

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

      11. lift-neg.f32 N/A

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

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

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

      13. pow2 N/A

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

      14. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left({\alpha}^{\color{blue}{\left(6 - 4\right)}}\right)\right) \cdot \log \left(1 - u0\right) \]

      15. pow-div N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{{\alpha}^{6}}{{\alpha}^{4}}}\right)\right) \cdot \log \left(1 - u0\right) \]

      16. lift-pow.f32 N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{{\alpha}^{6}}}{{\alpha}^{4}}\right)\right) \cdot \log \left(1 - u0\right) \]

      17. lift-pow.f32 N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{{\alpha}^{6}}{\color{blue}{{\alpha}^{4}}}\right)\right) \cdot \log \left(1 - u0\right) \]

      18. distribute-frac-neg N/A

          \[\leadsto \color{blue}{\frac{\mathsf{neg}\left({\alpha}^{6}\right)}{{\alpha}^{4}}} \cdot \log \left(1 - u0\right) \]

      19. lift-neg.f32 N/A

          \[\leadsto \frac{\color{blue}{-{\alpha}^{6}}}{{\alpha}^{4}} \cdot \log \left(1 - u0\right) \]

      20. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{{\alpha}^{4}}{-{\alpha}^{6}}}} \cdot \log \left(1 - u0\right) \]

      21. lower-/.f32 N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{{\alpha}^{4}}{-{\alpha}^{6}}}} \cdot \log \left(1 - u0\right) \]

      22. clear-num N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{-{\alpha}^{6}}{{\alpha}^{4}}}}} \cdot \log \left(1 - u0\right) \]

      23. lift-neg.f32 N/A

          \[\leadsto \frac{1}{\frac{1}{\frac{\color{blue}{\mathsf{neg}\left({\alpha}^{6}\right)}}{{\alpha}^{4}}}} \cdot \log \left(1 - u0\right) \]

      24. distribute-frac-neg N/A

          \[\leadsto \frac{1}{\frac{1}{\color{blue}{\mathsf{neg}\left(\frac{{\alpha}^{6}}{{\alpha}^{4}}\right)}}} \cdot \log \left(1 - u0\right) \]

      25. lift-pow.f32 N/A

          \[\leadsto \frac{1}{\frac{1}{\mathsf{neg}\left(\frac{\color{blue}{{\alpha}^{6}}}{{\alpha}^{4}}\right)}} \cdot \log \left(1 - u0\right) \]

      26. lift-pow.f32 N/A

          \[\leadsto \frac{1}{\frac{1}{\mathsf{neg}\left(\frac{{\alpha}^{6}}{\color{blue}{{\alpha}^{4}}}\right)}} \cdot \log \left(1 - u0\right) \]

      27. pow-div N/A

          \[\leadsto \frac{1}{\frac{1}{\mathsf{neg}\left(\color{blue}{{\alpha}^{\left(6 - 4\right)}}\right)}} \cdot \log \left(1 - u0\right) \]

   6. Applied rewrites 85.7%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.00011800000356743112:\\ \;\;\;\;\frac{u0}{{\alpha}^{-2}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \frac{1}{\frac{-1}{\alpha \cdot \alpha}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 63.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u0 \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\left(\left(\mathsf{log1p}\left(u0 \cdot u0 + u0\right) - \left(-{u0}^{3}\right)\right) \cdot \alpha\right) \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \frac{1}{\frac{-1}{\alpha \cdot \alpha}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= u0 4.999999873689376e-5)
   (* (* (- (log1p (+ (* u0 u0) u0)) (- (pow u0 3.0))) alpha) alpha)
   (* (log (- 1.0 u0)) (/ 1.0 (/ -1.0 (* alpha alpha))))))

C

float code(float alpha, float u0) {
	float tmp;
	if (u0 <= 4.999999873689376e-5f) {
		tmp = ((log1pf(((u0 * u0) + u0)) - -powf(u0, 3.0f)) * alpha) * alpha;
	} else {
		tmp = logf((1.0f - u0)) * (1.0f / (-1.0f / (alpha * alpha)));
	}
	return tmp;
}

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (u0 <= Float32(4.999999873689376e-5))
		tmp = Float32(Float32(Float32(log1p(Float32(Float32(u0 * u0) + u0)) - Float32(-(u0 ^ Float32(3.0)))) * alpha) * alpha);
	else
		tmp = Float32(log(Float32(Float32(1.0) - u0)) * Float32(Float32(1.0) / Float32(Float32(-1.0) / Float32(alpha * alpha))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u0 < 4.99999987e-5

   1. Initial program 34.4%

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

   3. Taylor expanded in alpha around 0

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

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

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

      8. mul-1-neg N/A

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

      9. lower-*.f32 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f32 N/A

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

      12. sub-neg N/A

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

      13. lower-log1p.f32 N/A

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

      14. lower-neg.f32 92.5

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

   5. Applied rewrites 92.5%

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

   7. Applied rewrites 92.3%

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

   8. Taylor expanded in u0 around 0

      \[\leadsto \left(\left(-\alpha\right) \cdot \left(-1 \cdot {u0}^{3} - \mathsf{log1p}\left(u0 \cdot u0 - \left(-u0\right)\right)\right)\right) \cdot \alpha \]
   9. Step-by-step derivation

   10. Applied rewrites 93.0%

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

   if 4.99999987e-5 < u0

   1. Initial program 84.7%

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

      1. lift-neg.f32 N/A

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

      2. neg-sub0 N/A

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

      3. flip-- N/A

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

      4. metadata-eval N/A

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

      5. neg-sub0 N/A

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

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

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

      7. lift-neg.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. div-inv N/A

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

      10. lower-*.f32 N/A

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

      11. +-lft-identity N/A

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

      12. lower-/.f32 84.6

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

   4. Applied rewrites 84.6%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-/.f32 N/A

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

      7. rgt-mult-inverse N/A

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

      8. *-commutative N/A

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

      9. *-rgt-identity 84.7

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

      10. lift-*.f32 N/A

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

      11. lift-neg.f32 N/A

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

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

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

      13. pow2 N/A

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

      14. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left({\alpha}^{\color{blue}{\left(6 - 4\right)}}\right)\right) \cdot \log \left(1 - u0\right) \]

      15. pow-div N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{{\alpha}^{6}}{{\alpha}^{4}}}\right)\right) \cdot \log \left(1 - u0\right) \]

      16. lift-pow.f32 N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{{\alpha}^{6}}}{{\alpha}^{4}}\right)\right) \cdot \log \left(1 - u0\right) \]

      17. lift-pow.f32 N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{{\alpha}^{6}}{\color{blue}{{\alpha}^{4}}}\right)\right) \cdot \log \left(1 - u0\right) \]

      18. distribute-frac-neg N/A

          \[\leadsto \color{blue}{\frac{\mathsf{neg}\left({\alpha}^{6}\right)}{{\alpha}^{4}}} \cdot \log \left(1 - u0\right) \]

      19. lift-neg.f32 N/A

          \[\leadsto \frac{\color{blue}{-{\alpha}^{6}}}{{\alpha}^{4}} \cdot \log \left(1 - u0\right) \]

      20. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{{\alpha}^{4}}{-{\alpha}^{6}}}} \cdot \log \left(1 - u0\right) \]

      21. lower-/.f32 N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{{\alpha}^{4}}{-{\alpha}^{6}}}} \cdot \log \left(1 - u0\right) \]

      22. clear-num N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{-{\alpha}^{6}}{{\alpha}^{4}}}}} \cdot \log \left(1 - u0\right) \]

      23. lift-neg.f32 N/A

          \[\leadsto \frac{1}{\frac{1}{\frac{\color{blue}{\mathsf{neg}\left({\alpha}^{6}\right)}}{{\alpha}^{4}}}} \cdot \log \left(1 - u0\right) \]

      24. distribute-frac-neg N/A

          \[\leadsto \frac{1}{\frac{1}{\color{blue}{\mathsf{neg}\left(\frac{{\alpha}^{6}}{{\alpha}^{4}}\right)}}} \cdot \log \left(1 - u0\right) \]

      25. lift-pow.f32 N/A

          \[\leadsto \frac{1}{\frac{1}{\mathsf{neg}\left(\frac{\color{blue}{{\alpha}^{6}}}{{\alpha}^{4}}\right)}} \cdot \log \left(1 - u0\right) \]

      26. lift-pow.f32 N/A

          \[\leadsto \frac{1}{\frac{1}{\mathsf{neg}\left(\frac{{\alpha}^{6}}{\color{blue}{{\alpha}^{4}}}\right)}} \cdot \log \left(1 - u0\right) \]

      27. pow-div N/A

          \[\leadsto \frac{1}{\frac{1}{\mathsf{neg}\left(\color{blue}{{\alpha}^{\left(6 - 4\right)}}\right)}} \cdot \log \left(1 - u0\right) \]

   6. Applied rewrites 84.8%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\left(\left(\mathsf{log1p}\left(u0 \cdot u0 + u0\right) - \left(-{u0}^{3}\right)\right) \cdot \alpha\right) \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \frac{1}{\frac{-1}{\alpha \cdot \alpha}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u0 \leq 0.00011800000356743112:\\ \;\;\;\;\frac{u0}{{\alpha}^{-2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(1 - u0\right) \cdot \left(-\alpha\right)\right) \cdot \alpha\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= u0 0.00011800000356743112)
   (/ u0 (pow alpha -2.0))
   (* (* (log (- 1.0 u0)) (- alpha)) alpha)))

C

float code(float alpha, float u0) {
	float tmp;
	if (u0 <= 0.00011800000356743112f) {
		tmp = u0 / powf(alpha, -2.0f);
	} else {
		tmp = (logf((1.0f - u0)) * -alpha) * 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 <= 0.00011800000356743112e0) then
        tmp = u0 / (alpha ** (-2.0e0))
    else
        tmp = (log((1.0e0 - u0)) * -alpha) * alpha
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.00011800000356743112))
		tmp = Float32(u0 / (alpha ^ Float32(-2.0)));
	else
		tmp = Float32(Float32(log(Float32(Float32(1.0) - u0)) * Float32(-alpha)) * alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if (u0 <= single(0.00011800000356743112))
		tmp = u0 / (alpha ^ single(-2.0));
	else
		tmp = (log((single(1.0) - u0)) * -alpha) * alpha;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u0 < 1.18000004e-4

   1. Initial program 35.7%

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

   3. Taylor expanded in u0 around 0

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 91.8

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

   5. Applied rewrites 91.8%

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

   7. Applied rewrites 91.7%

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

   9. Applied rewrites 91.6%

      \[\leadsto \frac{u0}{\color{blue}{\frac{\frac{1}{\alpha}}{\alpha}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 91.8%

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

   if 1.18000004e-4 < u0

   1. Initial program 85.6%

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

   3. Taylor expanded in alpha around 0

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

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

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

      8. mul-1-neg N/A

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

      9. lower-*.f32 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f32 N/A

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

      12. sub-neg N/A

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

      13. lower-log1p.f32 N/A

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

      14. lower-neg.f32 50.4

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

   5. Applied rewrites 50.4%

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

   7. Applied rewrites 85.7%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.00011800000356743112:\\ \;\;\;\;\frac{u0}{{\alpha}^{-2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(1 - u0\right) \cdot \left(-\alpha\right)\right) \cdot \alpha\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.8% accurate, 1.0× speedup

Math

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

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 55.2%

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

3. Taylor expanded in u0 around 0

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 75.6

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

5. Applied rewrites 75.6%

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

7. Applied rewrites 75.6%

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

9. Applied rewrites 75.5%

   \[\leadsto \frac{u0}{\color{blue}{\frac{\frac{1}{\alpha}}{\alpha}}} \]
10. Step-by-step derivation

11. Applied rewrites 75.6%

    \[\leadsto \frac{u0}{\color{blue}{{\alpha}^{-2}}} \]
12. Add Preprocessing

Alternative 5: 74.8% 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 55.2%

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

3. Taylor expanded in u0 around 0

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 75.6

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

5. Applied rewrites 75.6%

   \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right) \cdot u0} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024276 
(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))))