Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 55.5% → 89.8%

Time: 7.2s

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: 55.5% 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: 89.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u0 \leq 0.0001900000061141327:\\ \;\;\;\;\left(\alpha \cdot u0\right) \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \left(\left(\frac{-1}{\alpha \cdot \alpha} \cdot {\alpha}^{3}\right) \cdot \alpha\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= u0 0.0001900000061141327)
   (* (* alpha u0) alpha)
   (*
    (log (- 1.0 u0))
    (* (* (/ -1.0 (* alpha alpha)) (pow alpha 3.0)) alpha))))

C

float code(float alpha, float u0) {
	float tmp;
	if (u0 <= 0.0001900000061141327f) {
		tmp = (alpha * u0) * alpha;
	} else {
		tmp = logf((1.0f - u0)) * (((-1.0f / (alpha * alpha)) * powf(alpha, 3.0f)) * 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 <= 0.0001900000061141327e0) then
        tmp = (alpha * u0) * alpha
    else
        tmp = log((1.0e0 - u0)) * ((((-1.0e0) / (alpha * alpha)) * (alpha ** 3.0e0)) * alpha)
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.0001900000061141327))
		tmp = Float32(Float32(alpha * u0) * alpha);
	else
		tmp = Float32(log(Float32(Float32(1.0) - u0)) * Float32(Float32(Float32(Float32(-1.0) / Float32(alpha * alpha)) * (alpha ^ Float32(3.0))) * alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if (u0 <= single(0.0001900000061141327))
		tmp = (alpha * u0) * alpha;
	else
		tmp = log((single(1.0) - u0)) * (((single(-1.0) / (alpha * alpha)) * (alpha ^ single(3.0))) * alpha);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u0 < 1.90000006e-4

   1. Initial program 34.8%

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

   3. Taylor expanded in u0 around 0

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

      1. 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 91.0

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

   5. Applied rewrites 91.0%

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

   7. Applied rewrites 91.0%

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

   if 1.90000006e-4 < u0

   1. Initial program 87.5%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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. flip3-- N/A

         \[\leadsto \left(\color{blue}{\frac{{0}^{3} - {\alpha}^{3}}{0 \cdot 0 + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      4. frac-2neg N/A

         \[\leadsto \left(\color{blue}{\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\mathsf{neg}\left(\left(0 \cdot 0 + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)\right)\right)}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\mathsf{neg}\left(\left(\color{blue}{0} + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)\right)\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      6. +-lft-identity N/A

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\mathsf{neg}\left(\color{blue}{\left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      7. mul0-lft N/A

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\mathsf{neg}\left(\left(\alpha \cdot \alpha + \color{blue}{0}\right)\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      8. +-rgt-identity N/A

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\mathsf{neg}\left(\color{blue}{\alpha \cdot \alpha}\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

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

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      10. lift-neg.f32 N/A

          \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\color{blue}{\left(-\alpha\right)} \cdot \alpha} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      11. lift-*.f32 N/A

          \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\color{blue}{\left(-\alpha\right) \cdot \alpha}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      12. div-inv N/A

          \[\leadsto \left(\color{blue}{\left(\left(\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)\right) \cdot \frac{1}{\left(-\alpha\right) \cdot \alpha}\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\left(\mathsf{neg}\left(\left(\color{blue}{0} - {\alpha}^{3}\right)\right)\right) \cdot \frac{1}{\left(-\alpha\right) \cdot \alpha}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      14. sub0-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower-*.f32 N/A

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

      17. lower-pow.f32 N/A

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

      18. lower-/.f32 87.6

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

   4. Applied rewrites 87.6%

      \[\leadsto \left(\color{blue}{\left({\alpha}^{3} \cdot \frac{1}{\left(-\alpha\right) \cdot \alpha}\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   5. Step-by-step derivation

      1. lift-/.f32 N/A

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

      2. metadata-eval N/A

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

      3. lift-*.f32 N/A

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

      4. lift-neg.f32 N/A

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

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

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

      6. lift-*.f32 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f32 87.6

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

   6. Applied rewrites 87.6%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.0001900000061141327:\\ \;\;\;\;\left(\alpha \cdot u0\right) \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \left(\left(\frac{-1}{\alpha \cdot \alpha} \cdot {\alpha}^{3}\right) \cdot \alpha\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 46.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.999809980392456:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \left(\left(\frac{-1}{\alpha \cdot \alpha} \cdot {\alpha}^{3}\right) \cdot \alpha\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{log1p}\left(u0\right) + \mathsf{log1p}\left(u0 \cdot u0\right)\right) - \mathsf{log1p}\left(-{u0}^{4}\right)\right) \cdot \left(\alpha \cdot \alpha\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.999809980392456)
   (* (log (- 1.0 u0)) (* (* (/ -1.0 (* alpha alpha)) (pow alpha 3.0)) alpha))
   (*
    (- (+ (log1p u0) (log1p (* u0 u0))) (log1p (- (pow u0 4.0))))
    (* alpha alpha))))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.999809980392456f) {
		tmp = logf((1.0f - u0)) * (((-1.0f / (alpha * alpha)) * powf(alpha, 3.0f)) * alpha);
	} else {
		tmp = ((log1pf(u0) + log1pf((u0 * u0))) - log1pf(-powf(u0, 4.0f))) * (alpha * alpha);
	}
	return tmp;
}

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.999809980392456))
		tmp = Float32(log(Float32(Float32(1.0) - u0)) * Float32(Float32(Float32(Float32(-1.0) / Float32(alpha * alpha)) * (alpha ^ Float32(3.0))) * alpha));
	else
		tmp = Float32(Float32(Float32(log1p(u0) + log1p(Float32(u0 * u0))) - log1p(Float32(-(u0 ^ Float32(4.0))))) * Float32(alpha * alpha));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.99980998

   1. Initial program 87.5%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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. flip3-- N/A

         \[\leadsto \left(\color{blue}{\frac{{0}^{3} - {\alpha}^{3}}{0 \cdot 0 + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      4. frac-2neg N/A

         \[\leadsto \left(\color{blue}{\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\mathsf{neg}\left(\left(0 \cdot 0 + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)\right)\right)}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\mathsf{neg}\left(\left(\color{blue}{0} + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)\right)\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      6. +-lft-identity N/A

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\mathsf{neg}\left(\color{blue}{\left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      7. mul0-lft N/A

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\mathsf{neg}\left(\left(\alpha \cdot \alpha + \color{blue}{0}\right)\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      8. +-rgt-identity N/A

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\mathsf{neg}\left(\color{blue}{\alpha \cdot \alpha}\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

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

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      10. lift-neg.f32 N/A

          \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\color{blue}{\left(-\alpha\right)} \cdot \alpha} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      11. lift-*.f32 N/A

          \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\color{blue}{\left(-\alpha\right) \cdot \alpha}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      12. div-inv N/A

          \[\leadsto \left(\color{blue}{\left(\left(\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)\right) \cdot \frac{1}{\left(-\alpha\right) \cdot \alpha}\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\left(\mathsf{neg}\left(\left(\color{blue}{0} - {\alpha}^{3}\right)\right)\right) \cdot \frac{1}{\left(-\alpha\right) \cdot \alpha}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      14. sub0-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower-*.f32 N/A

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

      17. lower-pow.f32 N/A

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

      18. lower-/.f32 87.6

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

   4. Applied rewrites 87.6%

      \[\leadsto \left(\color{blue}{\left({\alpha}^{3} \cdot \frac{1}{\left(-\alpha\right) \cdot \alpha}\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   5. Step-by-step derivation

      1. lift-/.f32 N/A

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

      2. metadata-eval N/A

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

      3. lift-*.f32 N/A

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

      4. lift-neg.f32 N/A

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

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

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

      6. lift-*.f32 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f32 87.6

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

   6. Applied rewrites 87.6%

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

   if 0.99980998 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 34.8%

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

   4. Applied rewrites 91.1%

      \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\left(\mathsf{log1p}\left(-{u0}^{4}\right) - \left(\mathsf{log1p}\left(u0 \cdot u0\right) + \mathsf{log1p}\left(u0\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.999809980392456:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \left(\left(\frac{-1}{\alpha \cdot \alpha} \cdot {\alpha}^{3}\right) \cdot \alpha\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{log1p}\left(u0\right) + \mathsf{log1p}\left(u0 \cdot u0\right)\right) - \mathsf{log1p}\left(-{u0}^{4}\right)\right) \cdot \left(\alpha \cdot \alpha\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u0 \leq 0.0001900000061141327:\\ \;\;\;\;\left(\alpha \cdot u0\right) \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{\alpha} \cdot {\alpha}^{3}\right) \cdot \log \left(1 - u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= u0 0.0001900000061141327)
   (* (* alpha u0) alpha)
   (* (* (/ -1.0 alpha) (pow alpha 3.0)) (log (- 1.0 u0)))))

C

float code(float alpha, float u0) {
	float tmp;
	if (u0 <= 0.0001900000061141327f) {
		tmp = (alpha * u0) * alpha;
	} else {
		tmp = ((-1.0f / alpha) * powf(alpha, 3.0f)) * 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 <= 0.0001900000061141327e0) then
        tmp = (alpha * u0) * alpha
    else
        tmp = (((-1.0e0) / alpha) * (alpha ** 3.0e0)) * log((1.0e0 - u0))
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.0001900000061141327))
		tmp = Float32(Float32(alpha * u0) * alpha);
	else
		tmp = Float32(Float32(Float32(Float32(-1.0) / alpha) * (alpha ^ Float32(3.0))) * log(Float32(Float32(1.0) - u0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if (u0 <= single(0.0001900000061141327))
		tmp = (alpha * u0) * alpha;
	else
		tmp = ((single(-1.0) / alpha) * (alpha ^ single(3.0))) * log((single(1.0) - u0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u0 < 1.90000006e-4

   1. Initial program 34.8%

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

   3. Taylor expanded in u0 around 0

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

      1. 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 91.0

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

   5. Applied rewrites 91.0%

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

   7. Applied rewrites 91.0%

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

   if 1.90000006e-4 < u0

   1. Initial program 87.5%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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. clear-num N/A

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

      13. lower-/.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. lower-*.f32 87.4

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

   4. Applied rewrites 87.4%

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

      1. lift-/.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f32 N/A

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

      5. associate-/r/ N/A

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

      6. lift-*.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. associate-*l* N/A

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

      9. lift-neg.f32 N/A

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

      10. lift-*.f32 N/A

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

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

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

      12. lift-*.f32 N/A

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

      13. cube-mult N/A

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

      14. metadata-eval N/A

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

      15. remove-double-neg N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

      18. metadata-eval N/A

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

      19. lower-pow.f32 N/A

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

      20. metadata-eval N/A

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

      21. lift-neg.f32 N/A

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

      22. frac-2neg N/A

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

      23. lower-/.f32 87.6

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

   6. Applied rewrites 87.6%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.0001900000061141327:\\ \;\;\;\;\left(\alpha \cdot u0\right) \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{\alpha} \cdot {\alpha}^{3}\right) \cdot \log \left(1 - u0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u0 \leq 0.0001900000061141327:\\ \;\;\;\;\left(\alpha \cdot u0\right) \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\frac{-1}{\alpha}} \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= u0 0.0001900000061141327)
   (* (* alpha u0) alpha)
   (* (* (/ 1.0 (/ -1.0 alpha)) alpha) (log (- 1.0 u0)))))

C

float code(float alpha, float u0) {
	float tmp;
	if (u0 <= 0.0001900000061141327f) {
		tmp = (alpha * u0) * alpha;
	} else {
		tmp = ((1.0f / (-1.0f / 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 <= 0.0001900000061141327e0) then
        tmp = (alpha * u0) * alpha
    else
        tmp = ((1.0e0 / ((-1.0e0) / alpha)) * alpha) * log((1.0e0 - u0))
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.0001900000061141327))
		tmp = Float32(Float32(alpha * u0) * alpha);
	else
		tmp = Float32(Float32(Float32(Float32(1.0) / Float32(Float32(-1.0) / 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(0.0001900000061141327))
		tmp = (alpha * u0) * alpha;
	else
		tmp = ((single(1.0) / (single(-1.0) / alpha)) * alpha) * log((single(1.0) - u0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u0 < 1.90000006e-4

   1. Initial program 34.8%

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

   3. Taylor expanded in u0 around 0

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

      1. 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 91.0

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

   5. Applied rewrites 91.0%

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

   7. Applied rewrites 91.0%

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

   if 1.90000006e-4 < u0

   1. Initial program 87.5%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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. flip3-- N/A

         \[\leadsto \left(\color{blue}{\frac{{0}^{3} - {\alpha}^{3}}{0 \cdot 0 + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      4. frac-2neg N/A

         \[\leadsto \left(\color{blue}{\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\mathsf{neg}\left(\left(0 \cdot 0 + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)\right)\right)}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\mathsf{neg}\left(\left(\color{blue}{0} + \left(\alpha \cdot \alpha + 0 \cdot \alpha\right)\right)\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      6. +-lft-identity N/A

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\mathsf{neg}\left(\color{blue}{\left(\alpha \cdot \alpha + 0 \cdot \alpha\right)}\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      7. mul0-lft N/A

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\mathsf{neg}\left(\left(\alpha \cdot \alpha + \color{blue}{0}\right)\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      8. +-rgt-identity N/A

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\mathsf{neg}\left(\color{blue}{\alpha \cdot \alpha}\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

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

         \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\color{blue}{\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      10. lift-neg.f32 N/A

          \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\color{blue}{\left(-\alpha\right)} \cdot \alpha} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      11. lift-*.f32 N/A

          \[\leadsto \left(\frac{\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)}{\color{blue}{\left(-\alpha\right) \cdot \alpha}} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      12. div-inv N/A

          \[\leadsto \left(\color{blue}{\left(\left(\mathsf{neg}\left(\left({0}^{3} - {\alpha}^{3}\right)\right)\right) \cdot \frac{1}{\left(-\alpha\right) \cdot \alpha}\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\left(\mathsf{neg}\left(\left(\color{blue}{0} - {\alpha}^{3}\right)\right)\right) \cdot \frac{1}{\left(-\alpha\right) \cdot \alpha}\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      14. sub0-neg N/A

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

      15. remove-double-neg N/A

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

      16. lower-*.f32 N/A

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

      17. lower-pow.f32 N/A

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

      18. lower-/.f32 87.6

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

   4. Applied rewrites 87.6%

      \[\leadsto \left(\color{blue}{\left({\alpha}^{3} \cdot \frac{1}{\left(-\alpha\right) \cdot \alpha}\right)} \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   5. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. un-div-inv N/A

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

      4. lift-*.f32 N/A

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

      5. lift-neg.f32 N/A

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

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

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

      7. lift-*.f32 N/A

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

      8. distribute-frac-neg2 N/A

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

      9. lift-pow.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. pow2 N/A

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

      12. pow-div N/A

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

      13. metadata-eval N/A

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

      14. unpow1 N/A

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

      15. remove-double-div N/A

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

      16. lift-/.f32 N/A

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

      17. distribute-frac-neg2 N/A

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

      18. neg-mul-1 N/A

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

      19. lift-/.f32 N/A

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

      20. div-inv N/A

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

      21. lift-/.f32 N/A

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

      22. lower-/.f32 87.5

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

   6. Applied rewrites 87.5%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.0001900000061141327:\\ \;\;\;\;\left(\alpha \cdot u0\right) \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\frac{-1}{\alpha}} \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u0 \leq 0.0001900000061141327:\\ \;\;\;\;\left(\alpha \cdot u0\right) \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= u0 0.0001900000061141327)
   (* (* alpha u0) alpha)
   (* (* (- alpha) alpha) (log (- 1.0 u0)))))

C

float code(float alpha, float u0) {
	float tmp;
	if (u0 <= 0.0001900000061141327f) {
		tmp = (alpha * u0) * 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 <= 0.0001900000061141327e0) then
        tmp = (alpha * u0) * 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(0.0001900000061141327))
		tmp = Float32(Float32(alpha * u0) * 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(0.0001900000061141327))
		tmp = (alpha * u0) * 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 < 1.90000006e-4

   1. Initial program 34.8%

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

   3. Taylor expanded in u0 around 0

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

      1. 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 91.0

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

   5. Applied rewrites 91.0%

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

   7. Applied rewrites 91.0%

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

   if 1.90000006e-4 < u0

   1. Initial program 87.5%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.0001900000061141327:\\ \;\;\;\;\left(\alpha \cdot u0\right) \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \end{array} \]
5. 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 56.2%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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 73.9

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

5. Applied rewrites 73.9%

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

7. Applied rewrites 74.0%

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

8. Final simplification 74.0%

   \[\leadsto \left(\alpha \cdot u0\right) \cdot \alpha \]
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 56.2%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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 73.9

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

5. Applied rewrites 73.9%

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

Reproduce

herbie shell --seed 2024285 
(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))))