Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 56.2% → 97.9%

Time: 10.5s

Alternatives: 6

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: 56.2% 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: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9700000286102295:\\ \;\;\;\;\frac{\alpha \cdot \left(\alpha \cdot \alpha\right)}{-\alpha} \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot \left({u0}^{4} \cdot \left(\frac{\frac{0.5 + \frac{1}{u0}}{u0} - -0.3333333333333333}{u0} - -0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9700000286102295)
   (* (/ (* alpha (* alpha alpha)) (- alpha)) (log (- 1.0 u0)))
   (*
    (* alpha alpha)
    (*
     (pow u0 4.0)
     (- (/ (- (/ (+ 0.5 (/ 1.0 u0)) u0) -0.3333333333333333) u0) -0.25)))))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9700000286102295f) {
		tmp = ((alpha * (alpha * alpha)) / -alpha) * logf((1.0f - u0));
	} else {
		tmp = (alpha * alpha) * (powf(u0, 4.0f) * (((((0.5f + (1.0f / u0)) / u0) - -0.3333333333333333f) / u0) - -0.25f));
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: tmp
    if ((1.0e0 - u0) <= 0.9700000286102295e0) then
        tmp = ((alpha * (alpha * alpha)) / -alpha) * log((1.0e0 - u0))
    else
        tmp = (alpha * alpha) * ((u0 ** 4.0e0) * (((((0.5e0 + (1.0e0 / u0)) / u0) - (-0.3333333333333333e0)) / u0) - (-0.25e0)))
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9700000286102295))
		tmp = Float32(Float32(Float32(alpha * Float32(alpha * alpha)) / Float32(-alpha)) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(alpha * alpha) * Float32((u0 ^ Float32(4.0)) * Float32(Float32(Float32(Float32(Float32(Float32(0.5) + Float32(Float32(1.0) / u0)) / u0) - Float32(-0.3333333333333333)) / u0) - Float32(-0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9700000286102295))
		tmp = ((alpha * (alpha * alpha)) / -alpha) * log((single(1.0) - u0));
	else
		tmp = (alpha * alpha) * ((u0 ^ single(4.0)) * (((((single(0.5) + (single(1.0) / u0)) / u0) - single(-0.3333333333333333)) / u0) - single(-0.25)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.970000029

   1. Initial program 96.8%

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

      1. neg-sub0 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. neg-sub0 N/A

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

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

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

      6. lift-neg.f32 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-*.f32 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. +-lft-identity N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f32 N/A

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

      11. lower-*.f32 96.8

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 96.8

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

   4. Applied rewrites 96.8%

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

   if 0.970000029 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 50.5%

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

   4. Applied rewrites 22.6%

      \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\left(\log \left(\frac{1 - \left(u0 \cdot u0\right) \cdot \left(u0 \cdot u0\right)}{1 + u0}\right) - \mathsf{log1p}\left(u0 \cdot u0\right)\right)} \]

   5. Taylor expanded in u0 around 0

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \color{blue}{-1}\right)\right) \]

      4. lower-fma.f32 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}, -1\right)}\right) \]

      5. sub-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \color{blue}{\frac{-1}{2}}, -1\right)\right) \]

      7. lower-fma.f32 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{4} \cdot u0 - \frac{1}{3}, \frac{-1}{2}\right)}, -1\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\frac{-1}{4} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, \frac{-1}{2}\right), -1\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{-1}{4}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), \frac{-1}{2}\right), -1\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, u0 \cdot \frac{-1}{4} + \color{blue}{\frac{-1}{3}}, \frac{-1}{2}\right), -1\right)\right) \]

      11. lower-fma.f32 81.4

          \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)}, -0.5\right), -1\right)\right) \]

   7. Applied rewrites 81.4%

      \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right)} \]

   8. Taylor expanded in u0 around inf

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left({u0}^{4} \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{u0}}{{u0}^{2}} - \left(\frac{1}{4} + \frac{1}{3} \cdot \frac{1}{u0}\right)\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left({u0}^{4} \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{u0}}{{u0}^{2}} - \left(\frac{1}{4} + \frac{1}{3} \cdot \frac{1}{u0}\right)\right)\right)} \]

      2. lower-pow.f32 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(\color{blue}{{u0}^{4}} \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{u0}}{{u0}^{2}} - \left(\frac{1}{4} + \frac{1}{3} \cdot \frac{1}{u0}\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left({u0}^{4} \cdot \color{blue}{\left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{u0}}{{u0}^{2}} + \left(\mathsf{neg}\left(\left(\frac{1}{4} + \frac{1}{3} \cdot \frac{1}{u0}\right)\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left({u0}^{4} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{4} + \frac{1}{3} \cdot \frac{1}{u0}\right)\right)\right) + -1 \cdot \frac{\frac{1}{2} + \frac{1}{u0}}{{u0}^{2}}\right)}\right) \]

      5. distribute-neg-in N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left({u0}^{4} \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{u0}\right)\right)\right)} + -1 \cdot \frac{\frac{1}{2} + \frac{1}{u0}}{{u0}^{2}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left({u0}^{4} \cdot \left(\left(\color{blue}{\frac{-1}{4}} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{u0}\right)\right)\right) + -1 \cdot \frac{\frac{1}{2} + \frac{1}{u0}}{{u0}^{2}}\right)\right) \]

      7. associate-+l+ N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left({u0}^{4} \cdot \color{blue}{\left(\frac{-1}{4} + \left(\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{u0}\right)\right) + -1 \cdot \frac{\frac{1}{2} + \frac{1}{u0}}{{u0}^{2}}\right)\right)}\right) \]

      8. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left({u0}^{4} \cdot \left(\frac{-1}{4} + \color{blue}{\left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{u0}}{{u0}^{2}} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{u0}\right)\right)\right)}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left({u0}^{4} \cdot \left(\frac{-1}{4} + \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{u0}}{\color{blue}{u0 \cdot u0}} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{u0}\right)\right)\right)\right)\right) \]

      10. associate-/r* N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left({u0}^{4} \cdot \left(\frac{-1}{4} + \left(-1 \cdot \color{blue}{\frac{\frac{\frac{1}{2} + \frac{1}{u0}}{u0}}{u0}} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{u0}\right)\right)\right)\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left({u0}^{4} \cdot \left(\frac{-1}{4} + \left(\color{blue}{\frac{-1 \cdot \frac{\frac{1}{2} + \frac{1}{u0}}{u0}}{u0}} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{u0}\right)\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left({u0}^{4} \cdot \left(\frac{-1}{4} + \color{blue}{\left(\frac{-1 \cdot \frac{\frac{1}{2} + \frac{1}{u0}}{u0}}{u0} - \frac{1}{3} \cdot \frac{1}{u0}\right)}\right)\right) \]

      13. associate-*r/ N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left({u0}^{4} \cdot \left(\frac{-1}{4} + \left(\frac{-1 \cdot \frac{\frac{1}{2} + \frac{1}{u0}}{u0}}{u0} - \color{blue}{\frac{\frac{1}{3} \cdot 1}{u0}}\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left({u0}^{4} \cdot \left(\frac{-1}{4} + \left(\frac{-1 \cdot \frac{\frac{1}{2} + \frac{1}{u0}}{u0}}{u0} - \frac{\color{blue}{\frac{1}{3}}}{u0}\right)\right)\right) \]

   10. Applied rewrites 98.1%

       \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\left({u0}^{4} \cdot \left(-0.25 + \frac{-0.3333333333333333 - \frac{0.5 + \frac{1}{u0}}{u0}}{u0}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9700000286102295:\\ \;\;\;\;\frac{\alpha \cdot \left(\alpha \cdot \alpha\right)}{-\alpha} \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot \left({u0}^{4} \cdot \left(\frac{\frac{0.5 + \frac{1}{u0}}{u0} - -0.3333333333333333}{u0} - -0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u0 \leq 0.005799999926239252:\\ \;\;\;\;\left(\alpha \cdot \left(-\alpha\right)\right) \cdot \left(u0 \cdot \left(\left(-1 + u0 \cdot \left(u0 \cdot \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)\right)\right) + u0 \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha \cdot \left(\alpha \cdot \alpha\right)}{-\alpha} \cdot \log \left(1 - u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= u0 0.005799999926239252)
   (*
    (* alpha (- alpha))
    (*
     u0
     (+
      (+ -1.0 (* u0 (* u0 (fma u0 -0.25 -0.3333333333333333))))
      (* u0 -0.5))))
   (* (/ (* alpha (* alpha alpha)) (- alpha)) (log (- 1.0 u0)))))

C

float code(float alpha, float u0) {
	float tmp;
	if (u0 <= 0.005799999926239252f) {
		tmp = (alpha * -alpha) * (u0 * ((-1.0f + (u0 * (u0 * fmaf(u0, -0.25f, -0.3333333333333333f)))) + (u0 * -0.5f)));
	} else {
		tmp = ((alpha * (alpha * alpha)) / -alpha) * logf((1.0f - u0));
	}
	return tmp;
}

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.005799999926239252))
		tmp = Float32(Float32(alpha * Float32(-alpha)) * Float32(u0 * Float32(Float32(Float32(-1.0) + Float32(u0 * Float32(u0 * fma(u0, Float32(-0.25), Float32(-0.3333333333333333))))) + Float32(u0 * Float32(-0.5)))));
	else
		tmp = Float32(Float32(Float32(alpha * Float32(alpha * alpha)) / Float32(-alpha)) * log(Float32(Float32(1.0) - u0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.00579999993

   1. Initial program 46.6%

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

   4. Applied rewrites 22.9%

      \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\left(\log \left(\frac{1 - \left(u0 \cdot u0\right) \cdot \left(u0 \cdot u0\right)}{1 + u0}\right) - \mathsf{log1p}\left(u0 \cdot u0\right)\right)} \]

   5. Taylor expanded in u0 around 0

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \color{blue}{-1}\right)\right) \]

      4. lower-fma.f32 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}, -1\right)}\right) \]

      5. sub-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \color{blue}{\frac{-1}{2}}, -1\right)\right) \]

      7. lower-fma.f32 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{4} \cdot u0 - \frac{1}{3}, \frac{-1}{2}\right)}, -1\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\frac{-1}{4} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, \frac{-1}{2}\right), -1\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{-1}{4}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), \frac{-1}{2}\right), -1\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, u0 \cdot \frac{-1}{4} + \color{blue}{\frac{-1}{3}}, \frac{-1}{2}\right), -1\right)\right) \]

      11. lower-fma.f32 84.7

          \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)}, -0.5\right), -1\right)\right) \]

   7. Applied rewrites 84.7%

      \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right)} \]
   8. Step-by-step derivation

      1. lift-fma.f32 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right)} + \frac{-1}{2}\right) + -1\right)\right) \]

      2. lift-fma.f32 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(u0 \cdot \color{blue}{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right)} + -1\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\left(-1 + u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right)\right)}\right) \]

      4. lift-fma.f32 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(-1 + u0 \cdot \color{blue}{\left(u0 \cdot \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right) + \frac{-1}{2}\right)}\right)\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(-1 + \color{blue}{\left(u0 \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right)\right) + u0 \cdot \frac{-1}{2}\right)}\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\left(\left(-1 + u0 \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right)\right)\right) + u0 \cdot \frac{-1}{2}\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(\left(-1 + u0 \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right)\right)\right) + \color{blue}{\frac{-1}{2} \cdot u0}\right)\right) \]

      8. lower-+.f32 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\left(\left(-1 + u0 \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right)\right)\right) + \frac{-1}{2} \cdot u0\right)}\right) \]

      9. lower-+.f32 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(\color{blue}{\left(-1 + u0 \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right)\right)\right)} + \frac{-1}{2} \cdot u0\right)\right) \]

      10. lower-*.f32 N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(\left(-1 + \color{blue}{u0 \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right)\right)}\right) + \frac{-1}{2} \cdot u0\right)\right) \]

      11. lower-*.f32 N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(\left(-1 + u0 \cdot \color{blue}{\left(u0 \cdot \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right)\right)}\right) + \frac{-1}{2} \cdot u0\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(\left(-1 + u0 \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right)\right)\right) + \color{blue}{u0 \cdot \frac{-1}{2}}\right)\right) \]

      13. lower-*.f32 98.7

          \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(\left(-1 + u0 \cdot \left(u0 \cdot \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)\right)\right) + \color{blue}{u0 \cdot -0.5}\right)\right) \]

   9. Applied rewrites 98.2%

      \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\left(\left(-1 + u0 \cdot \left(u0 \cdot \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)\right)\right) + u0 \cdot -0.5\right)}\right) \]

   if 0.00579999993 < u0

   1. Initial program 94.0%

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

      1. neg-sub0 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. neg-sub0 N/A

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

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

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

      6. lift-neg.f32 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-*.f32 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. +-lft-identity N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f32 N/A

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

      11. lower-*.f32 94.0

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 94.0

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

   4. Applied rewrites 94.0%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.005799999926239252:\\ \;\;\;\;\left(\alpha \cdot \left(-\alpha\right)\right) \cdot \left(u0 \cdot \left(\left(-1 + u0 \cdot \left(u0 \cdot \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)\right)\right) + u0 \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha \cdot \left(\alpha \cdot \alpha\right)}{-\alpha} \cdot \log \left(1 - u0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha \cdot \left(-\alpha\right)\\ \mathbf{if}\;u0 \leq 0.005799999926239252:\\ \;\;\;\;t\_0 \cdot \left(u0 \cdot \left(\left(-1 + u0 \cdot \left(u0 \cdot \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)\right)\right) + u0 \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 - u0\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (let* ((t_0 (* alpha (- alpha))))
   (if (<= u0 0.005799999926239252)
     (*
      t_0
      (*
       u0
       (+
        (+ -1.0 (* u0 (* u0 (fma u0 -0.25 -0.3333333333333333))))
        (* u0 -0.5))))
     (* (log (- 1.0 u0)) t_0))))

C

float code(float alpha, float u0) {
	float t_0 = alpha * -alpha;
	float tmp;
	if (u0 <= 0.005799999926239252f) {
		tmp = t_0 * (u0 * ((-1.0f + (u0 * (u0 * fmaf(u0, -0.25f, -0.3333333333333333f)))) + (u0 * -0.5f)));
	} else {
		tmp = logf((1.0f - u0)) * t_0;
	}
	return tmp;
}

Julia

function code(alpha, u0)
	t_0 = Float32(alpha * Float32(-alpha))
	tmp = Float32(0.0)
	if (u0 <= Float32(0.005799999926239252))
		tmp = Float32(t_0 * Float32(u0 * Float32(Float32(Float32(-1.0) + Float32(u0 * Float32(u0 * fma(u0, Float32(-0.25), Float32(-0.3333333333333333))))) + Float32(u0 * Float32(-0.5)))));
	else
		tmp = Float32(log(Float32(Float32(1.0) - u0)) * t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.00579999993

   1. Initial program 46.6%

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

   4. Applied rewrites 22.9%

      \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\left(\log \left(\frac{1 - \left(u0 \cdot u0\right) \cdot \left(u0 \cdot u0\right)}{1 + u0}\right) - \mathsf{log1p}\left(u0 \cdot u0\right)\right)} \]

   5. Taylor expanded in u0 around 0

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \color{blue}{-1}\right)\right) \]

      4. lower-fma.f32 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}, -1\right)}\right) \]

      5. sub-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \color{blue}{\frac{-1}{2}}, -1\right)\right) \]

      7. lower-fma.f32 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{4} \cdot u0 - \frac{1}{3}, \frac{-1}{2}\right)}, -1\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\frac{-1}{4} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, \frac{-1}{2}\right), -1\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{-1}{4}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), \frac{-1}{2}\right), -1\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, u0 \cdot \frac{-1}{4} + \color{blue}{\frac{-1}{3}}, \frac{-1}{2}\right), -1\right)\right) \]

      11. lower-fma.f32 84.7

          \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)}, -0.5\right), -1\right)\right) \]

   7. Applied rewrites 84.7%

      \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right)} \]
   8. Step-by-step derivation

      1. lift-fma.f32 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right)} + \frac{-1}{2}\right) + -1\right)\right) \]

      2. lift-fma.f32 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(u0 \cdot \color{blue}{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right)} + -1\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\left(-1 + u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right)\right)}\right) \]

      4. lift-fma.f32 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(-1 + u0 \cdot \color{blue}{\left(u0 \cdot \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right) + \frac{-1}{2}\right)}\right)\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(-1 + \color{blue}{\left(u0 \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right)\right) + u0 \cdot \frac{-1}{2}\right)}\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\left(\left(-1 + u0 \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right)\right)\right) + u0 \cdot \frac{-1}{2}\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(\left(-1 + u0 \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right)\right)\right) + \color{blue}{\frac{-1}{2} \cdot u0}\right)\right) \]

      8. lower-+.f32 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\left(\left(-1 + u0 \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right)\right)\right) + \frac{-1}{2} \cdot u0\right)}\right) \]

      9. lower-+.f32 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(\color{blue}{\left(-1 + u0 \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right)\right)\right)} + \frac{-1}{2} \cdot u0\right)\right) \]

      10. lower-*.f32 N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(\left(-1 + \color{blue}{u0 \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right)\right)}\right) + \frac{-1}{2} \cdot u0\right)\right) \]

      11. lower-*.f32 N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(\left(-1 + u0 \cdot \color{blue}{\left(u0 \cdot \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right)\right)}\right) + \frac{-1}{2} \cdot u0\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(\left(-1 + u0 \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right)\right)\right) + \color{blue}{u0 \cdot \frac{-1}{2}}\right)\right) \]

      13. lower-*.f32 98.7

          \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(\left(-1 + u0 \cdot \left(u0 \cdot \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)\right)\right) + \color{blue}{u0 \cdot -0.5}\right)\right) \]

   9. Applied rewrites 98.2%

      \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\left(\left(-1 + u0 \cdot \left(u0 \cdot \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)\right)\right) + u0 \cdot -0.5\right)}\right) \]

   if 0.00579999993 < u0

   1. Initial program 94.0%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.005799999926239252:\\ \;\;\;\;\left(\alpha \cdot \left(-\alpha\right)\right) \cdot \left(u0 \cdot \left(\left(-1 + u0 \cdot \left(u0 \cdot \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)\right)\right) + u0 \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \left(\alpha \cdot \left(-\alpha\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \left(u0 \cdot \left(\left(-1 + u0 \cdot \left(u0 \cdot \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)\right)\right) + u0 \cdot -0.5\right)\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (*
  (* alpha (- alpha))
  (*
   u0
   (+ (+ -1.0 (* u0 (* u0 (fma u0 -0.25 -0.3333333333333333)))) (* u0 -0.5)))))

C

float code(float alpha, float u0) {
	return (alpha * -alpha) * (u0 * ((-1.0f + (u0 * (u0 * fmaf(u0, -0.25f, -0.3333333333333333f)))) + (u0 * -0.5f)));
}

Julia

function code(alpha, u0)
	return Float32(Float32(alpha * Float32(-alpha)) * Float32(u0 * Float32(Float32(Float32(-1.0) + Float32(u0 * Float32(u0 * fma(u0, Float32(-0.25), Float32(-0.3333333333333333))))) + Float32(u0 * Float32(-0.5)))))
end

Derivation

1. Initial program 57.1%

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

4. Applied rewrites 22.4%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\left(\log \left(\frac{1 - \left(u0 \cdot u0\right) \cdot \left(u0 \cdot u0\right)}{1 + u0}\right) - \mathsf{log1p}\left(u0 \cdot u0\right)\right)} \]

5. Taylor expanded in u0 around 0

   \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)} \]
6. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

   3. metadata-eval N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}\right) + \color{blue}{-1}\right)\right) \]

   4. lower-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{1}{2}, -1\right)}\right) \]

   5. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, u0 \cdot \left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) + \color{blue}{\frac{-1}{2}}, -1\right)\right) \]

   7. lower-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{4} \cdot u0 - \frac{1}{3}, \frac{-1}{2}\right)}, -1\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\frac{-1}{4} \cdot u0 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, \frac{-1}{2}\right), -1\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{u0 \cdot \frac{-1}{4}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), \frac{-1}{2}\right), -1\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, u0 \cdot \frac{-1}{4} + \color{blue}{\frac{-1}{3}}, \frac{-1}{2}\right), -1\right)\right) \]

   11. lower-fma.f32 74.8

       \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \color{blue}{\mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)}, -0.5\right), -1\right)\right) \]

7. Applied rewrites 74.8%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\left(u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right)} \]
8. Step-by-step derivation

   1. lift-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot \color{blue}{\mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right)} + \frac{-1}{2}\right) + -1\right)\right) \]

   2. lift-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(u0 \cdot \color{blue}{\mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right)} + -1\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\left(-1 + u0 \cdot \mathsf{fma}\left(u0, \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right), \frac{-1}{2}\right)\right)}\right) \]

   4. lift-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(-1 + u0 \cdot \color{blue}{\left(u0 \cdot \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right) + \frac{-1}{2}\right)}\right)\right) \]

   5. distribute-lft-in N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(-1 + \color{blue}{\left(u0 \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right)\right) + u0 \cdot \frac{-1}{2}\right)}\right)\right) \]

   6. associate-+r+ N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\left(\left(-1 + u0 \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right)\right)\right) + u0 \cdot \frac{-1}{2}\right)}\right) \]

   7. *-commutative N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(\left(-1 + u0 \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right)\right)\right) + \color{blue}{\frac{-1}{2} \cdot u0}\right)\right) \]

   8. lower-+.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\left(\left(-1 + u0 \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right)\right)\right) + \frac{-1}{2} \cdot u0\right)}\right) \]

   9. lower-+.f32 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(\color{blue}{\left(-1 + u0 \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right)\right)\right)} + \frac{-1}{2} \cdot u0\right)\right) \]

   10. lower-*.f32 N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(\left(-1 + \color{blue}{u0 \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right)\right)}\right) + \frac{-1}{2} \cdot u0\right)\right) \]

   11. lower-*.f32 N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(\left(-1 + u0 \cdot \color{blue}{\left(u0 \cdot \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right)\right)}\right) + \frac{-1}{2} \cdot u0\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(\left(-1 + u0 \cdot \left(u0 \cdot \mathsf{fma}\left(u0, \frac{-1}{4}, \frac{-1}{3}\right)\right)\right) + \color{blue}{u0 \cdot \frac{-1}{2}}\right)\right) \]

   13. lower-*.f32 91.4

       \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(u0 \cdot \left(\left(-1 + u0 \cdot \left(u0 \cdot \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)\right)\right) + \color{blue}{u0 \cdot -0.5}\right)\right) \]

9. Applied rewrites 91.4%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(u0 \cdot \color{blue}{\left(\left(-1 + u0 \cdot \left(u0 \cdot \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)\right)\right) + u0 \cdot -0.5\right)}\right) \]

10. Final simplification 91.4%

    \[\leadsto \left(\alpha \cdot \left(-\alpha\right)\right) \cdot \left(u0 \cdot \left(\left(-1 + u0 \cdot \left(u0 \cdot \mathsf{fma}\left(u0, -0.25, -0.3333333333333333\right)\right)\right) + u0 \cdot -0.5\right)\right) \]
11. Add Preprocessing

Alternative 5: 74.2% accurate, 3.9× speedup

Math

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

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)) * Float32(-u0))
end

MATLAB

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

Derivation

1. Initial program 57.1%

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

   1. neg-sub0 N/A

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

   2. flip-- N/A

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

   3. metadata-eval N/A

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

   4. neg-sub0 N/A

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

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

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

   6. lift-neg.f32 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-*.f32 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. +-lft-identity N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f32 N/A

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

   11. lower-*.f32 57.1

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 57.1

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

4. Applied rewrites 57.1%

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

5. Taylor expanded in u0 around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 74.7

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

7. Applied rewrites 74.7%

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

9. Applied rewrites 74.6%

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

11. Applied rewrites 74.8%

    \[\leadsto \color{blue}{\frac{\alpha}{\frac{-1}{\alpha}}} \cdot \left(-u0\right) \]
12. Add Preprocessing

Alternative 6: 74.3% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.1%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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 74.8

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

5. Applied rewrites 74.8%

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

Reproduce

herbie shell --seed 2024216 
(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))))