Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 55.8% → 98.3%

Time: 7.2s

Alternatives: 8

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: 55.8% 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: 98.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-\alpha\right) \cdot \alpha\\ \mathbf{if}\;1 - u0 \leq 0.9599999785423279:\\ \;\;\;\;t\_0 \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(\left(\left(-0.25 \cdot u0 - 0.3333333333333333\right) \cdot u0 - 0.5\right) \cdot u0 - 1\right) \cdot u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (let* ((t_0 (* (- alpha) alpha)))
   (if (<= (- 1.0 u0) 0.9599999785423279)
     (* t_0 (log (- 1.0 u0)))
     (*
      t_0
      (*
       (- (* (- (* (- (* -0.25 u0) 0.3333333333333333) u0) 0.5) u0) 1.0)
       u0)))))

C

float code(float alpha, float u0) {
	float t_0 = -alpha * alpha;
	float tmp;
	if ((1.0f - u0) <= 0.9599999785423279f) {
		tmp = t_0 * logf((1.0f - u0));
	} else {
		tmp = t_0 * (((((((-0.25f * u0) - 0.3333333333333333f) * u0) - 0.5f) * u0) - 1.0f) * u0);
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -alpha * alpha
    if ((1.0e0 - u0) <= 0.9599999785423279e0) then
        tmp = t_0 * log((1.0e0 - u0))
    else
        tmp = t_0 * ((((((((-0.25e0) * u0) - 0.3333333333333333e0) * u0) - 0.5e0) * u0) - 1.0e0) * u0)
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	t_0 = Float32(Float32(-alpha) * alpha)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9599999785423279))
		tmp = Float32(t_0 * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(t_0 * Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(-0.25) * u0) - Float32(0.3333333333333333)) * u0) - Float32(0.5)) * u0) - Float32(1.0)) * u0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	t_0 = -alpha * alpha;
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9599999785423279))
		tmp = t_0 * log((single(1.0) - u0));
	else
		tmp = t_0 * (((((((single(-0.25) * u0) - single(0.3333333333333333)) * u0) - single(0.5)) * u0) - single(1.0)) * u0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.959999979

   1. Initial program 97.2%

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

   if 0.959999979 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 49.1%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. metadata-eval N/A

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

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

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

      11. lower--.f32 N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower-*.f32 98.6

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

   5. Applied rewrites 98.6%

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

Alternative 2: 98.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9599999785423279:\\ \;\;\;\;\left(\log \left(1 - u0\right) \cdot \left(-\alpha\right)\right) \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(\left(\left(-0.25 \cdot u0 - 0.3333333333333333\right) \cdot u0 - 0.5\right) \cdot u0 - 1\right) \cdot u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9599999785423279)
   (* (* (log (- 1.0 u0)) (- alpha)) alpha)
   (*
    (* (- alpha) alpha)
    (* (- (* (- (* (- (* -0.25 u0) 0.3333333333333333) u0) 0.5) u0) 1.0) u0))))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9599999785423279f) {
		tmp = (logf((1.0f - u0)) * -alpha) * alpha;
	} else {
		tmp = (-alpha * alpha) * (((((((-0.25f * u0) - 0.3333333333333333f) * u0) - 0.5f) * u0) - 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 ((1.0e0 - u0) <= 0.9599999785423279e0) then
        tmp = (log((1.0e0 - u0)) * -alpha) * alpha
    else
        tmp = (-alpha * alpha) * ((((((((-0.25e0) * u0) - 0.3333333333333333e0) * u0) - 0.5e0) * u0) - 1.0e0) * u0)
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9599999785423279))
		tmp = Float32(Float32(log(Float32(Float32(1.0) - u0)) * Float32(-alpha)) * alpha);
	else
		tmp = Float32(Float32(Float32(-alpha) * alpha) * Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(-0.25) * u0) - Float32(0.3333333333333333)) * u0) - Float32(0.5)) * u0) - Float32(1.0)) * u0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9599999785423279))
		tmp = (log((single(1.0) - u0)) * -alpha) * alpha;
	else
		tmp = (-alpha * alpha) * (((((((single(-0.25) * u0) - single(0.3333333333333333)) * u0) - single(0.5)) * u0) - single(1.0)) * u0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.959999979

   1. Initial program 97.2%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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-log.f32 N/A

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

      3. log-pow-rev N/A

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

      4. sqr-pow N/A

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

      5. pow-prod-down N/A

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

      6. log-pow N/A

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

      7. lower-*.f32 N/A

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

      8. frac-2neg N/A

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

      9. lift-*.f32 N/A

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

      10. distribute-rgt-neg-out N/A

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

      11. lift-neg.f32 N/A

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

      12. sqr-neg-rev N/A

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

      13. lower-/.f32 N/A

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

      14. lower-*.f32 N/A

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

      15. metadata-eval N/A

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

      16. lower-log.f32 N/A

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

      17. pow2 N/A

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

      18. lower-pow.f32 96.5

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

   4. Applied rewrites 96.5%

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

      1. lift-*.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. log-pow-rev N/A

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

      4. lift-pow.f32 N/A

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

      5. unpow2 N/A

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

      6. unpow-prod-down N/A

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

      7. lift-/.f32 N/A

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

      8. frac-2neg N/A

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

      9. lift-*.f32 N/A

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

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

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

      11. lift-neg.f32 N/A

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

      12. lift-*.f32 N/A

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

      13. metadata-eval N/A

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

      14. lift-/.f32 N/A

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

      15. frac-2neg N/A

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

      16. lift-*.f32 N/A

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

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

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

      18. lift-neg.f32 N/A

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

      19. lift-*.f32 N/A

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

      20. metadata-eval N/A

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

   6. Applied rewrites 97.1%

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

   if 0.959999979 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 49.1%

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

   3. Taylor expanded in u0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. metadata-eval N/A

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

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

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

      11. lower--.f32 N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower-*.f32 98.6

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

   5. Applied rewrites 98.6%

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

Alternative 3: 93.4% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = (-alpha * alpha) * ((((((((-0.25e0) * u0) - 0.3333333333333333e0) * u0) - 0.5e0) * u0) - 1.0e0) * u0)
end function

Julia

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

MATLAB

function tmp = code(alpha, u0)
	tmp = (-alpha * alpha) * (((((((single(-0.25) * u0) - single(0.3333333333333333)) * u0) - single(0.5)) * u0) - single(1.0)) * 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

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. metadata-eval N/A

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

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

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

   11. lower--.f32 N/A

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

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

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

   13. metadata-eval N/A

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

   14. lower-*.f32 93.0

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

5. Applied rewrites 93.0%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\left(\left(\left(\left(-0.25 \cdot u0 - 0.3333333333333333\right) \cdot u0 - 0.5\right) \cdot u0 - 1\right) \cdot u0\right)} \]
6. Add Preprocessing

Alternative 4: 93.4% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = (((((((((-0.25e0) * u0) - 0.3333333333333333e0) * u0) - 0.5e0) * u0) - 1.0e0) * u0) * -alpha) * alpha
end function

Julia

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

MATLAB

function tmp = code(alpha, u0)
	tmp = ((((((((single(-0.25) * u0) - single(0.3333333333333333)) * u0) - single(0.5)) * u0) - single(1.0)) * u0) * -alpha) * 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

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. metadata-eval N/A

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

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

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

   11. lower--.f32 N/A

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

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

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

   13. metadata-eval N/A

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

   14. lower-*.f32 93.0

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

5. Applied rewrites 93.0%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 93.0

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

7. Applied rewrites 93.0%

   \[\leadsto \color{blue}{\left(\left(\left(\left(\left(-0.25 \cdot u0 - 0.3333333333333333\right) \cdot u0 - 0.5\right) \cdot u0 - 1\right) \cdot u0\right) \cdot \left(-\alpha\right)\right) \cdot \alpha} \]
8. Add Preprocessing

Alternative 5: 91.6% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = (((u0 * alpha) * ((0.5e0 - ((-0.3333333333333333e0) * u0)) * alpha)) + (alpha * alpha)) * u0
end function

Julia

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

MATLAB

function tmp = code(alpha, u0)
	tmp = (((u0 * alpha) * ((single(0.5) - (single(-0.3333333333333333) * u0)) * alpha)) + (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

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. distribute-rgt-out N/A

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

   8. lower-*.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-fma.f32 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f32 74.1

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

5. Applied rewrites 73.9%

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

7. Applied rewrites 86.1%

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

9. Applied rewrites 91.2%

   \[\leadsto \left(\left(u0 \cdot \alpha\right) \cdot \left(\left(0.5 - -0.3333333333333333 \cdot u0\right) \cdot \alpha\right) + \alpha \cdot \alpha\right) \cdot u0 \]
10. Add Preprocessing

Alternative 6: 91.4% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = (((((((-0.3333333333333333e0) * u0) - 0.5e0) * u0) - 1.0e0) * u0) * -alpha) * alpha
end function

Julia

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

MATLAB

function tmp = code(alpha, u0)
	tmp = ((((((single(-0.3333333333333333) * u0) - single(0.5)) * u0) - single(1.0)) * u0) * -alpha) * 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

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. metadata-eval N/A

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

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

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

   11. lower--.f32 N/A

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

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

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

   13. metadata-eval N/A

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

   14. lower-*.f32 93.0

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

5. Applied rewrites 93.0%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 93.0

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

7. Applied rewrites 93.0%

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

8. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 N/A

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

   7. lower-*.f32 90.9

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

10. Applied rewrites 90.9%

    \[\leadsto \left(\color{blue}{\left(\left(\left(-0.3333333333333333 \cdot u0 - 0.5\right) \cdot u0 - 1\right) \cdot u0\right)} \cdot \left(-\alpha\right)\right) \cdot \alpha \]
11. Add Preprocessing

Alternative 7: 87.4% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \left(\left(\alpha - -0.5 \cdot \left(\alpha \cdot u0\right)\right) \cdot u0\right) \cdot \alpha \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (* (* (- alpha (* -0.5 (* alpha u0))) u0) alpha))

C

float code(float alpha, float u0) {
	return ((alpha - (-0.5f * (alpha * u0))) * u0) * alpha;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = ((alpha - ((-0.5e0) * (alpha * u0))) * u0) * alpha
end function

Julia

function code(alpha, u0)
	return Float32(Float32(Float32(alpha - Float32(Float32(-0.5) * Float32(alpha * u0))) * u0) * alpha)
end

MATLAB

function tmp = code(alpha, u0)
	tmp = ((alpha - (single(-0.5) * (alpha * u0))) * 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 alpha around 0

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

   1. mul-1-neg N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

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

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

   6. lower-*.f32 N/A

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

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

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

   8. mul-1-neg N/A

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

   9. lower-*.f32 N/A

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

   10. mul-1-neg N/A

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

   11. lower-neg.f32 N/A

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

   12. remove-double-neg N/A

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

   13. mul-1-neg N/A

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

   14. *-commutative N/A

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

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

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

   16. fp-cancel-sign-sub N/A

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

   17. *-commutative N/A

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

   18. mul-1-neg N/A

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

   19. lower-log1p.f32 N/A

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

   20. lower-neg.f32 74.0

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

5. Applied rewrites 74.0%

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

6. Taylor expanded in u0 around 0

   \[\leadsto \left(u0 \cdot \left(\alpha + \frac{1}{2} \cdot \left(\alpha \cdot u0\right)\right)\right) \cdot \alpha \]
7. Step-by-step derivation

8. Applied rewrites 73.8%

   \[\leadsto \left(\mathsf{fma}\left(u0 \cdot \alpha, 0.5, \alpha\right) \cdot u0\right) \cdot \alpha \]
9. Step-by-step derivation

10. Applied rewrites 87.0%

    \[\leadsto \left(\left(\alpha - -0.5 \cdot \left(\alpha \cdot u0\right)\right) \cdot u0\right) \cdot \alpha \]
11. Add Preprocessing

Alternative 8: 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 74.1

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

5. Applied rewrites 74.1%

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

Reproduce

herbie shell --seed 2024326 
(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))))