Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 56.0% → 97.7%

Time: 6.4s

Alternatives: 7

Speedup: 10.5×

Specification

\[\left(0.0001 \leq \alpha \land \alpha \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\]

Math

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

FPCore

(FPCore (alpha u0)
 :precision binary32
 (* (* (- alpha) alpha) (log (- 1.0 u0))))

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 56.0% 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: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9639599919319153:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \frac{{\alpha}^{3}}{-\alpha}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(u0 \cdot u0\right) \cdot \left(\left(\frac{-0.3333333333333333 - \frac{\frac{1}{u0} - -0.5}{u0}}{u0} - 0.25\right) \cdot u0\right)\right) \cdot u0\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9639599919319153)
   (* (log (- 1.0 u0)) (/ (pow alpha 3.0) (- alpha)))
   (*
    (*
     (*
      (* u0 u0)
      (*
       (- (/ (- -0.3333333333333333 (/ (- (/ 1.0 u0) -0.5) u0)) u0) 0.25)
       u0))
     u0)
    (* (- alpha) alpha))))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9639599919319153f) {
		tmp = logf((1.0f - u0)) * (powf(alpha, 3.0f) / -alpha);
	} else {
		tmp = (((u0 * u0) * ((((-0.3333333333333333f - (((1.0f / u0) - -0.5f) / u0)) / u0) - 0.25f) * u0)) * u0) * (-alpha * alpha);
	}
	return tmp;
}

Fortran

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: tmp
    if ((1.0e0 - u0) <= 0.9639599919319153e0) then
        tmp = log((1.0e0 - u0)) * ((alpha ** 3.0e0) / -alpha)
    else
        tmp = (((u0 * u0) * (((((-0.3333333333333333e0) - (((1.0e0 / u0) - (-0.5e0)) / u0)) / u0) - 0.25e0) * u0)) * u0) * (-alpha * alpha)
    end if
    code = tmp
end function

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9639599919319153))
		tmp = Float32(log(Float32(Float32(1.0) - u0)) * Float32((alpha ^ Float32(3.0)) / Float32(-alpha)));
	else
		tmp = Float32(Float32(Float32(Float32(u0 * u0) * Float32(Float32(Float32(Float32(Float32(-0.3333333333333333) - Float32(Float32(Float32(Float32(1.0) / u0) - Float32(-0.5)) / u0)) / u0) - Float32(0.25)) * u0)) * u0) * Float32(Float32(-alpha) * alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9639599919319153))
		tmp = log((single(1.0) - u0)) * ((alpha ^ single(3.0)) / -alpha);
	else
		tmp = (((u0 * u0) * ((((single(-0.3333333333333333) - (((single(1.0) / u0) - single(-0.5)) / u0)) / u0) - single(0.25)) * u0)) * u0) * (-alpha * alpha);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.963959992

   1. Initial program 96.9%

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

      1. lift-*.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. neg-sub0 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. neg-sub0 N/A

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

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

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

      8. lift-neg.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. +-lft-identity N/A

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

      11. associate-*l/ N/A

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

      12. clear-num N/A

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

      13. lower-/.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. lower-*.f32 96.5

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

   4. Applied rewrites 96.5%

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

      1. lift-/.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. clear-num N/A

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

      4. frac-2neg N/A

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

      5. lift-*.f32 N/A

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

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

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

      7. lift-*.f32 N/A

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

      8. lift-neg.f32 N/A

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

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

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

      10. lift-*.f32 N/A

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

      11. remove-double-neg N/A

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

      12. lift-*.f32 N/A

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

      13. unpow3 N/A

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

      14. lift-pow.f32 N/A

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

      15. lift-neg.f32 N/A

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

      16. lower-/.f32 97.1

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

   6. Applied rewrites 97.1%

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

   if 0.963959992 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 50.3%

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

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f32 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f32 N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f32 80.2

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

   5. Applied rewrites 80.2%

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

   6. Taylor expanded in u0 around inf

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

   8. Applied rewrites 98.1%

      \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(\left(\frac{-0.3333333333333333 - \frac{\frac{1}{u0} - -0.5}{u0}}{u0} - 0.25\right) \cdot {u0}^{3}\right) \cdot u0\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 98.3%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9639599919319153:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \frac{{\alpha}^{3}}{-\alpha}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(u0 \cdot u0\right) \cdot \left(\left(\frac{-0.3333333333333333 - \frac{\frac{1}{u0} - -0.5}{u0}}{u0} - 0.25\right) \cdot u0\right)\right) \cdot u0\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-\alpha\right) \cdot \alpha\\ \mathbf{if}\;1 - u0 \leq 0.9639599919319153:\\ \;\;\;\;t\_0 \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(u0 \cdot u0\right) \cdot \left(\left(\frac{-0.3333333333333333 - \frac{\frac{1}{u0} - -0.5}{u0}}{u0} - 0.25\right) \cdot u0\right)\right) \cdot u0\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (let* ((t_0 (* (- alpha) alpha)))
   (if (<= (- 1.0 u0) 0.9639599919319153)
     (* t_0 (log (- 1.0 u0)))
     (*
      (*
       (*
        (* u0 u0)
        (*
         (- (/ (- -0.3333333333333333 (/ (- (/ 1.0 u0) -0.5) u0)) u0) 0.25)
         u0))
       u0)
      t_0))))

C

float code(float alpha, float u0) {
	float t_0 = -alpha * alpha;
	float tmp;
	if ((1.0f - u0) <= 0.9639599919319153f) {
		tmp = t_0 * logf((1.0f - u0));
	} else {
		tmp = (((u0 * u0) * ((((-0.3333333333333333f - (((1.0f / u0) - -0.5f) / u0)) / u0) - 0.25f) * u0)) * u0) * t_0;
	}
	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) <= 0.9639599919319153e0) then
        tmp = t_0 * log((1.0e0 - u0))
    else
        tmp = (((u0 * u0) * (((((-0.3333333333333333e0) - (((1.0e0 / u0) - (-0.5e0)) / u0)) / u0) - 0.25e0) * u0)) * u0) * t_0
    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(0.9639599919319153))
		tmp = Float32(t_0 * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(Float32(Float32(u0 * u0) * Float32(Float32(Float32(Float32(Float32(-0.3333333333333333) - Float32(Float32(Float32(Float32(1.0) / u0) - Float32(-0.5)) / u0)) / u0) - Float32(0.25)) * u0)) * u0) * t_0);
	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(0.9639599919319153))
		tmp = t_0 * log((single(1.0) - u0));
	else
		tmp = (((u0 * u0) * ((((single(-0.3333333333333333) - (((single(1.0) / u0) - single(-0.5)) / u0)) / u0) - single(0.25)) * u0)) * u0) * t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.963959992

   1. Initial program 96.9%

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

   if 0.963959992 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 50.3%

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

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f32 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f32 N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f32 80.2

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

   5. Applied rewrites 79.8%

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

   6. Taylor expanded in u0 around inf

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

   8. Applied rewrites 98.1%

      \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(\left(\frac{-0.3333333333333333 - \frac{\frac{1}{u0} - -0.5}{u0}}{u0} - 0.25\right) \cdot {u0}^{3}\right) \cdot u0\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 98.3%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9639599919319153:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(u0 \cdot u0\right) \cdot \left(\left(\frac{-0.3333333333333333 - \frac{\frac{1}{u0} - -0.5}{u0}}{u0} - 0.25\right) \cdot u0\right)\right) \cdot u0\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f32 N/A

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f32 N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f32 73.5

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

5. Applied rewrites 73.1%

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

6. Taylor expanded in u0 around inf

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

8. Applied rewrites 92.5%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(\left(\frac{-0.3333333333333333 - \frac{\frac{1}{u0} - -0.5}{u0}}{u0} - 0.25\right) \cdot {u0}^{3}\right) \cdot u0\right) \]
9. Step-by-step derivation

10. Applied rewrites 92.7%

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

11. Final simplification 92.7%

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

Alternative 4: 92.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f32 N/A

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f32 N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f32 73.5

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

5. Applied rewrites 73.1%

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

6. Taylor expanded in u0 around inf

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

8. Applied rewrites 92.5%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(\left(\frac{-0.3333333333333333 - \frac{\frac{1}{u0} - -0.5}{u0}}{u0} - 0.25\right) \cdot {u0}^{3}\right) \cdot u0\right) \]
9. Step-by-step derivation

10. Applied rewrites 92.7%

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

12. Applied rewrites 92.7%

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

13. Final simplification 92.7%

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

Alternative 5: 58.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.4%

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

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f32 N/A

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f32 N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f32 73.5

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

5. Applied rewrites 73.1%

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

7. Applied rewrites 86.2%

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

9. Applied rewrites 91.1%

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

10. Final simplification 91.1%

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

Alternative 6: 86.9% accurate, 4.5× speedup

Math

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

FPCore

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

C

float code(float alpha, float u0) {
	return (((-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 = ((((-0.5e0) * u0) + (-1.0e0)) * u0) * (-alpha * alpha)
end function

Julia

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

MATLAB

function tmp = code(alpha, u0)
	tmp = (((single(-0.5) * u0) + single(-1.0)) * u0) * (-alpha * alpha);
end

Derivation

1. Initial program 57.4%

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

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f32 N/A

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f32 N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f32 73.5

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

5. Applied rewrites 73.1%

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

7. Applied rewrites 86.2%

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

8. Taylor expanded in u0 around 0

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

10. Applied rewrites 86.4%

    \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\left(-0.5 \cdot u0 + -1\right) \cdot u0\right) \]

11. Final simplification 86.4%

    \[\leadsto \left(\left(-0.5 \cdot u0 + -1\right) \cdot u0\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right) \]
12. Add Preprocessing

Alternative 7: 74.3% 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 57.4%

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

3. Taylor expanded in u0 around 0

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 73.5

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

5. Applied rewrites 73.5%

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

Reproduce

herbie shell --seed 2024295 
(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))))