Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 55.3% → 74.9%

Time: 6.9s

Alternatives: 5

Speedup: 10.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 55.3% 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: 74.9% 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 56.3%

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

3. Taylor expanded in u0 around 0

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 75.0

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

5. Applied rewrites 75.0%

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

Alternative 2: 63.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9987499713897705:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \frac{{\alpha}^{3}}{-\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, u0, -0.3333333333333333\right), u0, -0.5\right) \cdot \alpha\right) \cdot u0 - \alpha\right) \cdot u0}{\frac{-1}{\alpha}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9987499713897705)
   (* (log (- 1.0 u0)) (/ (pow alpha 3.0) (- alpha)))
   (/
    (*
     (-
      (* (* (fma (fma -0.25 u0 -0.3333333333333333) u0 -0.5) alpha) u0)
      alpha)
     u0)
    (/ -1.0 alpha))))

C

float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9987499713897705f) {
		tmp = logf((1.0f - u0)) * (powf(alpha, 3.0f) / -alpha);
	} else {
		tmp = ((((fmaf(fmaf(-0.25f, u0, -0.3333333333333333f), u0, -0.5f) * alpha) * u0) - alpha) * u0) / (-1.0f / alpha);
	}
	return tmp;
}

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9987499713897705))
		tmp = Float32(log(Float32(Float32(1.0) - u0)) * Float32((alpha ^ Float32(3.0)) / Float32(-alpha)));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(fma(fma(Float32(-0.25), u0, Float32(-0.3333333333333333)), u0, Float32(-0.5)) * alpha) * u0) - alpha) * u0) / Float32(Float32(-1.0) / alpha));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u0) < 0.998749971

   1. Initial program 90.1%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 90.1

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

   4. Applied rewrites 90.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{\alpha}{\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \alpha}}} \cdot \log \left(1 - u0\right) \]
   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. distribute-rgt-neg-in N/A

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

      9. lift-neg.f32 N/A

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

      10. sqr-neg N/A

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

      11. unpow3 N/A

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

      12. lift-pow.f32 N/A

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

      13. lift-neg.f32 N/A

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

      14. lower-/.f32 90.3

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

   6. Applied rewrites 90.3%

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

   if 0.998749971 < (-.f32 #s(literal 1 binary32) u0)

   1. Initial program 43.3%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 43.3

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

   4. Applied rewrites 43.3%

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

   5. Taylor expanded in u0 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 85.9

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

   7. Applied rewrites 85.9%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f32 N/A

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

      8. div-inv N/A

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

      9. times-frac N/A

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

      10. associate-*l/ N/A

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

      11. associate-/l* N/A

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

      12. *-lft-identity N/A

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

      13. lower-/.f32 N/A

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

   9. Applied rewrites 86.1%

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

   10. Taylor expanded in u0 around 0

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

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

   12. Applied rewrites 97.9%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9987499713897705:\\ \;\;\;\;\log \left(1 - u0\right) \cdot \frac{{\alpha}^{3}}{-\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, u0, -0.3333333333333333\right), u0, -0.5\right) \cdot \alpha\right) \cdot u0 - \alpha\right) \cdot u0}{\frac{-1}{\alpha}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u0 \leq 0.0012499999720603228:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, u0, -0.3333333333333333\right), u0, -0.5\right) \cdot \alpha\right) \cdot u0 - \alpha\right) \cdot u0}{\frac{-1}{\alpha}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \alpha}{\alpha} \cdot \log \left(1 - u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= u0 0.0012499999720603228)
   (/
    (*
     (-
      (* (* (fma (fma -0.25 u0 -0.3333333333333333) u0 -0.5) alpha) u0)
      alpha)
     u0)
    (/ -1.0 alpha))
   (* (/ (* (* (- alpha) alpha) alpha) alpha) (log (- 1.0 u0)))))

C

float code(float alpha, float u0) {
	float tmp;
	if (u0 <= 0.0012499999720603228f) {
		tmp = ((((fmaf(fmaf(-0.25f, u0, -0.3333333333333333f), u0, -0.5f) * alpha) * u0) - alpha) * u0) / (-1.0f / alpha);
	} else {
		tmp = (((-alpha * alpha) * alpha) / alpha) * logf((1.0f - u0));
	}
	return tmp;
}

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.0012499999720603228))
		tmp = Float32(Float32(Float32(Float32(Float32(fma(fma(Float32(-0.25), u0, Float32(-0.3333333333333333)), u0, Float32(-0.5)) * alpha) * u0) - alpha) * u0) / Float32(Float32(-1.0) / alpha));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(-alpha) * alpha) * alpha) / alpha) * log(Float32(Float32(1.0) - u0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.00124999997

   1. Initial program 43.3%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 43.3

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

   4. Applied rewrites 43.3%

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

   5. Taylor expanded in u0 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 85.9

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

   7. Applied rewrites 85.9%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f32 N/A

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

      8. div-inv N/A

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

      9. times-frac N/A

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

      10. associate-*l/ N/A

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

      11. associate-/l* N/A

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

      12. *-lft-identity N/A

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

      13. lower-/.f32 N/A

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

   9. Applied rewrites 86.1%

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

   10. Taylor expanded in u0 around 0

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

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

   12. Applied rewrites 97.9%

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

   if 0.00124999997 < u0

   1. Initial program 90.1%

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

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

      13. lower-*.f32 90.2

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

   4. Applied rewrites 90.2%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.0012499999720603228:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, u0, -0.3333333333333333\right), u0, -0.5\right) \cdot \alpha\right) \cdot u0 - \alpha\right) \cdot u0}{\frac{-1}{\alpha}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \alpha}{\alpha} \cdot \log \left(1 - u0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u0 \leq 0.0012499999720603228:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, u0, -0.3333333333333333\right), u0, -0.5\right) \cdot \alpha\right) \cdot u0 - \alpha\right) \cdot u0}{\frac{-1}{\alpha}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (if (<= u0 0.0012499999720603228)
   (/
    (*
     (-
      (* (* (fma (fma -0.25 u0 -0.3333333333333333) u0 -0.5) alpha) u0)
      alpha)
     u0)
    (/ -1.0 alpha))
   (* (* (- alpha) alpha) (log (- 1.0 u0)))))

C

float code(float alpha, float u0) {
	float tmp;
	if (u0 <= 0.0012499999720603228f) {
		tmp = ((((fmaf(fmaf(-0.25f, u0, -0.3333333333333333f), u0, -0.5f) * alpha) * u0) - alpha) * u0) / (-1.0f / alpha);
	} else {
		tmp = (-alpha * alpha) * logf((1.0f - u0));
	}
	return tmp;
}

Julia

function code(alpha, u0)
	tmp = Float32(0.0)
	if (u0 <= Float32(0.0012499999720603228))
		tmp = Float32(Float32(Float32(Float32(Float32(fma(fma(Float32(-0.25), u0, Float32(-0.3333333333333333)), u0, Float32(-0.5)) * alpha) * u0) - alpha) * u0) / Float32(Float32(-1.0) / alpha));
	else
		tmp = Float32(Float32(Float32(-alpha) * alpha) * log(Float32(Float32(1.0) - u0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u0 < 0.00124999997

   1. Initial program 43.3%

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   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 43.3

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

   4. Applied rewrites 43.3%

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

   5. Taylor expanded in u0 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 85.9

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

   7. Applied rewrites 85.9%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f32 N/A

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

      8. div-inv N/A

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

      9. times-frac N/A

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

      10. associate-*l/ N/A

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

      11. associate-/l* N/A

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

      12. *-lft-identity N/A

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

      13. lower-/.f32 N/A

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

   9. Applied rewrites 86.1%

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

   10. Taylor expanded in u0 around 0

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

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

   12. Applied rewrites 97.9%

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

   if 0.00124999997 < u0

   1. Initial program 90.1%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u0 \leq 0.0012499999720603228:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, u0, -0.3333333333333333\right), u0, -0.5\right) \cdot \alpha\right) \cdot u0 - \alpha\right) \cdot u0}{\frac{-1}{\alpha}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 44.4% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.3%

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
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 56.3

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

4. Applied rewrites 56.3%

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

5. Taylor expanded in u0 around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 74.8

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

7. Applied rewrites 74.8%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. lift-/.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-/r* N/A

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

   7. lift-/.f32 N/A

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

   8. div-inv N/A

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

   9. times-frac N/A

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

   10. associate-*l/ N/A

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

   11. associate-/l* N/A

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

   12. *-lft-identity N/A

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

   13. lower-/.f32 N/A

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

9. Applied rewrites 74.9%

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

10. Taylor expanded in u0 around 0

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

    1. *-commutative N/A

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

    2. lower-*.f32 N/A

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

12. Applied rewrites 88.6%

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

13. Final simplification 88.9%

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

Reproduce

herbie shell --seed 2024304 
(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))))