Beckmann Distribution sample, tan2theta, alphax == alphay

Percentage Accurate: 56.4% → 99.0%

Time: 9.4s

Alternatives: 13

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: 56.4% 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: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{log1p}\left(-u0\right) \cdot \left(-\alpha\right)\right) \cdot \alpha \end{array} \]

FPCore

(FPCore (alpha u0) :precision binary32 (* (* (log1p (- u0)) (- alpha)) alpha))

C

float code(float alpha, float u0) {
	return (log1pf(-u0) * -alpha) * alpha;
}

Julia

function code(alpha, u0)
	return Float32(Float32(log1p(Float32(-u0)) * Float32(-alpha)) * alpha)
end

Derivation

1. Initial program 51.6%

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

3. Taylor expanded in alpha around 0

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

   1. mul-1-neg N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. distribute-rgt-neg-in N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

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

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

   8. mul-1-neg N/A

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

   9. lower-*.f32 N/A

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

   10. mul-1-neg N/A

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

   11. lower-neg.f32 N/A

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

   12. sub-neg N/A

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

   13. lower-log1p.f32 N/A

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

   14. lower-neg.f32 99.1

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

5. Applied rewrites 99.1%

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

6. Final simplification 99.1%

   \[\leadsto \left(\mathsf{log1p}\left(-u0\right) \cdot \left(-\alpha\right)\right) \cdot \alpha \]
7. Add Preprocessing

Alternative 2: 93.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, u0, -0.3333333333333333\right) - \left(u0 \cdot u0\right) \cdot 0.1875, u0, -0.5\right), u0, -1\right) \cdot u0\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right) \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (*
  (*
   (fma
    (fma (- (fma -0.25 u0 -0.3333333333333333) (* (* u0 u0) 0.1875)) u0 -0.5)
    u0
    -1.0)
   u0)
  (* (- alpha) alpha)))

C

float code(float alpha, float u0) {
	return (fmaf(fmaf((fmaf(-0.25f, u0, -0.3333333333333333f) - ((u0 * u0) * 0.1875f)), u0, -0.5f), u0, -1.0f) * u0) * (-alpha * alpha);
}

Julia

function code(alpha, u0)
	return Float32(Float32(fma(fma(Float32(fma(Float32(-0.25), u0, Float32(-0.3333333333333333)) - Float32(Float32(u0 * u0) * Float32(0.1875))), u0, Float32(-0.5)), u0, Float32(-1.0)) * u0) * Float32(Float32(-alpha) * alpha))
end

Derivation

1. Initial program 51.6%

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

3. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f32 N/A

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f32 N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f32 94.2

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

5. Applied rewrites 94.2%

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

7. Applied rewrites 94.6%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.1111111111111111}{\mathsf{fma}\left(0.25, u0, -0.3333333333333333\right)} - \frac{0.0625 \cdot \left(u0 \cdot u0\right)}{\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right)}, u0, -0.5\right), u0, -1\right) \cdot u0\right) \]

8. Taylor expanded in u0 around 0

   \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{-1}{4} \cdot u0 - \frac{1}{3}\right) - \frac{\frac{1}{16} \cdot \left(u0 \cdot u0\right)}{\mathsf{fma}\left(\frac{1}{4}, u0, \frac{1}{3}\right)}, u0, \frac{-1}{2}\right), u0, -1\right) \cdot u0\right) \]
9. Step-by-step derivation

10. Applied rewrites 94.9%

    \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, u0, -0.3333333333333333\right) - \frac{0.0625 \cdot \left(u0 \cdot u0\right)}{\mathsf{fma}\left(0.25, u0, 0.3333333333333333\right)}, u0, -0.5\right), u0, -1\right) \cdot u0\right) \]

11. Taylor expanded in u0 around 0

    \[\leadsto \left(\left(\mathsf{neg}\left(\alpha\right)\right) \cdot \alpha\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4}, u0, \frac{-1}{3}\right) - \frac{3}{16} \cdot {u0}^{2}, u0, \frac{-1}{2}\right), u0, -1\right) \cdot u0\right) \]
12. Step-by-step derivation

13. Applied rewrites 95.0%

    \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, u0, -0.3333333333333333\right) - 0.1875 \cdot \left(u0 \cdot u0\right), u0, -0.5\right), u0, -1\right) \cdot u0\right) \]

14. Final simplification 95.0%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, u0, -0.3333333333333333\right) - \left(u0 \cdot u0\right) \cdot 0.1875, u0, -0.5\right), u0, -1\right) \cdot u0\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right) \]
15. Add Preprocessing

Alternative 3: 93.2% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 51.6%

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

3. Taylor expanded in alpha around 0

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

   1. mul-1-neg N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. distribute-rgt-neg-in N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

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

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

   8. mul-1-neg N/A

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

   9. lower-*.f32 N/A

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

   10. mul-1-neg N/A

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

   11. lower-neg.f32 N/A

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

   12. sub-neg N/A

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

   13. lower-log1p.f32 N/A

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

   14. lower-neg.f32 99.1

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in u0 around 0

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

8. Applied rewrites 94.6%

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

10. Applied rewrites 94.7%

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

11. Final simplification 94.7%

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

Alternative 4: 93.1% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 51.6%

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

3. Taylor expanded in alpha around 0

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

   1. mul-1-neg N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. distribute-rgt-neg-in N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

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

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

   8. mul-1-neg N/A

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

   9. lower-*.f32 N/A

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

   10. mul-1-neg N/A

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

   11. lower-neg.f32 N/A

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

   12. sub-neg N/A

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

   13. lower-log1p.f32 N/A

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

   14. lower-neg.f32 99.1

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in u0 around 0

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

8. Applied rewrites 94.6%

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

10. Applied rewrites 94.6%

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

11. Final simplification 94.6%

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

Alternative 5: 93.1% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 51.6%

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

3. Taylor expanded in alpha around 0

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

   1. mul-1-neg N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. distribute-rgt-neg-in N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

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

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

   8. mul-1-neg N/A

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

   9. lower-*.f32 N/A

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

   10. mul-1-neg N/A

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

   11. lower-neg.f32 N/A

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

   12. sub-neg N/A

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

   13. lower-log1p.f32 N/A

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

   14. lower-neg.f32 99.1

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in u0 around 0

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

8. Applied rewrites 94.6%

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

9. Taylor expanded in u0 around 0

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

11. Applied rewrites 94.6%

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

Alternative 6: 92.8% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 51.6%

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

3. Taylor expanded in alpha around 0

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

   1. mul-1-neg N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. distribute-rgt-neg-in N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

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

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

   8. mul-1-neg N/A

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

   9. lower-*.f32 N/A

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

   10. mul-1-neg N/A

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

   11. lower-neg.f32 N/A

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

   12. sub-neg N/A

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

   13. lower-log1p.f32 N/A

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

   14. lower-neg.f32 99.1

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in u0 around 0

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

8. Applied rewrites 94.6%

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

9. Taylor expanded in alpha around -inf

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

11. Applied rewrites 94.5%

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

12. Taylor expanded in alpha around 0

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

14. Applied rewrites 94.5%

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

15. Final simplification 94.5%

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

Alternative 7: 91.2% accurate, 3.5× speedup

Math

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

FPCore

(FPCore (alpha u0)
 :precision binary32
 (*
  (fma alpha alpha (* (* (* (fma 0.3333333333333333 u0 0.5) u0) alpha) alpha))
  u0))

C

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

Julia

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

Derivation

1. Initial program 51.6%

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

3. Taylor expanded in alpha around 0

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

   1. mul-1-neg N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. distribute-rgt-neg-in N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

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

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

   8. mul-1-neg N/A

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

   9. lower-*.f32 N/A

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

   10. mul-1-neg N/A

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

   11. lower-neg.f32 N/A

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

   12. sub-neg N/A

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

   13. lower-log1p.f32 N/A

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

   14. lower-neg.f32 99.1

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in u0 around 0

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

8. Applied rewrites 92.6%

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

10. Applied rewrites 93.0%

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

Alternative 8: 91.0% accurate, 4.1× speedup

Math

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

FPCore

(FPCore (alpha u0)
 :precision binary32
 (* (* (fma (* (fma 0.3333333333333333 u0 0.5) alpha) u0 alpha) u0) alpha))

C

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

Julia

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

Derivation

1. Initial program 51.6%

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

3. Taylor expanded in alpha around 0

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

   1. mul-1-neg N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. distribute-rgt-neg-in N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

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

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

   8. mul-1-neg N/A

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

   9. lower-*.f32 N/A

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

   10. mul-1-neg N/A

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

   11. lower-neg.f32 N/A

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

   12. sub-neg N/A

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

   13. lower-log1p.f32 N/A

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

   14. lower-neg.f32 99.1

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in u0 around 0

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

8. Applied rewrites 92.9%

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

9. Final simplification 92.9%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u0, 0.5\right) \cdot \alpha, u0, \alpha\right) \cdot u0\right) \cdot \alpha \]
10. Add Preprocessing

Alternative 9: 86.9% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 51.6%

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

3. Taylor expanded in alpha around 0

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

   1. mul-1-neg N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. distribute-rgt-neg-in N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

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

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

   8. mul-1-neg N/A

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

   9. lower-*.f32 N/A

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

   10. mul-1-neg N/A

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

   11. lower-neg.f32 N/A

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

   12. sub-neg N/A

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

   13. lower-log1p.f32 N/A

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

   14. lower-neg.f32 99.1

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in u0 around 0

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

8. Applied rewrites 89.8%

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

10. Applied rewrites 89.9%

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

11. Final simplification 89.9%

    \[\leadsto \mathsf{fma}\left(u0, \alpha, \left(\left(u0 \cdot \alpha\right) \cdot 0.5\right) \cdot u0\right) \cdot \alpha \]
12. Add Preprocessing

Alternative 10: 86.8% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 51.6%

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

3. Taylor expanded in alpha around 0

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

   1. mul-1-neg N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. distribute-rgt-neg-in N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

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

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

   8. mul-1-neg N/A

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

   9. lower-*.f32 N/A

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

   10. mul-1-neg N/A

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

   11. lower-neg.f32 N/A

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

   12. sub-neg N/A

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

   13. lower-log1p.f32 N/A

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

   14. lower-neg.f32 99.1

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in u0 around 0

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

8. Applied rewrites 89.8%

   \[\leadsto \left(\mathsf{fma}\left(u0 \cdot \alpha, 0.5, \alpha\right) \cdot u0\right) \cdot \alpha \]
9. Add Preprocessing

Alternative 11: 86.6% accurate, 5.3× speedup

Math

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

FPCore

(FPCore (alpha u0)
 :precision binary32
 (* (* (* (fma 0.5 u0 1.0) alpha) u0) alpha))

C

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

Julia

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

Derivation

1. Initial program 51.6%

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

3. Taylor expanded in alpha around 0

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

   1. mul-1-neg N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. distribute-rgt-neg-in N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

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

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

   8. mul-1-neg N/A

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

   9. lower-*.f32 N/A

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

   10. mul-1-neg N/A

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

   11. lower-neg.f32 N/A

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

   12. sub-neg N/A

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

   13. lower-log1p.f32 N/A

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

   14. lower-neg.f32 99.1

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in u0 around 0

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

8. Applied rewrites 89.8%

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

9. Taylor expanded in u0 around 0

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

11. Applied rewrites 89.6%

    \[\leadsto \left(\left(\mathsf{fma}\left(0.5, u0, 1\right) \cdot \alpha\right) \cdot u0\right) \cdot \alpha \]
12. Add Preprocessing

Alternative 12: 74.0% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 51.6%

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

3. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 77.5

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

5. Applied rewrites 77.5%

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

7. Applied rewrites 77.6%

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

Alternative 13: 74.0% 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 51.6%

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

3. Taylor expanded in u0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 77.5

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

5. Applied rewrites 77.5%

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

6. Final simplification 77.5%

   \[\leadsto \left(\alpha \cdot \alpha\right) \cdot u0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024235 
(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))))