Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.9% → 96.7%

Time: 8.9s

Alternatives: 7

Speedup: 1.2×

Specification

\[\left(0 \leq s \land s \leq 256\right) \land \left(0.25 \leq u \land u \leq 1\right)\]

Math

\[\begin{array}{l} \\ \left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* 3.0 s) (log (/ 1.0 (- 1.0 (/ (- u 0.25) 0.75))))))

C

float code(float s, float u) {
	return (3.0f * s) * logf((1.0f / (1.0f - ((u - 0.25f) / 0.75f))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (3.0e0 * s) * log((1.0e0 / (1.0e0 - ((u - 0.25e0) / 0.75e0))))
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(3.0) * s) * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(u - Float32(0.25)) / Float32(0.75))))))
end

MATLAB

function tmp = code(s, u)
	tmp = (single(3.0) * s) * log((single(1.0) / (single(1.0) - ((u - single(0.25)) / single(0.75)))));
end

Initial Program: 95.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* 3.0 s) (log (/ 1.0 (- 1.0 (/ (- u 0.25) 0.75))))))

C

float code(float s, float u) {
	return (3.0f * s) * logf((1.0f / (1.0f - ((u - 0.25f) / 0.75f))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (3.0e0 * s) * log((1.0e0 / (1.0e0 - ((u - 0.25e0) / 0.75e0))))
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(3.0) * s) * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(u - Float32(0.25)) / Float32(0.75))))))
end

MATLAB

function tmp = code(s, u)
	tmp = (single(3.0) * s) * log((single(1.0) / (single(1.0) - ((u - single(0.25)) / single(0.75)))));
end

Alternative 1: 96.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(\log \left(1 + \frac{u - 0.25}{-0.75}\right) \cdot -3\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* s (* (log (+ 1.0 (/ (- u 0.25) -0.75))) -3.0)))

C

float code(float s, float u) {
	return s * (logf((1.0f + ((u - 0.25f) / -0.75f))) * -3.0f);
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * (log((1.0e0 + ((u - 0.25e0) / (-0.75e0)))) * (-3.0e0))
end function

Julia

function code(s, u)
	return Float32(s * Float32(log(Float32(Float32(1.0) + Float32(Float32(u - Float32(0.25)) / Float32(-0.75)))) * Float32(-3.0)))
end

MATLAB

function tmp = code(s, u)
	tmp = s * (log((single(1.0) + ((u - single(0.25)) / single(-0.75)))) * single(-3.0));
end

Derivation

1. Initial program 96.1%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   2. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(3 \cdot s\right)} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right)} \]

   4. *-commutative N/A

      \[\leadsto 3 \cdot \color{blue}{\left(\log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \cdot s\right)} \]

   5. associate-*r* N/A

      \[\leadsto \color{blue}{\left(3 \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \cdot s} \]

   6. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(3 \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \cdot s} \]

4. Applied rewrites 34.0%

   \[\leadsto \color{blue}{\left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right)\right) \cdot s} \]
5. Step-by-step derivation

   1. lift-log1p.f32 N/A

      \[\leadsto \left(-3 \cdot \color{blue}{\log \left(1 + \frac{-4}{3} \cdot \left(u - \frac{1}{4}\right)\right)}\right) \cdot s \]

   2. +-commutative N/A

      \[\leadsto \left(-3 \cdot \log \color{blue}{\left(\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right) + 1\right)}\right) \cdot s \]

   3. lift-+.f32 N/A

      \[\leadsto \left(-3 \cdot \log \color{blue}{\left(\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right) + 1\right)}\right) \cdot s \]

   4. lower-log.f32 96.4

      \[\leadsto \left(-3 \cdot \color{blue}{\log \left(-1.3333333333333333 \cdot \left(u - 0.25\right) + 1\right)}\right) \cdot s \]

   5. lift-+.f32 N/A

      \[\leadsto \left(-3 \cdot \log \color{blue}{\left(\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right) + 1\right)}\right) \cdot s \]

   6. +-commutative N/A

      \[\leadsto \left(-3 \cdot \log \color{blue}{\left(1 + \frac{-4}{3} \cdot \left(u - \frac{1}{4}\right)\right)}\right) \cdot s \]

   7. lift-*.f32 N/A

      \[\leadsto \left(-3 \cdot \log \left(1 + \color{blue}{\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right)}\right)\right) \cdot s \]

   8. *-commutative N/A

      \[\leadsto \left(-3 \cdot \log \left(1 + \color{blue}{\left(u - \frac{1}{4}\right) \cdot \frac{-4}{3}}\right)\right) \cdot s \]

   9. metadata-eval N/A

      \[\leadsto \left(-3 \cdot \log \left(1 + \left(u - \frac{1}{4}\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{4}{3}\right)\right)}\right)\right) \cdot s \]

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

       \[\leadsto \left(-3 \cdot \log \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(u - \frac{1}{4}\right) \cdot \frac{4}{3}\right)\right)}\right)\right) \cdot s \]

   11. metadata-eval N/A

       \[\leadsto \left(-3 \cdot \log \left(1 + \left(\mathsf{neg}\left(\left(u - \frac{1}{4}\right) \cdot \color{blue}{\frac{1}{\frac{3}{4}}}\right)\right)\right)\right) \cdot s \]

   12. div-inv N/A

       \[\leadsto \left(-3 \cdot \log \left(1 + \left(\mathsf{neg}\left(\color{blue}{\frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right)\right)\right) \cdot s \]

   13. lift--.f32 N/A

       \[\leadsto \left(-3 \cdot \log \left(1 + \left(\mathsf{neg}\left(\frac{\color{blue}{u - \frac{1}{4}}}{\frac{3}{4}}\right)\right)\right)\right) \cdot s \]

   14. sub-neg N/A

       \[\leadsto \left(-3 \cdot \log \color{blue}{\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}\right) \cdot s \]

   15. div-sub N/A

       \[\leadsto \left(-3 \cdot \log \left(1 - \color{blue}{\left(\frac{u}{\frac{3}{4}} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)}\right)\right) \cdot s \]

   16. div-inv N/A

       \[\leadsto \left(-3 \cdot \log \left(1 - \left(\color{blue}{u \cdot \frac{1}{\frac{3}{4}}} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right) \cdot s \]

   17. metadata-eval N/A

       \[\leadsto \left(-3 \cdot \log \left(1 - \left(u \cdot \color{blue}{\frac{4}{3}} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right) \cdot s \]

   18. *-commutative N/A

       \[\leadsto \left(-3 \cdot \log \left(1 - \left(\color{blue}{\frac{4}{3} \cdot u} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right) \cdot s \]

   19. sub-neg N/A

       \[\leadsto \left(-3 \cdot \log \left(1 - \color{blue}{\left(\frac{4}{3} \cdot u + \left(\mathsf{neg}\left(\frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right)}\right)\right) \cdot s \]

   20. metadata-eval N/A

       \[\leadsto \left(-3 \cdot \log \left(1 - \left(\frac{4}{3} \cdot u + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{3}}\right)\right)\right)\right)\right) \cdot s \]

   21. metadata-eval N/A

       \[\leadsto \left(-3 \cdot \log \left(1 - \left(\frac{4}{3} \cdot u + \color{blue}{\frac{-1}{3}}\right)\right)\right) \cdot s \]

   22. +-commutative N/A

       \[\leadsto \left(-3 \cdot \log \left(1 - \color{blue}{\left(\frac{-1}{3} + \frac{4}{3} \cdot u\right)}\right)\right) \cdot s \]

   23. associate--r+ N/A

       \[\leadsto \left(-3 \cdot \log \color{blue}{\left(\left(1 - \frac{-1}{3}\right) - \frac{4}{3} \cdot u\right)}\right) \cdot s \]

   24. metadata-eval N/A

       \[\leadsto \left(-3 \cdot \log \left(\color{blue}{\frac{4}{3}} - \frac{4}{3} \cdot u\right)\right) \cdot s \]

   25. lower--.f32 N/A

       \[\leadsto \left(-3 \cdot \log \color{blue}{\left(\frac{4}{3} - \frac{4}{3} \cdot u\right)}\right) \cdot s \]

   26. lower-*.f32 96.1

       \[\leadsto \left(-3 \cdot \log \left(1.3333333333333333 - \color{blue}{1.3333333333333333 \cdot u}\right)\right) \cdot s \]

6. Applied rewrites 96.1%

   \[\leadsto \left(-3 \cdot \color{blue}{\log \left(1.3333333333333333 - 1.3333333333333333 \cdot u\right)}\right) \cdot s \]
7. Step-by-step derivation

   1. lift--.f32 N/A

      \[\leadsto \left(-3 \cdot \log \color{blue}{\left(\frac{4}{3} - \frac{4}{3} \cdot u\right)}\right) \cdot s \]

   2. metadata-eval N/A

      \[\leadsto \left(-3 \cdot \log \left(\color{blue}{\left(1 - \frac{-1}{3}\right)} - \frac{4}{3} \cdot u\right)\right) \cdot s \]

   3. associate--r+ N/A

      \[\leadsto \left(-3 \cdot \log \color{blue}{\left(1 - \left(\frac{-1}{3} + \frac{4}{3} \cdot u\right)\right)}\right) \cdot s \]

   4. +-commutative N/A

      \[\leadsto \left(-3 \cdot \log \left(1 - \color{blue}{\left(\frac{4}{3} \cdot u + \frac{-1}{3}\right)}\right)\right) \cdot s \]

   5. metadata-eval N/A

      \[\leadsto \left(-3 \cdot \log \left(1 - \left(\frac{4}{3} \cdot u + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right)\right)\right) \cdot s \]

   6. metadata-eval N/A

      \[\leadsto \left(-3 \cdot \log \left(1 - \left(\frac{4}{3} \cdot u + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{4}}{\frac{3}{4}}}\right)\right)\right)\right)\right) \cdot s \]

   7. sub-neg N/A

      \[\leadsto \left(-3 \cdot \log \left(1 - \color{blue}{\left(\frac{4}{3} \cdot u - \frac{\frac{1}{4}}{\frac{3}{4}}\right)}\right)\right) \cdot s \]

   8. lift-*.f32 N/A

      \[\leadsto \left(-3 \cdot \log \left(1 - \left(\color{blue}{\frac{4}{3} \cdot u} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right) \cdot s \]

   9. *-commutative N/A

      \[\leadsto \left(-3 \cdot \log \left(1 - \left(\color{blue}{u \cdot \frac{4}{3}} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right) \cdot s \]

   10. metadata-eval N/A

       \[\leadsto \left(-3 \cdot \log \left(1 - \left(u \cdot \color{blue}{\frac{1}{\frac{3}{4}}} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right) \cdot s \]

   11. div-inv N/A

       \[\leadsto \left(-3 \cdot \log \left(1 - \left(\color{blue}{\frac{u}{\frac{3}{4}}} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right) \cdot s \]

   12. div-sub N/A

       \[\leadsto \left(-3 \cdot \log \left(1 - \color{blue}{\frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \cdot s \]

   13. sub-neg N/A

       \[\leadsto \left(-3 \cdot \log \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right)}\right) \cdot s \]

   14. +-commutative N/A

       \[\leadsto \left(-3 \cdot \log \color{blue}{\left(\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right) + 1\right)}\right) \cdot s \]

   15. lower-+.f32 N/A

       \[\leadsto \left(-3 \cdot \log \color{blue}{\left(\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right) + 1\right)}\right) \cdot s \]

   16. div-inv N/A

       \[\leadsto \left(-3 \cdot \log \left(\left(\mathsf{neg}\left(\color{blue}{\left(u - \frac{1}{4}\right) \cdot \frac{1}{\frac{3}{4}}}\right)\right) + 1\right)\right) \cdot s \]

   17. metadata-eval N/A

       \[\leadsto \left(-3 \cdot \log \left(\left(\mathsf{neg}\left(\left(u - \frac{1}{4}\right) \cdot \color{blue}{\frac{4}{3}}\right)\right) + 1\right)\right) \cdot s \]

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

       \[\leadsto \left(-3 \cdot \log \left(\color{blue}{\left(u - \frac{1}{4}\right) \cdot \left(\mathsf{neg}\left(\frac{4}{3}\right)\right)} + 1\right)\right) \cdot s \]

   19. lower-*.f32 N/A

       \[\leadsto \left(-3 \cdot \log \left(\color{blue}{\left(u - \frac{1}{4}\right) \cdot \left(\mathsf{neg}\left(\frac{4}{3}\right)\right)} + 1\right)\right) \cdot s \]

   20. lower--.f32 N/A

       \[\leadsto \left(-3 \cdot \log \left(\color{blue}{\left(u - \frac{1}{4}\right)} \cdot \left(\mathsf{neg}\left(\frac{4}{3}\right)\right) + 1\right)\right) \cdot s \]

   21. metadata-eval 96.4

       \[\leadsto \left(-3 \cdot \log \left(\left(u - 0.25\right) \cdot \color{blue}{-1.3333333333333333} + 1\right)\right) \cdot s \]

8. Applied rewrites 96.4%

   \[\leadsto \left(-3 \cdot \log \color{blue}{\left(\left(u - 0.25\right) \cdot -1.3333333333333333 + 1\right)}\right) \cdot s \]
9. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \left(-3 \cdot \log \left(\color{blue}{\left(u - \frac{1}{4}\right) \cdot \frac{-4}{3}} + 1\right)\right) \cdot s \]

   2. metadata-eval N/A

      \[\leadsto \left(-3 \cdot \log \left(\left(u - \frac{1}{4}\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{4}{3}\right)\right)} + 1\right)\right) \cdot s \]

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

      \[\leadsto \left(-3 \cdot \log \left(\color{blue}{\left(\mathsf{neg}\left(\left(u - \frac{1}{4}\right) \cdot \frac{4}{3}\right)\right)} + 1\right)\right) \cdot s \]

   4. metadata-eval N/A

      \[\leadsto \left(-3 \cdot \log \left(\left(\mathsf{neg}\left(\left(u - \frac{1}{4}\right) \cdot \color{blue}{\frac{1}{\frac{3}{4}}}\right)\right) + 1\right)\right) \cdot s \]

   5. div-inv N/A

      \[\leadsto \left(-3 \cdot \log \left(\left(\mathsf{neg}\left(\color{blue}{\frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) + 1\right)\right) \cdot s \]

   6. distribute-neg-frac2 N/A

      \[\leadsto \left(-3 \cdot \log \left(\color{blue}{\frac{u - \frac{1}{4}}{\mathsf{neg}\left(\frac{3}{4}\right)}} + 1\right)\right) \cdot s \]

   7. lower-/.f32 N/A

      \[\leadsto \left(-3 \cdot \log \left(\color{blue}{\frac{u - \frac{1}{4}}{\mathsf{neg}\left(\frac{3}{4}\right)}} + 1\right)\right) \cdot s \]

   8. metadata-eval 96.6

      \[\leadsto \left(-3 \cdot \log \left(\frac{u - 0.25}{\color{blue}{-0.75}} + 1\right)\right) \cdot s \]

10. Applied rewrites 96.6%

    \[\leadsto \left(-3 \cdot \log \left(\color{blue}{\frac{u - 0.25}{-0.75}} + 1\right)\right) \cdot s \]

11. Final simplification 96.6%

    \[\leadsto s \cdot \left(\log \left(1 + \frac{u - 0.25}{-0.75}\right) \cdot -3\right) \]
12. Add Preprocessing

Alternative 2: 96.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(\log \left(-1.3333333333333333 \cdot \left(u - 0.25\right) + 1\right) \cdot -3\right) \cdot s \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* (log (+ (* -1.3333333333333333 (- u 0.25)) 1.0)) -3.0) s))

C

float code(float s, float u) {
	return (logf(((-1.3333333333333333f * (u - 0.25f)) + 1.0f)) * -3.0f) * s;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (log((((-1.3333333333333333e0) * (u - 0.25e0)) + 1.0e0)) * (-3.0e0)) * s
end function

Julia

function code(s, u)
	return Float32(Float32(log(Float32(Float32(Float32(-1.3333333333333333) * Float32(u - Float32(0.25))) + Float32(1.0))) * Float32(-3.0)) * s)
end

MATLAB

function tmp = code(s, u)
	tmp = (log(((single(-1.3333333333333333) * (u - single(0.25))) + single(1.0))) * single(-3.0)) * s;
end

Derivation

1. Initial program 96.1%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   2. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(3 \cdot s\right)} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right)} \]

   4. *-commutative N/A

      \[\leadsto 3 \cdot \color{blue}{\left(\log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \cdot s\right)} \]

   5. associate-*r* N/A

      \[\leadsto \color{blue}{\left(3 \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \cdot s} \]

   6. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(3 \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \cdot s} \]

4. Applied rewrites 34.0%

   \[\leadsto \color{blue}{\left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right)\right) \cdot s} \]
5. Step-by-step derivation

   1. lift-log1p.f32 N/A

      \[\leadsto \left(-3 \cdot \color{blue}{\log \left(1 + \frac{-4}{3} \cdot \left(u - \frac{1}{4}\right)\right)}\right) \cdot s \]

   2. +-commutative N/A

      \[\leadsto \left(-3 \cdot \log \color{blue}{\left(\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right) + 1\right)}\right) \cdot s \]

   3. lift-+.f32 N/A

      \[\leadsto \left(-3 \cdot \log \color{blue}{\left(\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right) + 1\right)}\right) \cdot s \]

   4. lower-log.f32 96.4

      \[\leadsto \left(-3 \cdot \color{blue}{\log \left(-1.3333333333333333 \cdot \left(u - 0.25\right) + 1\right)}\right) \cdot s \]

   5. lift-+.f32 N/A

      \[\leadsto \left(-3 \cdot \log \color{blue}{\left(\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right) + 1\right)}\right) \cdot s \]

   6. +-commutative N/A

      \[\leadsto \left(-3 \cdot \log \color{blue}{\left(1 + \frac{-4}{3} \cdot \left(u - \frac{1}{4}\right)\right)}\right) \cdot s \]

   7. lift-*.f32 N/A

      \[\leadsto \left(-3 \cdot \log \left(1 + \color{blue}{\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right)}\right)\right) \cdot s \]

   8. *-commutative N/A

      \[\leadsto \left(-3 \cdot \log \left(1 + \color{blue}{\left(u - \frac{1}{4}\right) \cdot \frac{-4}{3}}\right)\right) \cdot s \]

   9. metadata-eval N/A

      \[\leadsto \left(-3 \cdot \log \left(1 + \left(u - \frac{1}{4}\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{4}{3}\right)\right)}\right)\right) \cdot s \]

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

       \[\leadsto \left(-3 \cdot \log \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(u - \frac{1}{4}\right) \cdot \frac{4}{3}\right)\right)}\right)\right) \cdot s \]

   11. metadata-eval N/A

       \[\leadsto \left(-3 \cdot \log \left(1 + \left(\mathsf{neg}\left(\left(u - \frac{1}{4}\right) \cdot \color{blue}{\frac{1}{\frac{3}{4}}}\right)\right)\right)\right) \cdot s \]

   12. div-inv N/A

       \[\leadsto \left(-3 \cdot \log \left(1 + \left(\mathsf{neg}\left(\color{blue}{\frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right)\right)\right) \cdot s \]

   13. lift--.f32 N/A

       \[\leadsto \left(-3 \cdot \log \left(1 + \left(\mathsf{neg}\left(\frac{\color{blue}{u - \frac{1}{4}}}{\frac{3}{4}}\right)\right)\right)\right) \cdot s \]

   14. sub-neg N/A

       \[\leadsto \left(-3 \cdot \log \color{blue}{\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}\right) \cdot s \]

   15. div-sub N/A

       \[\leadsto \left(-3 \cdot \log \left(1 - \color{blue}{\left(\frac{u}{\frac{3}{4}} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)}\right)\right) \cdot s \]

   16. div-inv N/A

       \[\leadsto \left(-3 \cdot \log \left(1 - \left(\color{blue}{u \cdot \frac{1}{\frac{3}{4}}} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right) \cdot s \]

   17. metadata-eval N/A

       \[\leadsto \left(-3 \cdot \log \left(1 - \left(u \cdot \color{blue}{\frac{4}{3}} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right) \cdot s \]

   18. *-commutative N/A

       \[\leadsto \left(-3 \cdot \log \left(1 - \left(\color{blue}{\frac{4}{3} \cdot u} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right) \cdot s \]

   19. sub-neg N/A

       \[\leadsto \left(-3 \cdot \log \left(1 - \color{blue}{\left(\frac{4}{3} \cdot u + \left(\mathsf{neg}\left(\frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right)}\right)\right) \cdot s \]

   20. metadata-eval N/A

       \[\leadsto \left(-3 \cdot \log \left(1 - \left(\frac{4}{3} \cdot u + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{3}}\right)\right)\right)\right)\right) \cdot s \]

   21. metadata-eval N/A

       \[\leadsto \left(-3 \cdot \log \left(1 - \left(\frac{4}{3} \cdot u + \color{blue}{\frac{-1}{3}}\right)\right)\right) \cdot s \]

   22. +-commutative N/A

       \[\leadsto \left(-3 \cdot \log \left(1 - \color{blue}{\left(\frac{-1}{3} + \frac{4}{3} \cdot u\right)}\right)\right) \cdot s \]

   23. associate--r+ N/A

       \[\leadsto \left(-3 \cdot \log \color{blue}{\left(\left(1 - \frac{-1}{3}\right) - \frac{4}{3} \cdot u\right)}\right) \cdot s \]

   24. metadata-eval N/A

       \[\leadsto \left(-3 \cdot \log \left(\color{blue}{\frac{4}{3}} - \frac{4}{3} \cdot u\right)\right) \cdot s \]

   25. lower--.f32 N/A

       \[\leadsto \left(-3 \cdot \log \color{blue}{\left(\frac{4}{3} - \frac{4}{3} \cdot u\right)}\right) \cdot s \]

   26. lower-*.f32 96.1

       \[\leadsto \left(-3 \cdot \log \left(1.3333333333333333 - \color{blue}{1.3333333333333333 \cdot u}\right)\right) \cdot s \]

6. Applied rewrites 96.1%

   \[\leadsto \left(-3 \cdot \color{blue}{\log \left(1.3333333333333333 - 1.3333333333333333 \cdot u\right)}\right) \cdot s \]
7. Step-by-step derivation

   1. lift--.f32 N/A

      \[\leadsto \left(-3 \cdot \log \color{blue}{\left(\frac{4}{3} - \frac{4}{3} \cdot u\right)}\right) \cdot s \]

   2. metadata-eval N/A

      \[\leadsto \left(-3 \cdot \log \left(\color{blue}{\left(1 - \frac{-1}{3}\right)} - \frac{4}{3} \cdot u\right)\right) \cdot s \]

   3. associate--r+ N/A

      \[\leadsto \left(-3 \cdot \log \color{blue}{\left(1 - \left(\frac{-1}{3} + \frac{4}{3} \cdot u\right)\right)}\right) \cdot s \]

   4. +-commutative N/A

      \[\leadsto \left(-3 \cdot \log \left(1 - \color{blue}{\left(\frac{4}{3} \cdot u + \frac{-1}{3}\right)}\right)\right) \cdot s \]

   5. metadata-eval N/A

      \[\leadsto \left(-3 \cdot \log \left(1 - \left(\frac{4}{3} \cdot u + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right)\right)\right) \cdot s \]

   6. metadata-eval N/A

      \[\leadsto \left(-3 \cdot \log \left(1 - \left(\frac{4}{3} \cdot u + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{4}}{\frac{3}{4}}}\right)\right)\right)\right)\right) \cdot s \]

   7. sub-neg N/A

      \[\leadsto \left(-3 \cdot \log \left(1 - \color{blue}{\left(\frac{4}{3} \cdot u - \frac{\frac{1}{4}}{\frac{3}{4}}\right)}\right)\right) \cdot s \]

   8. lift-*.f32 N/A

      \[\leadsto \left(-3 \cdot \log \left(1 - \left(\color{blue}{\frac{4}{3} \cdot u} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right) \cdot s \]

   9. *-commutative N/A

      \[\leadsto \left(-3 \cdot \log \left(1 - \left(\color{blue}{u \cdot \frac{4}{3}} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right) \cdot s \]

   10. metadata-eval N/A

       \[\leadsto \left(-3 \cdot \log \left(1 - \left(u \cdot \color{blue}{\frac{1}{\frac{3}{4}}} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right) \cdot s \]

   11. div-inv N/A

       \[\leadsto \left(-3 \cdot \log \left(1 - \left(\color{blue}{\frac{u}{\frac{3}{4}}} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right) \cdot s \]

   12. div-sub N/A

       \[\leadsto \left(-3 \cdot \log \left(1 - \color{blue}{\frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \cdot s \]

   13. sub-neg N/A

       \[\leadsto \left(-3 \cdot \log \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right)}\right) \cdot s \]

   14. +-commutative N/A

       \[\leadsto \left(-3 \cdot \log \color{blue}{\left(\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right) + 1\right)}\right) \cdot s \]

   15. lower-+.f32 N/A

       \[\leadsto \left(-3 \cdot \log \color{blue}{\left(\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right) + 1\right)}\right) \cdot s \]

   16. div-inv N/A

       \[\leadsto \left(-3 \cdot \log \left(\left(\mathsf{neg}\left(\color{blue}{\left(u - \frac{1}{4}\right) \cdot \frac{1}{\frac{3}{4}}}\right)\right) + 1\right)\right) \cdot s \]

   17. metadata-eval N/A

       \[\leadsto \left(-3 \cdot \log \left(\left(\mathsf{neg}\left(\left(u - \frac{1}{4}\right) \cdot \color{blue}{\frac{4}{3}}\right)\right) + 1\right)\right) \cdot s \]

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

       \[\leadsto \left(-3 \cdot \log \left(\color{blue}{\left(u - \frac{1}{4}\right) \cdot \left(\mathsf{neg}\left(\frac{4}{3}\right)\right)} + 1\right)\right) \cdot s \]

   19. lower-*.f32 N/A

       \[\leadsto \left(-3 \cdot \log \left(\color{blue}{\left(u - \frac{1}{4}\right) \cdot \left(\mathsf{neg}\left(\frac{4}{3}\right)\right)} + 1\right)\right) \cdot s \]

   20. lower--.f32 N/A

       \[\leadsto \left(-3 \cdot \log \left(\color{blue}{\left(u - \frac{1}{4}\right)} \cdot \left(\mathsf{neg}\left(\frac{4}{3}\right)\right) + 1\right)\right) \cdot s \]

   21. metadata-eval 96.4

       \[\leadsto \left(-3 \cdot \log \left(\left(u - 0.25\right) \cdot \color{blue}{-1.3333333333333333} + 1\right)\right) \cdot s \]

8. Applied rewrites 96.4%

   \[\leadsto \left(-3 \cdot \log \color{blue}{\left(\left(u - 0.25\right) \cdot -1.3333333333333333 + 1\right)}\right) \cdot s \]

9. Final simplification 96.4%

   \[\leadsto \left(\log \left(-1.3333333333333333 \cdot \left(u - 0.25\right) + 1\right) \cdot -3\right) \cdot s \]
10. Add Preprocessing

Alternative 3: 96.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(\log \left(1.3333333333333333 - 1.3333333333333333 \cdot u\right) \cdot -3\right) \cdot s \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* (log (- 1.3333333333333333 (* 1.3333333333333333 u))) -3.0) s))

C

float code(float s, float u) {
	return (logf((1.3333333333333333f - (1.3333333333333333f * u))) * -3.0f) * s;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (log((1.3333333333333333e0 - (1.3333333333333333e0 * u))) * (-3.0e0)) * s
end function

Julia

function code(s, u)
	return Float32(Float32(log(Float32(Float32(1.3333333333333333) - Float32(Float32(1.3333333333333333) * u))) * Float32(-3.0)) * s)
end

MATLAB

function tmp = code(s, u)
	tmp = (log((single(1.3333333333333333) - (single(1.3333333333333333) * u))) * single(-3.0)) * s;
end

Derivation

1. Initial program 96.1%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   2. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(3 \cdot s\right)} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right)} \]

   4. *-commutative N/A

      \[\leadsto 3 \cdot \color{blue}{\left(\log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \cdot s\right)} \]

   5. associate-*r* N/A

      \[\leadsto \color{blue}{\left(3 \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \cdot s} \]

   6. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(3 \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \cdot s} \]

4. Applied rewrites 34.1%

   \[\leadsto \color{blue}{\left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right)\right) \cdot s} \]
5. Step-by-step derivation

   1. lift-log1p.f32 N/A

      \[\leadsto \left(-3 \cdot \color{blue}{\log \left(1 + \frac{-4}{3} \cdot \left(u - \frac{1}{4}\right)\right)}\right) \cdot s \]

   2. +-commutative N/A

      \[\leadsto \left(-3 \cdot \log \color{blue}{\left(\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right) + 1\right)}\right) \cdot s \]

   3. lift-+.f32 N/A

      \[\leadsto \left(-3 \cdot \log \color{blue}{\left(\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right) + 1\right)}\right) \cdot s \]

   4. lower-log.f32 96.4

      \[\leadsto \left(-3 \cdot \color{blue}{\log \left(-1.3333333333333333 \cdot \left(u - 0.25\right) + 1\right)}\right) \cdot s \]

   5. lift-+.f32 N/A

      \[\leadsto \left(-3 \cdot \log \color{blue}{\left(\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right) + 1\right)}\right) \cdot s \]

   6. +-commutative N/A

      \[\leadsto \left(-3 \cdot \log \color{blue}{\left(1 + \frac{-4}{3} \cdot \left(u - \frac{1}{4}\right)\right)}\right) \cdot s \]

   7. lift-*.f32 N/A

      \[\leadsto \left(-3 \cdot \log \left(1 + \color{blue}{\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right)}\right)\right) \cdot s \]

   8. *-commutative N/A

      \[\leadsto \left(-3 \cdot \log \left(1 + \color{blue}{\left(u - \frac{1}{4}\right) \cdot \frac{-4}{3}}\right)\right) \cdot s \]

   9. metadata-eval N/A

      \[\leadsto \left(-3 \cdot \log \left(1 + \left(u - \frac{1}{4}\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{4}{3}\right)\right)}\right)\right) \cdot s \]

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

       \[\leadsto \left(-3 \cdot \log \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(u - \frac{1}{4}\right) \cdot \frac{4}{3}\right)\right)}\right)\right) \cdot s \]

   11. metadata-eval N/A

       \[\leadsto \left(-3 \cdot \log \left(1 + \left(\mathsf{neg}\left(\left(u - \frac{1}{4}\right) \cdot \color{blue}{\frac{1}{\frac{3}{4}}}\right)\right)\right)\right) \cdot s \]

   12. div-inv N/A

       \[\leadsto \left(-3 \cdot \log \left(1 + \left(\mathsf{neg}\left(\color{blue}{\frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right)\right)\right) \cdot s \]

   13. lift--.f32 N/A

       \[\leadsto \left(-3 \cdot \log \left(1 + \left(\mathsf{neg}\left(\frac{\color{blue}{u - \frac{1}{4}}}{\frac{3}{4}}\right)\right)\right)\right) \cdot s \]

   14. sub-neg N/A

       \[\leadsto \left(-3 \cdot \log \color{blue}{\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}\right) \cdot s \]

   15. div-sub N/A

       \[\leadsto \left(-3 \cdot \log \left(1 - \color{blue}{\left(\frac{u}{\frac{3}{4}} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)}\right)\right) \cdot s \]

   16. div-inv N/A

       \[\leadsto \left(-3 \cdot \log \left(1 - \left(\color{blue}{u \cdot \frac{1}{\frac{3}{4}}} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right) \cdot s \]

   17. metadata-eval N/A

       \[\leadsto \left(-3 \cdot \log \left(1 - \left(u \cdot \color{blue}{\frac{4}{3}} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right) \cdot s \]

   18. *-commutative N/A

       \[\leadsto \left(-3 \cdot \log \left(1 - \left(\color{blue}{\frac{4}{3} \cdot u} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right) \cdot s \]

   19. sub-neg N/A

       \[\leadsto \left(-3 \cdot \log \left(1 - \color{blue}{\left(\frac{4}{3} \cdot u + \left(\mathsf{neg}\left(\frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right)}\right)\right) \cdot s \]

   20. metadata-eval N/A

       \[\leadsto \left(-3 \cdot \log \left(1 - \left(\frac{4}{3} \cdot u + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{3}}\right)\right)\right)\right)\right) \cdot s \]

   21. metadata-eval N/A

       \[\leadsto \left(-3 \cdot \log \left(1 - \left(\frac{4}{3} \cdot u + \color{blue}{\frac{-1}{3}}\right)\right)\right) \cdot s \]

   22. +-commutative N/A

       \[\leadsto \left(-3 \cdot \log \left(1 - \color{blue}{\left(\frac{-1}{3} + \frac{4}{3} \cdot u\right)}\right)\right) \cdot s \]

   23. associate--r+ N/A

       \[\leadsto \left(-3 \cdot \log \color{blue}{\left(\left(1 - \frac{-1}{3}\right) - \frac{4}{3} \cdot u\right)}\right) \cdot s \]

   24. metadata-eval N/A

       \[\leadsto \left(-3 \cdot \log \left(\color{blue}{\frac{4}{3}} - \frac{4}{3} \cdot u\right)\right) \cdot s \]

   25. lower--.f32 N/A

       \[\leadsto \left(-3 \cdot \log \color{blue}{\left(\frac{4}{3} - \frac{4}{3} \cdot u\right)}\right) \cdot s \]

   26. lower-*.f32 96.1

       \[\leadsto \left(-3 \cdot \log \left(1.3333333333333333 - \color{blue}{1.3333333333333333 \cdot u}\right)\right) \cdot s \]

6. Applied rewrites 96.1%

   \[\leadsto \left(-3 \cdot \color{blue}{\log \left(1.3333333333333333 - 1.3333333333333333 \cdot u\right)}\right) \cdot s \]

7. Final simplification 96.1%

   \[\leadsto \left(\log \left(1.3333333333333333 - 1.3333333333333333 \cdot u\right) \cdot -3\right) \cdot s \]
8. Add Preprocessing

Alternative 4: 30.3% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \left(\left(u \cdot u\right) \cdot s\right) \cdot \left(\frac{3}{u} + 1.5\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (* (* u u) s) (+ (/ 3.0 u) 1.5)))

C

float code(float s, float u) {
	return ((u * u) * s) * ((3.0f / u) + 1.5f);
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = ((u * u) * s) * ((3.0e0 / u) + 1.5e0)
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(u * u) * s) * Float32(Float32(Float32(3.0) / u) + Float32(1.5)))
end

MATLAB

function tmp = code(s, u)
	tmp = ((u * u) * s) * ((single(3.0) / u) + single(1.5));
end

Derivation

1. Initial program 96.1%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \frac{3}{4}\right) + u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right) + 3 \cdot s\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right) + 3 \cdot s\right) + 3 \cdot \left(s \cdot \log \frac{3}{4}\right)} \]

   2. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\left(\frac{3}{2} \cdot \left(s \cdot u\right)\right) \cdot u + \left(3 \cdot s\right) \cdot u\right)} + 3 \cdot \left(s \cdot \log \frac{3}{4}\right) \]

   3. associate-+l+ N/A

      \[\leadsto \color{blue}{\left(\frac{3}{2} \cdot \left(s \cdot u\right)\right) \cdot u + \left(\left(3 \cdot s\right) \cdot u + 3 \cdot \left(s \cdot \log \frac{3}{4}\right)\right)} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right)\right)} + \left(\left(3 \cdot s\right) \cdot u + 3 \cdot \left(s \cdot \log \frac{3}{4}\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(3 \cdot s\right) \cdot u + 3 \cdot \left(s \cdot \log \frac{3}{4}\right)\right) + u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right)\right)} \]

   6. associate-*r* N/A

      \[\leadsto \left(\color{blue}{3 \cdot \left(s \cdot u\right)} + 3 \cdot \left(s \cdot \log \frac{3}{4}\right)\right) + u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right)\right) \]

   7. distribute-lft-out N/A

      \[\leadsto \color{blue}{3 \cdot \left(s \cdot u + s \cdot \log \frac{3}{4}\right)} + u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \color{blue}{\left(s \cdot u + s \cdot \log \frac{3}{4}\right) \cdot 3} + u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right)\right) \]

   9. distribute-lft-out N/A

      \[\leadsto \color{blue}{\left(s \cdot \left(u + \log \frac{3}{4}\right)\right)} \cdot 3 + u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \color{blue}{s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right)} + u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right)\right) \]

   11. *-commutative N/A

       \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \color{blue}{\left(\frac{3}{2} \cdot \left(s \cdot u\right)\right) \cdot u} \]

   12. *-commutative N/A

       \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \color{blue}{\left(\left(s \cdot u\right) \cdot \frac{3}{2}\right)} \cdot u \]

   13. associate-*l* N/A

       \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \color{blue}{\left(s \cdot u\right) \cdot \left(\frac{3}{2} \cdot u\right)} \]

   14. associate-*l* N/A

       \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \color{blue}{s \cdot \left(u \cdot \left(\frac{3}{2} \cdot u\right)\right)} \]

   15. distribute-lft-out N/A

       \[\leadsto \color{blue}{s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3 + u \cdot \left(\frac{3}{2} \cdot u\right)\right)} \]

5. Applied rewrites 14.9%

   \[\leadsto \color{blue}{s \cdot \mathsf{fma}\left(\log 0.75 + u, 3, \left(1.5 \cdot u\right) \cdot u\right)} \]

6. Taylor expanded in u around inf

   \[\leadsto \frac{3}{2} \cdot \color{blue}{\left(s \cdot {u}^{2}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 26.5%

   \[\leadsto \left(\left(u \cdot s\right) \cdot 1.5\right) \cdot \color{blue}{u} \]

9. Taylor expanded in u around inf

   \[\leadsto {u}^{2} \cdot \color{blue}{\left(\frac{3}{2} \cdot s + 3 \cdot \frac{s}{u}\right)} \]
10. Step-by-step derivation

11. Applied rewrites 30.4%

    \[\leadsto \left(s \cdot \left(\frac{3}{u} + 1.5\right)\right) \cdot \color{blue}{\left(u \cdot u\right)} \]
12. Step-by-step derivation

13. Applied rewrites 30.4%

    \[\leadsto \left(\frac{3}{u} + 1.5\right) \cdot \left(\left(u \cdot u\right) \cdot \color{blue}{s}\right) \]

14. Final simplification 30.4%

    \[\leadsto \left(\left(u \cdot u\right) \cdot s\right) \cdot \left(\frac{3}{u} + 1.5\right) \]
15. Add Preprocessing

Alternative 5: 30.3% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\frac{3}{u} + 1.5\right) \cdot u\right) \cdot u\right) \cdot s \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (* (* (+ (/ 3.0 u) 1.5) u) u) s))

C

float code(float s, float u) {
	return ((((3.0f / u) + 1.5f) * u) * u) * s;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = ((((3.0e0 / u) + 1.5e0) * u) * u) * s
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(Float32(Float32(Float32(3.0) / u) + Float32(1.5)) * u) * u) * s)
end

MATLAB

function tmp = code(s, u)
	tmp = ((((single(3.0) / u) + single(1.5)) * u) * u) * s;
end

Derivation

1. Initial program 96.1%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \frac{3}{4}\right) + u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right) + 3 \cdot s\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right) + 3 \cdot s\right) + 3 \cdot \left(s \cdot \log \frac{3}{4}\right)} \]

   2. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\left(\frac{3}{2} \cdot \left(s \cdot u\right)\right) \cdot u + \left(3 \cdot s\right) \cdot u\right)} + 3 \cdot \left(s \cdot \log \frac{3}{4}\right) \]

   3. associate-+l+ N/A

      \[\leadsto \color{blue}{\left(\frac{3}{2} \cdot \left(s \cdot u\right)\right) \cdot u + \left(\left(3 \cdot s\right) \cdot u + 3 \cdot \left(s \cdot \log \frac{3}{4}\right)\right)} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right)\right)} + \left(\left(3 \cdot s\right) \cdot u + 3 \cdot \left(s \cdot \log \frac{3}{4}\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(3 \cdot s\right) \cdot u + 3 \cdot \left(s \cdot \log \frac{3}{4}\right)\right) + u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right)\right)} \]

   6. associate-*r* N/A

      \[\leadsto \left(\color{blue}{3 \cdot \left(s \cdot u\right)} + 3 \cdot \left(s \cdot \log \frac{3}{4}\right)\right) + u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right)\right) \]

   7. distribute-lft-out N/A

      \[\leadsto \color{blue}{3 \cdot \left(s \cdot u + s \cdot \log \frac{3}{4}\right)} + u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \color{blue}{\left(s \cdot u + s \cdot \log \frac{3}{4}\right) \cdot 3} + u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right)\right) \]

   9. distribute-lft-out N/A

      \[\leadsto \color{blue}{\left(s \cdot \left(u + \log \frac{3}{4}\right)\right)} \cdot 3 + u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \color{blue}{s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right)} + u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right)\right) \]

   11. *-commutative N/A

       \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \color{blue}{\left(\frac{3}{2} \cdot \left(s \cdot u\right)\right) \cdot u} \]

   12. *-commutative N/A

       \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \color{blue}{\left(\left(s \cdot u\right) \cdot \frac{3}{2}\right)} \cdot u \]

   13. associate-*l* N/A

       \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \color{blue}{\left(s \cdot u\right) \cdot \left(\frac{3}{2} \cdot u\right)} \]

   14. associate-*l* N/A

       \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \color{blue}{s \cdot \left(u \cdot \left(\frac{3}{2} \cdot u\right)\right)} \]

   15. distribute-lft-out N/A

       \[\leadsto \color{blue}{s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3 + u \cdot \left(\frac{3}{2} \cdot u\right)\right)} \]

5. Applied rewrites 14.9%

   \[\leadsto \color{blue}{s \cdot \mathsf{fma}\left(\log 0.75 + u, 3, \left(1.5 \cdot u\right) \cdot u\right)} \]

6. Taylor expanded in u around inf

   \[\leadsto \frac{3}{2} \cdot \color{blue}{\left(s \cdot {u}^{2}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 26.5%

   \[\leadsto \left(\left(u \cdot s\right) \cdot 1.5\right) \cdot \color{blue}{u} \]

9. Taylor expanded in u around inf

   \[\leadsto {u}^{2} \cdot \color{blue}{\left(\frac{3}{2} \cdot s + 3 \cdot \frac{s}{u}\right)} \]
10. Step-by-step derivation

11. Applied rewrites 30.4%

    \[\leadsto \left(s \cdot \left(\frac{3}{u} + 1.5\right)\right) \cdot \color{blue}{\left(u \cdot u\right)} \]
12. Step-by-step derivation

13. Applied rewrites 30.4%

    \[\leadsto \left(\left(\left(\frac{3}{u} + 1.5\right) \cdot u\right) \cdot u\right) \cdot s \]
14. Add Preprocessing

Alternative 6: 30.3% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \left(\left(\frac{3}{u} + 1.5\right) \cdot s\right) \cdot \left(u \cdot u\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (* (+ (/ 3.0 u) 1.5) s) (* u u)))

C

float code(float s, float u) {
	return (((3.0f / u) + 1.5f) * s) * (u * u);
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (((3.0e0 / u) + 1.5e0) * s) * (u * u)
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(Float32(Float32(3.0) / u) + Float32(1.5)) * s) * Float32(u * u))
end

MATLAB

function tmp = code(s, u)
	tmp = (((single(3.0) / u) + single(1.5)) * s) * (u * u);
end

Derivation

1. Initial program 96.1%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \frac{3}{4}\right) + u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right) + 3 \cdot s\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right) + 3 \cdot s\right) + 3 \cdot \left(s \cdot \log \frac{3}{4}\right)} \]

   2. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\left(\frac{3}{2} \cdot \left(s \cdot u\right)\right) \cdot u + \left(3 \cdot s\right) \cdot u\right)} + 3 \cdot \left(s \cdot \log \frac{3}{4}\right) \]

   3. associate-+l+ N/A

      \[\leadsto \color{blue}{\left(\frac{3}{2} \cdot \left(s \cdot u\right)\right) \cdot u + \left(\left(3 \cdot s\right) \cdot u + 3 \cdot \left(s \cdot \log \frac{3}{4}\right)\right)} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right)\right)} + \left(\left(3 \cdot s\right) \cdot u + 3 \cdot \left(s \cdot \log \frac{3}{4}\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(3 \cdot s\right) \cdot u + 3 \cdot \left(s \cdot \log \frac{3}{4}\right)\right) + u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right)\right)} \]

   6. associate-*r* N/A

      \[\leadsto \left(\color{blue}{3 \cdot \left(s \cdot u\right)} + 3 \cdot \left(s \cdot \log \frac{3}{4}\right)\right) + u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right)\right) \]

   7. distribute-lft-out N/A

      \[\leadsto \color{blue}{3 \cdot \left(s \cdot u + s \cdot \log \frac{3}{4}\right)} + u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \color{blue}{\left(s \cdot u + s \cdot \log \frac{3}{4}\right) \cdot 3} + u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right)\right) \]

   9. distribute-lft-out N/A

      \[\leadsto \color{blue}{\left(s \cdot \left(u + \log \frac{3}{4}\right)\right)} \cdot 3 + u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \color{blue}{s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right)} + u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right)\right) \]

   11. *-commutative N/A

       \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \color{blue}{\left(\frac{3}{2} \cdot \left(s \cdot u\right)\right) \cdot u} \]

   12. *-commutative N/A

       \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \color{blue}{\left(\left(s \cdot u\right) \cdot \frac{3}{2}\right)} \cdot u \]

   13. associate-*l* N/A

       \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \color{blue}{\left(s \cdot u\right) \cdot \left(\frac{3}{2} \cdot u\right)} \]

   14. associate-*l* N/A

       \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \color{blue}{s \cdot \left(u \cdot \left(\frac{3}{2} \cdot u\right)\right)} \]

   15. distribute-lft-out N/A

       \[\leadsto \color{blue}{s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3 + u \cdot \left(\frac{3}{2} \cdot u\right)\right)} \]

5. Applied rewrites 15.0%

   \[\leadsto \color{blue}{s \cdot \mathsf{fma}\left(\log 0.75 + u, 3, \left(1.5 \cdot u\right) \cdot u\right)} \]

6. Taylor expanded in u around inf

   \[\leadsto \frac{3}{2} \cdot \color{blue}{\left(s \cdot {u}^{2}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 26.5%

   \[\leadsto \left(\left(u \cdot s\right) \cdot 1.5\right) \cdot \color{blue}{u} \]

9. Taylor expanded in u around inf

   \[\leadsto {u}^{2} \cdot \color{blue}{\left(\frac{3}{2} \cdot s + 3 \cdot \frac{s}{u}\right)} \]
10. Step-by-step derivation

11. Applied rewrites 30.4%

    \[\leadsto \left(s \cdot \left(\frac{3}{u} + 1.5\right)\right) \cdot \color{blue}{\left(u \cdot u\right)} \]

12. Final simplification 30.4%

    \[\leadsto \left(\left(\frac{3}{u} + 1.5\right) \cdot s\right) \cdot \left(u \cdot u\right) \]
13. Add Preprocessing

Alternative 7: 29.9% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \left(3 \cdot u\right) \cdot s \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (* 3.0 u) s))

C

float code(float s, float u) {
	return (3.0f * u) * s;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (3.0e0 * u) * s
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(3.0) * u) * s)
end

MATLAB

function tmp = code(s, u)
	tmp = (single(3.0) * u) * s;
end

Derivation

1. Initial program 96.1%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \frac{3}{4}\right) + u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right) + 3 \cdot s\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right) + 3 \cdot s\right) + 3 \cdot \left(s \cdot \log \frac{3}{4}\right)} \]

   2. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\left(\frac{3}{2} \cdot \left(s \cdot u\right)\right) \cdot u + \left(3 \cdot s\right) \cdot u\right)} + 3 \cdot \left(s \cdot \log \frac{3}{4}\right) \]

   3. associate-+l+ N/A

      \[\leadsto \color{blue}{\left(\frac{3}{2} \cdot \left(s \cdot u\right)\right) \cdot u + \left(\left(3 \cdot s\right) \cdot u + 3 \cdot \left(s \cdot \log \frac{3}{4}\right)\right)} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right)\right)} + \left(\left(3 \cdot s\right) \cdot u + 3 \cdot \left(s \cdot \log \frac{3}{4}\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(3 \cdot s\right) \cdot u + 3 \cdot \left(s \cdot \log \frac{3}{4}\right)\right) + u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right)\right)} \]

   6. associate-*r* N/A

      \[\leadsto \left(\color{blue}{3 \cdot \left(s \cdot u\right)} + 3 \cdot \left(s \cdot \log \frac{3}{4}\right)\right) + u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right)\right) \]

   7. distribute-lft-out N/A

      \[\leadsto \color{blue}{3 \cdot \left(s \cdot u + s \cdot \log \frac{3}{4}\right)} + u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \color{blue}{\left(s \cdot u + s \cdot \log \frac{3}{4}\right) \cdot 3} + u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right)\right) \]

   9. distribute-lft-out N/A

      \[\leadsto \color{blue}{\left(s \cdot \left(u + \log \frac{3}{4}\right)\right)} \cdot 3 + u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \color{blue}{s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right)} + u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right)\right) \]

   11. *-commutative N/A

       \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \color{blue}{\left(\frac{3}{2} \cdot \left(s \cdot u\right)\right) \cdot u} \]

   12. *-commutative N/A

       \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \color{blue}{\left(\left(s \cdot u\right) \cdot \frac{3}{2}\right)} \cdot u \]

   13. associate-*l* N/A

       \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \color{blue}{\left(s \cdot u\right) \cdot \left(\frac{3}{2} \cdot u\right)} \]

   14. associate-*l* N/A

       \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \color{blue}{s \cdot \left(u \cdot \left(\frac{3}{2} \cdot u\right)\right)} \]

   15. distribute-lft-out N/A

       \[\leadsto \color{blue}{s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3 + u \cdot \left(\frac{3}{2} \cdot u\right)\right)} \]

5. Applied rewrites 14.9%

   \[\leadsto \color{blue}{s \cdot \mathsf{fma}\left(\log 0.75 + u, 3, \left(1.5 \cdot u\right) \cdot u\right)} \]

6. Taylor expanded in u around inf

   \[\leadsto \frac{3}{2} \cdot \color{blue}{\left(s \cdot {u}^{2}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 26.5%

   \[\leadsto \left(\left(u \cdot s\right) \cdot 1.5\right) \cdot \color{blue}{u} \]

9. Taylor expanded in u around inf

   \[\leadsto {u}^{2} \cdot \color{blue}{\left(\frac{3}{2} \cdot s + 3 \cdot \frac{s}{u}\right)} \]
10. Step-by-step derivation

11. Applied rewrites 30.4%

    \[\leadsto \left(s \cdot \left(\frac{3}{u} + 1.5\right)\right) \cdot \color{blue}{\left(u \cdot u\right)} \]

12. Taylor expanded in u around 0

    \[\leadsto 3 \cdot \left(s \cdot \color{blue}{u}\right) \]
13. Step-by-step derivation

14. Applied rewrites 30.3%

    \[\leadsto \left(3 \cdot u\right) \cdot s \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024259 
(FPCore (s u)
  :name "Disney BSSRDF, sample scattering profile, upper"
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (<= 0.25 u) (<= u 1.0)))
  (* (* 3.0 s) (log (/ 1.0 (- 1.0 (/ (- u 0.25) 0.75))))))