Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.9% → 96.2%

Time: 7.6s

Alternatives: 4

Speedup: 1.1×

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: 96.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.9%

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

   1. lift--.f32 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lower-+.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. distribute-neg-frac2 N/A

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

   7. div-inv N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval 95.8

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

4. Applied rewrites 95.8%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

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

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

   6. div-inv N/A

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

   7. clear-num N/A

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

   8. distribute-neg-frac N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f32 N/A

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

   11. lower-/.f32 95.6

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

6. Applied rewrites 95.6%

   \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{-1}{\frac{0.75}{u - 0.25}}} + 1}\right) \]
7. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. div-inv N/A

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

   5. lift-/.f32 N/A

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

   6. clear-num N/A

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

   7. lift--.f32 N/A

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

   8. neg-mul-1 N/A

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

   9. sub-neg N/A

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

   10. div-sub N/A

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

   11. div-inv N/A

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

   12. metadata-eval N/A

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

   13. sub-neg N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. +-commutative N/A

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

   17. associate--r+ N/A

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

   18. metadata-eval N/A

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

   19. lower--.f32 N/A

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

   20. *-commutative N/A

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

   21. lower-*.f32 95.4

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

8. Applied rewrites 95.4%

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

   1. lift-log.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. frac-2neg N/A

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

   4. metadata-eval N/A

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

   5. frac-2neg N/A

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

   6. metadata-eval N/A

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

   7. log-rec N/A

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

   8. lower-neg.f32 N/A

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

   9. lower-log.f32 N/A

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

   10. lower-neg.f32 N/A

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

   11. lift--.f32 N/A

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

   12. sub-neg N/A

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

   13. distribute-neg-in N/A

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

   14. unsub-neg N/A

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

   15. lower--.f32 N/A

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

   16. metadata-eval N/A

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

   17. lift-*.f32 N/A

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

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

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

   19. lower-*.f32 N/A

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

   20. metadata-eval 96.0

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

10. Applied rewrites 96.0%

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

11. Final simplification 96.0%

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

Alternative 2: 28.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \left(\log 0.6666666666666666 \cdot -3\right) \cdot s \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (* (log 0.6666666666666666) -3.0) s))

C

float code(float s, float u) {
	return (logf(0.6666666666666666f) * -3.0f) * s;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (log(0.6666666666666666e0) * (-3.0e0)) * s
end function

Julia

function code(s, u)
	return Float32(Float32(log(Float32(0.6666666666666666)) * Float32(-3.0)) * s)
end

MATLAB

function tmp = code(s, u)
	tmp = (log(single(0.6666666666666666)) * single(-3.0)) * s;
end

Derivation

1. Initial program 95.9%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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. +-lft-identity N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(0 + \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right)} \]

   3. metadata-eval N/A

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

   4. distribute-rgt-in N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. lower-fma.f32 N/A

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

   8. metadata-eval 7.1

      \[\leadsto \mathsf{fma}\left(\color{blue}{0}, 3 \cdot s, \left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right)\right) \]

   9. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(0, \color{blue}{3 \cdot s}, \left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(0, \color{blue}{s \cdot 3}, \left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \]

   11. lower-*.f32 7.1

       \[\leadsto \mathsf{fma}\left(0, \color{blue}{s \cdot 3}, \left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right)\right) \]

   12. lift-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(0, s \cdot 3, \color{blue}{\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)}\right) \]

   13. lift-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(0, s \cdot 3, \color{blue}{\left(3 \cdot s\right)} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \]

   14. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(0, s \cdot 3, \color{blue}{3 \cdot \left(s \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right)}\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(0, s \cdot 3, 3 \cdot \color{blue}{\left(\log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \cdot s\right)}\right) \]

   16. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(0, s \cdot 3, \color{blue}{\left(3 \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \cdot s}\right) \]

   17. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(0, s \cdot 3, \color{blue}{\left(3 \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \cdot s}\right) \]

4. Applied rewrites 33.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0, s \cdot 3, \left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right)\right) \cdot s\right)} \]

6. Applied rewrites 27.5%

   \[\leadsto \color{blue}{\left(\mathsf{log1p}\left(\mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)\right) \cdot s\right) \cdot -3} \]

7. Taylor expanded in u around 0

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

   1. lower-log.f32 27.9

      \[\leadsto \left(\color{blue}{\log 0.6666666666666666} \cdot s\right) \cdot -3 \]

9. Applied rewrites 27.9%

   \[\leadsto \left(\color{blue}{\log 0.6666666666666666} \cdot s\right) \cdot -3 \]
10. Step-by-step derivation

    1. lift-*.f32 N/A

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

    2. *-commutative N/A

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

    3. lift-*.f32 N/A

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

    4. associate-*r* N/A

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

    5. lower-*.f32 N/A

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

    6. lower-*.f32 27.9

       \[\leadsto \color{blue}{\left(-3 \cdot \log 0.6666666666666666\right)} \cdot s \]

11. Applied rewrites 27.9%

    \[\leadsto \color{blue}{\left(-3 \cdot \log 0.6666666666666666\right) \cdot s} \]

12. Final simplification 27.9%

    \[\leadsto \left(\log 0.6666666666666666 \cdot -3\right) \cdot s \]
13. Add Preprocessing

Alternative 3: 28.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \left(-3 \cdot s\right) \cdot \log 0.6666666666666666 \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (* -3.0 s) (log 0.6666666666666666)))

C

float code(float s, float u) {
	return (-3.0f * s) * logf(0.6666666666666666f);
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = ((-3.0e0) * s) * log(0.6666666666666666e0)
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(-3.0) * s) * log(Float32(0.6666666666666666)))
end

MATLAB

function tmp = code(s, u)
	tmp = (single(-3.0) * s) * log(single(0.6666666666666666));
end

Derivation

1. Initial program 95.9%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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. +-lft-identity N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(0 + \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right)} \]

   3. metadata-eval N/A

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

   4. distribute-rgt-in N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. lower-fma.f32 N/A

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

   8. metadata-eval 7.1

      \[\leadsto \mathsf{fma}\left(\color{blue}{0}, 3 \cdot s, \left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right)\right) \]

   9. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(0, \color{blue}{3 \cdot s}, \left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(0, \color{blue}{s \cdot 3}, \left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \]

   11. lower-*.f32 7.1

       \[\leadsto \mathsf{fma}\left(0, \color{blue}{s \cdot 3}, \left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right)\right) \]

   12. lift-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(0, s \cdot 3, \color{blue}{\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)}\right) \]

   13. lift-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(0, s \cdot 3, \color{blue}{\left(3 \cdot s\right)} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \]

   14. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(0, s \cdot 3, \color{blue}{3 \cdot \left(s \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right)}\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(0, s \cdot 3, 3 \cdot \color{blue}{\left(\log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \cdot s\right)}\right) \]

   16. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(0, s \cdot 3, \color{blue}{\left(3 \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \cdot s}\right) \]

   17. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(0, s \cdot 3, \color{blue}{\left(3 \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \cdot s}\right) \]

4. Applied rewrites 33.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0, s \cdot 3, \left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right)\right) \cdot s\right)} \]

6. Applied rewrites 27.5%

   \[\leadsto \color{blue}{\left(\mathsf{log1p}\left(\mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)\right) \cdot s\right) \cdot -3} \]

7. Taylor expanded in u around 0

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

   1. lower-log.f32 27.9

      \[\leadsto \left(\color{blue}{\log 0.6666666666666666} \cdot s\right) \cdot -3 \]

9. Applied rewrites 27.9%

   \[\leadsto \left(\color{blue}{\log 0.6666666666666666} \cdot s\right) \cdot -3 \]
10. Step-by-step derivation

    1. lift-*.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. associate-*l* N/A

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

    4. *-commutative N/A

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

    5. lower-*.f32 N/A

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

    6. *-commutative N/A

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

    7. lower-*.f32 27.9

       \[\leadsto \color{blue}{\left(-3 \cdot s\right)} \cdot \log 0.6666666666666666 \]

11. Applied rewrites 27.9%

    \[\leadsto \color{blue}{\left(-3 \cdot s\right) \cdot \log 0.6666666666666666} \]
12. Add Preprocessing

Alternative 4: 7.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \left(\log 0.75 \cdot s\right) \cdot 3 \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (* (log 0.75) s) 3.0))

C

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

Fortran

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

Julia

function code(s, u)
	return Float32(Float32(log(Float32(0.75)) * s) * Float32(3.0))
end

MATLAB

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

Derivation

1. Initial program 95.9%

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

3. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-log.f32 7.4

      \[\leadsto \left(\color{blue}{\log 0.75} \cdot s\right) \cdot 3 \]

5. Applied rewrites 7.4%

   \[\leadsto \color{blue}{\left(\log 0.75 \cdot s\right) \cdot 3} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024242 
(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))))))