Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.9% → 96.1%

Time: 7.7s

Alternatives: 6

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: 96.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.8%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 95.6

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

4. Applied rewrites 95.6%

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

   1. lift-log.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. frac-2neg N/A

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

   4. metadata-eval N/A

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

   5. frac-2neg N/A

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

   6. metadata-eval N/A

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

   7. log-rec N/A

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

   8. lower-neg.f32 N/A

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

   9. lower-log.f32 N/A

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

   10. lower-neg.f32 N/A

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

   11. lift-+.f32 N/A

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

   12. distribute-neg-in N/A

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

6. Applied rewrites 10.4%

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

   1. lift-fma.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-neg N/A

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

   4. distribute-lft-in N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. lift-*.f32 N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. sub-neg N/A

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

   11. associate-+l- N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower--.f32 96.2

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

   15. lift-*.f32 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f32 96.2

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

8. Applied rewrites 96.2%

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

9. Final simplification 96.2%

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

Alternative 2: 28.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.8%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 95.6

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

4. Applied rewrites 95.6%

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

6. Applied rewrites 28.5%

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

7. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-log.f32 28.3

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

9. Applied rewrites 28.3%

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

10. Final simplification 28.3%

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

Alternative 3: 5.1% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \left(\left(\mathsf{fma}\left(0.5, u, 1\right) \cdot u\right) \cdot 3\right) \cdot s \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (* (* (fma 0.5 u 1.0) u) 3.0) s))

C

float code(float s, float u) {
	return ((fmaf(0.5f, u, 1.0f) * u) * 3.0f) * s;
}

Julia

function code(s, u)
	return Float32(Float32(Float32(fma(Float32(0.5), u, Float32(1.0)) * u) * Float32(3.0)) * s)
end

Derivation

1. Initial program 95.8%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(u \cdot \left(1 + \frac{1}{2} \cdot u\right) + \log \frac{3}{4}\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\left(1 + \frac{1}{2} \cdot u\right) \cdot u} + \log \frac{3}{4}\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\mathsf{fma}\left(1 + \frac{1}{2} \cdot u, u, \log \frac{3}{4}\right)} \]

   4. +-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot u + 1}, u, \log \frac{3}{4}\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, u, 1\right)}, u, \log \frac{3}{4}\right) \]

   6. lower-log.f32 10.9

      \[\leadsto \left(3 \cdot s\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, u, 1\right), u, \color{blue}{\log 0.75}\right) \]

5. Applied rewrites 10.9%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, u, 1\right), u, \log 0.75\right)} \]

6. Taylor expanded in u around inf

   \[\leadsto \left(3 \cdot s\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{{u}^{2}}\right) \]
7. Step-by-step derivation

8. Applied rewrites 26.1%

   \[\leadsto \left(3 \cdot s\right) \cdot \left(\left(u \cdot u\right) \cdot \color{blue}{0.5}\right) \]
9. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \left(\left(u \cdot u\right) \cdot \frac{1}{2}\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(u \cdot u\right) \cdot \frac{1}{2}\right) \cdot \left(3 \cdot s\right)} \]

   3. lift-*.f32 N/A

      \[\leadsto \left(\left(u \cdot u\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(3 \cdot s\right)} \]

   4. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\left(u \cdot u\right) \cdot \frac{1}{2}\right) \cdot 3\right) \cdot s} \]

   5. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\left(\left(u \cdot u\right) \cdot \frac{1}{2}\right) \cdot 3\right) \cdot s} \]

   6. lower-*.f32 26.1

      \[\leadsto \color{blue}{\left(\left(\left(u \cdot u\right) \cdot 0.5\right) \cdot 3\right)} \cdot s \]

10. Applied rewrites 26.1%

    \[\leadsto \color{blue}{\left(\left(\left(u \cdot u\right) \cdot 0.5\right) \cdot 3\right) \cdot s} \]

11. Taylor expanded in u around inf

    \[\leadsto \left(\left({u}^{2} \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{u}\right)}\right) \cdot 3\right) \cdot s \]
12. Step-by-step derivation

13. Applied rewrites 30.2%

    \[\leadsto \left(\left(\mathsf{fma}\left(0.5, u, 1\right) \cdot \color{blue}{u}\right) \cdot 3\right) \cdot s \]
14. Add Preprocessing

Alternative 4: 5.1% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(0.5, u, 1\right) \cdot u\right) \cdot \left(s \cdot 3\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (* (fma 0.5 u 1.0) u) (* s 3.0)))

C

float code(float s, float u) {
	return (fmaf(0.5f, u, 1.0f) * u) * (s * 3.0f);
}

Julia

function code(s, u)
	return Float32(Float32(fma(Float32(0.5), u, Float32(1.0)) * u) * Float32(s * Float32(3.0)))
end

Derivation

1. Initial program 95.8%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(u \cdot \left(1 + \frac{1}{2} \cdot u\right) + \log \frac{3}{4}\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\left(1 + \frac{1}{2} \cdot u\right) \cdot u} + \log \frac{3}{4}\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\mathsf{fma}\left(1 + \frac{1}{2} \cdot u, u, \log \frac{3}{4}\right)} \]

   4. +-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot u + 1}, u, \log \frac{3}{4}\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, u, 1\right)}, u, \log \frac{3}{4}\right) \]

   6. lower-log.f32 11.0

      \[\leadsto \left(3 \cdot s\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, u, 1\right), u, \color{blue}{\log 0.75}\right) \]

5. Applied rewrites 10.9%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, u, 1\right), u, \log 0.75\right)} \]

6. Taylor expanded in u around inf

   \[\leadsto \left(3 \cdot s\right) \cdot \left({u}^{2} \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{u}\right)}\right) \]
7. Step-by-step derivation

8. Applied rewrites 30.2%

   \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{fma}\left(0.5, u, 1\right) \cdot \color{blue}{u}\right) \]

9. Final simplification 30.3%

   \[\leadsto \left(\mathsf{fma}\left(0.5, u, 1\right) \cdot u\right) \cdot \left(s \cdot 3\right) \]
10. Add Preprocessing

Alternative 5: 26.4% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(u \cdot u\right) \cdot 0.5\right) \cdot s\right) \cdot 3 \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (* (* (* u u) 0.5) s) 3.0))

C

float code(float s, float u) {
	return (((u * u) * 0.5f) * s) * 3.0f;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (((u * u) * 0.5e0) * s) * 3.0e0
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(Float32(u * u) * Float32(0.5)) * s) * Float32(3.0))
end

MATLAB

function tmp = code(s, u)
	tmp = (((u * u) * single(0.5)) * s) * single(3.0);
end

Derivation

1. Initial program 95.8%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(u \cdot \left(1 + \frac{1}{2} \cdot u\right) + \log \frac{3}{4}\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\left(1 + \frac{1}{2} \cdot u\right) \cdot u} + \log \frac{3}{4}\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\mathsf{fma}\left(1 + \frac{1}{2} \cdot u, u, \log \frac{3}{4}\right)} \]

   4. +-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot u + 1}, u, \log \frac{3}{4}\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, u, 1\right)}, u, \log \frac{3}{4}\right) \]

   6. lower-log.f32 11.0

      \[\leadsto \left(3 \cdot s\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, u, 1\right), u, \color{blue}{\log 0.75}\right) \]

5. Applied rewrites 10.8%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, u, 1\right), u, \log 0.75\right)} \]

6. Taylor expanded in u around inf

   \[\leadsto \left(3 \cdot s\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{{u}^{2}}\right) \]
7. Step-by-step derivation

8. Applied rewrites 26.1%

   \[\leadsto \left(3 \cdot s\right) \cdot \left(\left(u \cdot u\right) \cdot \color{blue}{0.5}\right) \]
9. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \left(\left(u \cdot u\right) \cdot \frac{1}{2}\right)} \]

   2. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(3 \cdot s\right)} \cdot \left(\left(u \cdot u\right) \cdot \frac{1}{2}\right) \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{3 \cdot \left(s \cdot \left(\left(u \cdot u\right) \cdot \frac{1}{2}\right)\right)} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\left(s \cdot \left(\left(u \cdot u\right) \cdot \frac{1}{2}\right)\right) \cdot 3} \]

   5. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(s \cdot \left(\left(u \cdot u\right) \cdot \frac{1}{2}\right)\right) \cdot 3} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\left(u \cdot u\right) \cdot \frac{1}{2}\right) \cdot s\right)} \cdot 3 \]

   7. lower-*.f32 26.1

      \[\leadsto \color{blue}{\left(\left(\left(u \cdot u\right) \cdot 0.5\right) \cdot s\right)} \cdot 3 \]

10. Applied rewrites 26.1%

    \[\leadsto \color{blue}{\left(\left(\left(u \cdot u\right) \cdot 0.5\right) \cdot s\right) \cdot 3} \]
11. Add Preprocessing

Alternative 6: 26.4% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \left(\left(u \cdot u\right) \cdot 0.5\right) \cdot \left(s \cdot 3\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (* (* u u) 0.5) (* s 3.0)))

C

float code(float s, float u) {
	return ((u * u) * 0.5f) * (s * 3.0f);
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = ((u * u) * 0.5e0) * (s * 3.0e0)
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(u * u) * Float32(0.5)) * Float32(s * Float32(3.0)))
end

MATLAB

function tmp = code(s, u)
	tmp = ((u * u) * single(0.5)) * (s * single(3.0));
end

Derivation

1. Initial program 95.8%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(u \cdot \left(1 + \frac{1}{2} \cdot u\right) + \log \frac{3}{4}\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\left(1 + \frac{1}{2} \cdot u\right) \cdot u} + \log \frac{3}{4}\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\mathsf{fma}\left(1 + \frac{1}{2} \cdot u, u, \log \frac{3}{4}\right)} \]

   4. +-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot u + 1}, u, \log \frac{3}{4}\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, u, 1\right)}, u, \log \frac{3}{4}\right) \]

   6. lower-log.f32 11.0

      \[\leadsto \left(3 \cdot s\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, u, 1\right), u, \color{blue}{\log 0.75}\right) \]

5. Applied rewrites 10.9%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, u, 1\right), u, \log 0.75\right)} \]

6. Taylor expanded in u around inf

   \[\leadsto \left(3 \cdot s\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{{u}^{2}}\right) \]
7. Step-by-step derivation

8. Applied rewrites 26.1%

   \[\leadsto \left(3 \cdot s\right) \cdot \left(\left(u \cdot u\right) \cdot \color{blue}{0.5}\right) \]

9. Final simplification 26.1%

   \[\leadsto \left(\left(u \cdot u\right) \cdot 0.5\right) \cdot \left(s \cdot 3\right) \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024285 
(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))))))