Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.9% → 96.6%

Time: 9.0s

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: 96.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.4%

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

   1. lift-log.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift--.f32 N/A

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

   4. flip-- N/A

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

   5. associate-/r/ N/A

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

   6. flip-+ N/A

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

   7. lift--.f32 N/A

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

   8. frac-times N/A

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

   9. log-div N/A

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

   10. lower--.f32 N/A

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

4. Applied rewrites 95.9%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \left(1 \cdot \left(1 - {\left(\frac{u - 0.25}{0.75}\right)}^{2}\right)\right) - \log \left(\left(1 - {\left(\frac{u - 0.25}{0.75}\right)}^{2}\right) \cdot \left(1 - \frac{u - 0.25}{0.75}\right)\right)\right)} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

6. Applied rewrites 96.5%

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

7. Final simplification 96.5%

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

Alternative 2: 96.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

float code(float s, float u) {
	return (-s * 3.0f) * logf((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 = (-s * 3.0e0) * log((1.0e0 - ((u - 0.25e0) / 0.75e0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 95.4%

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

   1. lift-log.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift--.f32 N/A

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

   4. flip-- N/A

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

   5. associate-/r/ N/A

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

   6. flip-+ N/A

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

   7. lift--.f32 N/A

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

   8. frac-times N/A

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

   9. log-div N/A

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

   10. lower--.f32 N/A

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

4. Applied rewrites 95.9%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \left(1 \cdot \left(1 - {\left(\frac{u - 0.25}{0.75}\right)}^{2}\right)\right) - \log \left(\left(1 - {\left(\frac{u - 0.25}{0.75}\right)}^{2}\right) \cdot \left(1 - \frac{u - 0.25}{0.75}\right)\right)\right)} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 95.9

      \[\leadsto \color{blue}{\left(s \cdot 3\right)} \cdot \left(\log \left(1 \cdot \left(1 - {\left(\frac{u - 0.25}{0.75}\right)}^{2}\right)\right) - \log \left(\left(1 - {\left(\frac{u - 0.25}{0.75}\right)}^{2}\right) \cdot \left(1 - \frac{u - 0.25}{0.75}\right)\right)\right) \]

   4. lift--.f32 N/A

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

   5. lift-log.f32 N/A

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

   6. lift-log.f32 N/A

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

   7. diff-log N/A

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

   8. lift-*.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. times-frac N/A

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

6. Applied rewrites 96.5%

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

7. Final simplification 96.5%

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

Alternative 3: 94.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(\log \left(1.3333333333333333 - \frac{u}{0.75}\right) \cdot -3\right) \cdot s \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* (log (- 1.3333333333333333 (/ u 0.75))) -3.0) s))

C

float code(float s, float u) {
	return (logf((1.3333333333333333f - (u / 0.75f))) * -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 / 0.75e0))) * (-3.0e0)) * s
end function

Julia

function code(s, u)
	return Float32(Float32(log(Float32(Float32(1.3333333333333333) - Float32(u / Float32(0.75)))) * Float32(-3.0)) * s)
end

MATLAB

function tmp = code(s, u)
	tmp = (log((single(1.3333333333333333) - (u / single(0.75)))) * single(-3.0)) * s;
end

Derivation

1. Initial program 95.4%

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

   1. lift-log.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. inv-pow N/A

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

   4. sqr-pow N/A

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

   5. pow-prod-down N/A

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

   6. log-pow N/A

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

   7. lower-*.f32 N/A

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

   8. metadata-eval N/A

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

   9. lower-log.f32 N/A

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

   10. pow2 N/A

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

   11. lower-pow.f32 96.0

       \[\leadsto \left(3 \cdot s\right) \cdot \left(-0.5 \cdot \log \color{blue}{\left({\left(1 - \frac{u - 0.25}{0.75}\right)}^{2}\right)}\right) \]

4. Applied rewrites 96.0%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(-0.5 \cdot \log \left({\left(1 - \frac{u - 0.25}{0.75}\right)}^{2}\right)\right)} \]

6. Applied rewrites 94.5%

   \[\leadsto \color{blue}{\left(\left(-\log \left(1.3333333333333333 - \frac{u}{0.75}\right)\right) \cdot 3\right) \cdot s} \]
7. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-neg.f32 N/A

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

   3. distribute-lft-neg-out N/A

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

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

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

   5. lower-*.f32 N/A

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

   6. metadata-eval 94.5

      \[\leadsto \left(\log \left(1.3333333333333333 - \frac{u}{0.75}\right) \cdot \color{blue}{-3}\right) \cdot s \]

8. Applied rewrites 94.5%

   \[\leadsto \color{blue}{\left(\log \left(1.3333333333333333 - \frac{u}{0.75}\right) \cdot -3\right)} \cdot s \]
9. Add Preprocessing

Alternative 4: 25.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.4%

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

3. Taylor expanded in u around 0

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

   1. distribute-lft-out N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. distribute-lft-out N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f32 N/A

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

   9. lower-log.f32 25.0

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

5. Applied rewrites 25.0%

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

Alternative 5: 23.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ s \cdot \log \left(\mathsf{fma}\left(1.265625, u, 0.421875\right)\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (log (fma 1.265625 u 0.421875))))

C

float code(float s, float u) {
	return s * logf(fmaf(1.265625f, u, 0.421875f));
}

Julia

function code(s, u)
	return Float32(s * log(fma(Float32(1.265625), u, Float32(0.421875))))
end

Derivation

1. Initial program 95.4%

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

   1. lift-log.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. inv-pow N/A

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

   4. sqr-pow N/A

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

   5. pow-prod-down N/A

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

   6. log-pow N/A

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

   7. lower-*.f32 N/A

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

   8. metadata-eval N/A

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

   9. lower-log.f32 N/A

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

   10. pow2 N/A

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

   11. lower-pow.f32 96.0

       \[\leadsto \left(3 \cdot s\right) \cdot \left(-0.5 \cdot \log \color{blue}{\left({\left(1 - \frac{u - 0.25}{0.75}\right)}^{2}\right)}\right) \]

4. Applied rewrites 96.0%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(-0.5 \cdot \log \left({\left(1 - \frac{u - 0.25}{0.75}\right)}^{2}\right)\right)} \]

5. Taylor expanded in u around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f32 93.9

      \[\leadsto \left(3 \cdot s\right) \cdot \left(-0.5 \cdot \log \left({\left(\left(\color{blue}{\frac{1.3333333333333333}{u}} - 1.3333333333333333\right) \cdot u\right)}^{2}\right)\right) \]

7. Applied rewrites 93.9%

   \[\leadsto \left(3 \cdot s\right) \cdot \left(-0.5 \cdot \log \left({\color{blue}{\left(\left(\frac{1.3333333333333333}{u} - 1.3333333333333333\right) \cdot u\right)}}^{2}\right)\right) \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-log.f32 N/A

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

   8. log-pow-rev N/A

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

   9. log-pow-rev N/A

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

   10. lower-log.f32 N/A

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

   11. lower-pow.f32 N/A

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

9. Applied rewrites 94.1%

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

10. Taylor expanded in u around 0

    \[\leadsto s \cdot \log \color{blue}{\left(\frac{27}{64} + \frac{81}{64} \cdot u\right)} \]
11. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto s \cdot \log \color{blue}{\left(\frac{81}{64} \cdot u + \frac{27}{64}\right)} \]

    2. lower-fma.f32 7.0

       \[\leadsto s \cdot \log \color{blue}{\left(\mathsf{fma}\left(1.265625, u, 0.421875\right)\right)} \]

12. Applied rewrites 7.5%

    \[\leadsto s \cdot \log \color{blue}{\left(\mathsf{fma}\left(1.265625, u, 0.421875\right)\right)} \]
13. Add Preprocessing

Alternative 6: 7.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \log 0.421875 \cdot s \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (log 0.421875) s))

C

float code(float s, float u) {
	return logf(0.421875f) * s;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = log(0.421875e0) * s
end function

Julia

function code(s, u)
	return Float32(log(Float32(0.421875)) * s)
end

MATLAB

function tmp = code(s, u)
	tmp = log(single(0.421875)) * s;
end

Derivation

1. Initial program 95.4%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 7.6

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

5. Applied rewrites 7.6%

   \[\leadsto \color{blue}{\left(\log 0.75 \cdot s\right) \cdot 3} \]
6. Step-by-step derivation

7. Applied rewrites 7.6%

   \[\leadsto \color{blue}{\log 0.421875 \cdot s} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024339 
(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))))))