Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.9% → 96.7%

Time: 8.8s

Alternatives: 8

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: 96.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

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))) * s) * -3.0e0
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 95.7%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 96.5

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

4. Applied rewrites 96.5%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

6. Applied rewrites 96.5%

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

7. Final simplification 96.5%

   \[\leadsto \left(\log \left(1 - \frac{u - 0.25}{0.75}\right) \cdot s\right) \cdot \left(-3\right) \]
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 95.7%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 96.5

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

4. Applied rewrites 96.5%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 96.5

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

   4. lift-*.f32 N/A

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

   5. lift-log.f32 N/A

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

   6. log-pow-rev N/A

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

   7. lift-pow.f32 N/A

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

   8. pow-pow N/A

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

   9. metadata-eval N/A

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

   10. inv-pow N/A

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

   11. log-rec N/A

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

   12. lower-neg.f32 N/A

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

   13. lower-log.f32 96.5

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

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: 95.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(\left(-s\right) \cdot 3\right) \cdot \log \left(\left(1 - \frac{u}{0.75}\right) + 0.3333333333333333\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* (- s) 3.0) (log (+ (- 1.0 (/ u 0.75)) 0.3333333333333333))))

C

float code(float s, float u) {
	return (-s * 3.0f) * logf(((1.0f - (u / 0.75f)) + 0.3333333333333333f));
}

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 / 0.75e0)) + 0.3333333333333333e0))
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(-s) * Float32(3.0)) * log(Float32(Float32(Float32(1.0) - Float32(u / Float32(0.75))) + Float32(0.3333333333333333))))
end

MATLAB

function tmp = code(s, u)
	tmp = (-s * single(3.0)) * log(((single(1.0) - (u / single(0.75))) + single(0.3333333333333333)));
end

Derivation

1. Initial program 95.7%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 96.5

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

4. Applied rewrites 96.5%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 96.5

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

   4. lift-*.f32 N/A

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

   5. lift-log.f32 N/A

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

   6. log-pow-rev N/A

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

   7. lift-pow.f32 N/A

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

   8. pow-pow N/A

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

   9. metadata-eval N/A

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

   10. inv-pow N/A

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

   11. log-rec N/A

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

   12. lower-neg.f32 N/A

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

   13. lower-log.f32 96.5

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

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. Step-by-step derivation

   1. lift--.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift--.f32 N/A

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

   4. div-sub N/A

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

   5. lift-/.f32 N/A

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

   6. metadata-eval N/A

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

   7. associate-+l- N/A

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

   8. lift--.f32 N/A

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

   9. lift-+.f32 95.3

      \[\leadsto \left(s \cdot 3\right) \cdot \left(-\log \color{blue}{\left(\left(1 - \frac{u}{0.75}\right) + 0.3333333333333333\right)}\right) \]

8. Applied rewrites 95.3%

   \[\leadsto \left(s \cdot 3\right) \cdot \left(-\log \color{blue}{\left(\left(1 - \frac{u}{0.75}\right) + 0.3333333333333333\right)}\right) \]

9. Final simplification 95.3%

   \[\leadsto \left(\left(-s\right) \cdot 3\right) \cdot \log \left(\left(1 - \frac{u}{0.75}\right) + 0.3333333333333333\right) \]
10. Add Preprocessing

Alternative 4: 94.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

float code(float s, float u) {
	return (-s * logf(((u / -0.75f) - -1.3333333333333333f))) * 3.0f;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (-s * log(((u / (-0.75e0)) - (-1.3333333333333333e0)))) * 3.0e0
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 95.7%

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

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

   5. lower-*.f32 N/A

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

4. Applied rewrites 32.2%

   \[\leadsto \color{blue}{\left(\left(-s\right) \cdot \mathsf{log1p}\left(\frac{u - 0.25}{-0.75}\right)\right) \cdot 3} \]

6. Applied rewrites 94.7%

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

Alternative 5: 94.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

float code(float s, float u) {
	return (-3.0f * s) * logf(((u / -0.75f) - -1.3333333333333333f));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = ((-3.0e0) * s) * log(((u / (-0.75e0)) - (-1.3333333333333333e0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 95.7%

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

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

   5. lower-*.f32 N/A

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

4. Applied rewrites 32.8%

   \[\leadsto \color{blue}{\left(\left(-s\right) \cdot \mathsf{log1p}\left(\frac{u - 0.25}{-0.75}\right)\right) \cdot 3} \]

6. Applied rewrites 94.7%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. lower-*.f32 94.6

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

   8. lift-*.f32 N/A

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

   9. lift-neg.f32 N/A

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

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

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

   11. *-commutative N/A

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

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

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

   13. lower-*.f32 N/A

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

   14. metadata-eval 94.6

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

8. Applied rewrites 94.6%

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

Alternative 6: 25.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.7%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. lower-log.f32 25.5

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

5. Applied rewrites 25.5%

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

Alternative 7: 25.8% 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 95.7%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 25.5

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

5. Applied rewrites 25.5%

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

Alternative 8: 10.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \log 1 \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (log 1.0))

C

float code(float s, float u) {
	return logf(1.0f);
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = log(1.0e0)
end function

Julia

function code(s, u)
	return log(Float32(1.0))
end

MATLAB

function tmp = code(s, u)
	tmp = log(single(1.0));
end

Derivation

1. Initial program 95.7%

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

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

   5. lower-*.f32 N/A

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

4. Applied rewrites 32.5%

   \[\leadsto \color{blue}{\left(\left(-s\right) \cdot \mathsf{log1p}\left(\frac{u - 0.25}{-0.75}\right)\right) \cdot 3} \]

6. Applied rewrites 25.6%

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

7. Taylor expanded in s around 0

   \[\leadsto \log \color{blue}{1} \]
8. Step-by-step derivation

9. Applied rewrites 10.6%

   \[\leadsto \log \color{blue}{1} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024342 
(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))))))