Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.9% → 95.9%

Time: 8.4s

Alternatives: 6

Speedup: 1.0×

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: 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

Derivation

1. Initial program 96.3%

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

Alternative 2: 95.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 96.3%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 96.1

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

4. Applied rewrites 96.1%

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

Alternative 3: 95.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 96.3%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 96.1

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

4. Applied rewrites 96.1%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

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

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

   7. metadata-eval N/A

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

   8. div-inv N/A

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

   9. lift--.f32 N/A

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

   10. sub-neg N/A

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

   11. div-sub N/A

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

   12. div-inv N/A

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

   13. metadata-eval N/A

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

   14. *-commutative N/A

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

   15. sub-neg N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. +-commutative N/A

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

   19. associate--r+ N/A

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

   20. metadata-eval N/A

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

   21. lower--.f32 N/A

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

   22. lower-*.f32 95.7

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

6. Applied rewrites 95.7%

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

Alternative 4: 28.1% accurate, 1.3× speedup

Math

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

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(Float32(-3.0) * 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 96.3%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 96.1

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

4. Applied rewrites 96.1%

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

6. Applied rewrites 10.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(s, 3 \cdot \left(-\mathsf{log1p}\left(\mathsf{fma}\left(1.3333333333333333, u, -0.3333333333333333\right)\right)\right), 0\right)} \]

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 28.5

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

9. Applied rewrites 28.5%

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

11. Applied rewrites 28.5%

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

Alternative 5: 28.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 96.3%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 96.1

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

4. Applied rewrites 96.1%

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

6. Applied rewrites 10.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(s, 3 \cdot \left(-\mathsf{log1p}\left(\mathsf{fma}\left(1.3333333333333333, u, -0.3333333333333333\right)\right)\right), 0\right)} \]

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 28.5

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

9. Applied rewrites 28.5%

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

11. Applied rewrites 28.5%

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

Alternative 6: 7.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 96.3%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 7.5

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

5. Applied rewrites 7.5%

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

Reproduce

herbie shell --seed 2024322 
(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))))))