Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.9% → 95.9%

Time: 8.2s

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.5%

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

3. Final simplification 95.5%

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

Alternative 2: 95.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.5%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 95.2

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

4. Applied rewrites 95.2%

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

5. Final simplification 95.2%

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

Alternative 3: 95.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.5%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 95.2

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

4. Applied rewrites 95.2%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

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

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

   5. metadata-eval N/A

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

   6. div-inv N/A

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

   7. clear-num N/A

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

   8. distribute-neg-frac N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f32 N/A

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

   11. frac-2neg N/A

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

   12. lower-/.f32 N/A

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

   13. metadata-eval N/A

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

   14. lift--.f32 N/A

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

   15. sub-neg N/A

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

   16. distribute-neg-in N/A

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

   19. +-commutative N/A

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

   20. sub-neg N/A

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

   21. lift--.f32 95.2

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

6. Applied rewrites 95.2%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. frac-2neg N/A

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

   5. associate-/r/ N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. lift--.f32 N/A

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

   9. sub-neg N/A

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

   10. distribute-neg-in N/A

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

   11. remove-double-neg N/A

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

   12. +-commutative N/A

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

   13. distribute-lft-in N/A

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

   14. lift-*.f32 N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. associate-+l+ N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. lower-+.f32 N/A

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

   21. metadata-eval 95.1

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

8. Applied rewrites 95.1%

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

9. Final simplification 95.1%

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

Alternative 4: 7.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-3 \cdot s, \log 1.3333333333333333, \left(\mathsf{fma}\left(1.5, u, 3\right) \cdot s\right) \cdot u\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (fma (* -3.0 s) (log 1.3333333333333333) (* (* (fma 1.5 u 3.0) s) u)))

C

float code(float s, float u) {
	return fmaf((-3.0f * s), logf(1.3333333333333333f), ((fmaf(1.5f, u, 3.0f) * s) * u));
}

Julia

function code(s, u)
	return fma(Float32(Float32(-3.0) * s), log(Float32(1.3333333333333333)), Float32(Float32(fma(Float32(1.5), u, Float32(3.0)) * s) * u))
end

Derivation

1. Initial program 95.5%

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

   1. +-lft-identity N/A

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. lift-log.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. inv-pow N/A

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

   7. pow-to-exp N/A

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

   8. rem-log-exp N/A

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

   9. lower-fma.f32 N/A

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

   10. lift--.f32 N/A

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

   11. sub-neg N/A

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

   12. lower-log1p.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)}, -1, \log 1\right) \]

   13. lift-/.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \mathsf{fma}\left(\mathsf{log1p}\left(\mathsf{neg}\left(\color{blue}{\frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right), -1, \log 1\right) \]

   14. distribute-neg-frac2 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \mathsf{fma}\left(\mathsf{log1p}\left(\color{blue}{\frac{u - \frac{1}{4}}{\mathsf{neg}\left(\frac{3}{4}\right)}}\right), -1, \log 1\right) \]

   15. div-inv N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \mathsf{fma}\left(\mathsf{log1p}\left(\color{blue}{\left(u - \frac{1}{4}\right) \cdot \frac{1}{\mathsf{neg}\left(\frac{3}{4}\right)}}\right), -1, \log 1\right) \]

   16. *-commutative N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \mathsf{fma}\left(\mathsf{log1p}\left(\color{blue}{\frac{1}{\mathsf{neg}\left(\frac{3}{4}\right)} \cdot \left(u - \frac{1}{4}\right)}\right), -1, \log 1\right) \]

   17. lower-*.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \mathsf{fma}\left(\mathsf{log1p}\left(\color{blue}{\frac{1}{\mathsf{neg}\left(\frac{3}{4}\right)} \cdot \left(u - \frac{1}{4}\right)}\right), -1, \log 1\right) \]

   18. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \mathsf{fma}\left(\mathsf{log1p}\left(\frac{1}{\color{blue}{\frac{-3}{4}}} \cdot \left(u - \frac{1}{4}\right)\right), -1, \log 1\right) \]

   19. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \mathsf{fma}\left(\mathsf{log1p}\left(\color{blue}{\frac{-4}{3}} \cdot \left(u - \frac{1}{4}\right)\right), -1, \log 1\right) \]

   20. metadata-eval 10.8

       \[\leadsto \left(3 \cdot s\right) \cdot \mathsf{fma}\left(\mathsf{log1p}\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right), -1, \color{blue}{0}\right) \]

4. Applied rewrites 10.8%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{log1p}\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right), -1, 0\right)} \]

5. Taylor expanded in u around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-log.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. distribute-lft-out N/A

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

   13. lower-*.f32 N/A

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

   14. +-commutative N/A

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

   15. lower-fma.f32 7.1

       \[\leadsto \mathsf{fma}\left(-3 \cdot s, \log 1.3333333333333333, \left(s \cdot \color{blue}{\mathsf{fma}\left(1.5, u, 3\right)}\right) \cdot u\right) \]

7. Applied rewrites 7.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-3 \cdot s, \log 1.3333333333333333, \left(s \cdot \mathsf{fma}\left(1.5, u, 3\right)\right) \cdot u\right)} \]

8. Final simplification 7.1%

   \[\leadsto \mathsf{fma}\left(-3 \cdot s, \log 1.3333333333333333, \left(\mathsf{fma}\left(1.5, u, 3\right) \cdot s\right) \cdot u\right) \]
9. Add Preprocessing

Alternative 5: 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 95.5%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 25.9

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

5. Applied rewrites 25.9%

   \[\leadsto \color{blue}{\left(\left(\log 0.75 + u\right) \cdot s\right) \cdot 3} \]
6. 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 95.5%

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

Reproduce

herbie shell --seed 2024271 
(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))))))