Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.9% → 95.9%

Time: 8.1s

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 95.9%

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

Alternative 2: 95.7% 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 95.9%

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

   1. lift--.f32 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lower-+.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. distribute-neg-frac2 N/A

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

   7. div-inv N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval 95.6

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

4. Applied rewrites 95.6%

   \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{-1.3333333333333333 \cdot \left(u - 0.25\right) + 1}}\right) \]
5. 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 95.9%

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

   1. lift--.f32 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lower-+.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. distribute-neg-frac2 N/A

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

   7. div-inv N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval 95.6

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

4. Applied rewrites 95.6%

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

   1. lift-*.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. metadata-eval N/A

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

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

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

   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. lower-/.f32 95.6

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

6. Applied rewrites 95.6%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. associate-/r/ N/A

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

   6. metadata-eval N/A

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

   7. lift--.f32 N/A

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

   8. sub-neg N/A

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

   9. distribute-rgt-in N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. +-commutative N/A

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

   13. associate-+r+ N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lower-+.f32 N/A

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

   17. metadata-eval N/A

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

   18. *-commutative N/A

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

   19. lower-*.f32 95.3

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

8. Applied rewrites 95.3%

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

Alternative 4: 9.6% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 95.9%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. lower-log.f32 11.1

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

5. Applied rewrites 11.1%

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

6. Taylor expanded in u around inf

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

8. Applied rewrites 26.3%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 26.3

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

10. Applied rewrites 26.3%

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

11. Taylor expanded in u around inf

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

13. Applied rewrites 29.9%

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

Alternative 5: 9.6% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 95.9%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. lower-log.f32 10.9

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

5. Applied rewrites 11.0%

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

6. Taylor expanded in u around inf

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

8. Applied rewrites 26.3%

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

9. Taylor expanded in u around inf

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

11. Applied rewrites 29.9%

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

Alternative 6: 27.6% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(1.5 + u\right) \cdot u\right) \cdot u\right) \cdot s \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (* (* (+ 1.5 u) u) u) s))

C

float code(float s, float u) {
	return (((1.5f + u) * u) * u) * s;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (((1.5e0 + u) * u) * u) * s
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(Float32(Float32(1.5) + u) * u) * u) * s)
end

MATLAB

function tmp = code(s, u)
	tmp = (((single(1.5) + u) * u) * u) * s;
end

Derivation

1. Initial program 95.9%

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

3. Taylor expanded in u around 0

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

   1. distribute-rgt-in N/A

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

   2. associate-+r+ N/A

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

   3. +-commutative N/A

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

   4. associate-*r* N/A

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

   5. distribute-lft-out N/A

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

   6. *-commutative N/A

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

   7. distribute-lft-out N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. distribute-lft-out N/A

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

   13. unpow2 N/A

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

   14. associate-*l* N/A

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

   15. distribute-lft-out N/A

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

5. Applied rewrites 14.5%

   \[\leadsto \color{blue}{s \cdot \mathsf{fma}\left(\log 0.75 + u, 3, \left(1.5 + u\right) \cdot \left(u \cdot u\right)\right)} \]
6. Step-by-step derivation

7. Applied rewrites 15.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(1.5 + u\right) \cdot u, u, \left(\log 0.75 + u\right) \cdot 3\right) \cdot s} \]

8. Taylor expanded in u around inf

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

10. Applied rewrites 27.6%

    \[\leadsto \left(\left(\left(1.5 + u\right) \cdot u\right) \cdot u\right) \cdot s \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024313 
(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))))))