Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.4% → 99.3%

Time: 10.5s

Alternatives: 15

Speedup: 11.4×

Specification

\[\left(0 \leq s \land s \leq 256\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 0.25\right)\]

Math

\[\begin{array}{l} \\ s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))

C

float code(float s, float u) {
	return s * logf((1.0f / (1.0f - (4.0f * u))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * log((1.0e0 / (1.0e0 - (4.0e0 * u))))
end function

Julia

function code(s, u)
	return Float32(s * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(4.0) * u)))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * log((single(1.0) / (single(1.0) - (single(4.0) * u))));
end

Initial Program: 61.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))

C

float code(float s, float u) {
	return s * logf((1.0f / (1.0f - (4.0f * u))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * log((1.0e0 / (1.0e0 - (4.0e0 * u))))
end function

Julia

function code(s, u)
	return Float32(s * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(4.0) * u)))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * log((single(1.0) / (single(1.0) - (single(4.0) * u))));
end

Alternative 1: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{log1p}\left(u \cdot 4\right), s, \left(-s\right) \cdot \mathsf{log1p}\left(\left(-16 \cdot u\right) \cdot u\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (fma (log1p (* u 4.0)) s (* (- s) (log1p (* (* -16.0 u) u)))))

C

float code(float s, float u) {
	return fmaf(log1pf((u * 4.0f)), s, (-s * log1pf(((-16.0f * u) * u))));
}

Julia

function code(s, u)
	return fma(log1p(Float32(u * Float32(4.0))), s, Float32(Float32(-s) * log1p(Float32(Float32(Float32(-16.0) * u) * u))))
end

Derivation

1. Initial program 61.4%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 61.4

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

   4. lift-log.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. log-rec N/A

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

   7. lower-neg.f32 N/A

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

   8. lift--.f32 N/A

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

   9. sub-neg N/A

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

   10. lower-log1p.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(4 \cdot u\right)\right)}\right)\right) \cdot s \]

   11. lift-*.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(\color{blue}{4 \cdot u}\right)\right)\right)\right) \cdot s \]

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

       \[\leadsto \left(\mathsf{neg}\left(\mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot u}\right)\right)\right) \cdot s \]

   13. lower-*.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(\mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot u}\right)\right)\right) \cdot s \]

   14. metadata-eval 99.2

       \[\leadsto \left(-\mathsf{log1p}\left(\color{blue}{-4} \cdot u\right)\right) \cdot s \]

4. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\left(-\mathsf{log1p}\left(-4 \cdot u\right)\right) \cdot s} \]

6. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\left(\mathsf{log1p}\left(4 \cdot u\right) - \mathsf{log1p}\left(\left(-16 \cdot u\right) \cdot u\right)\right)} \cdot s \]
7. Step-by-step derivation

   1. lift-log1p.f32 N/A

      \[\leadsto \left(\mathsf{log1p}\left(4 \cdot u\right) - \color{blue}{\log \left(1 + \left(-16 \cdot u\right) \cdot u\right)}\right) \cdot s \]

   2. lower-log.f32 N/A

      \[\leadsto \left(\mathsf{log1p}\left(4 \cdot u\right) - \color{blue}{\log \left(1 + \left(-16 \cdot u\right) \cdot u\right)}\right) \cdot s \]

   3. +-commutative N/A

      \[\leadsto \left(\mathsf{log1p}\left(4 \cdot u\right) - \log \color{blue}{\left(\left(-16 \cdot u\right) \cdot u + 1\right)}\right) \cdot s \]

   4. lift-*.f32 N/A

      \[\leadsto \left(\mathsf{log1p}\left(4 \cdot u\right) - \log \left(\color{blue}{\left(-16 \cdot u\right) \cdot u} + 1\right)\right) \cdot s \]

   5. lift-*.f32 N/A

      \[\leadsto \left(\mathsf{log1p}\left(4 \cdot u\right) - \log \left(\color{blue}{\left(-16 \cdot u\right)} \cdot u + 1\right)\right) \cdot s \]

   6. associate-*l* N/A

      \[\leadsto \left(\mathsf{log1p}\left(4 \cdot u\right) - \log \left(\color{blue}{-16 \cdot \left(u \cdot u\right)} + 1\right)\right) \cdot s \]

   7. lift-*.f32 N/A

      \[\leadsto \left(\mathsf{log1p}\left(4 \cdot u\right) - \log \left(-16 \cdot \color{blue}{\left(u \cdot u\right)} + 1\right)\right) \cdot s \]

   8. *-commutative N/A

      \[\leadsto \left(\mathsf{log1p}\left(4 \cdot u\right) - \log \left(\color{blue}{\left(u \cdot u\right) \cdot -16} + 1\right)\right) \cdot s \]

   9. lower-fma.f32 89.3

      \[\leadsto \left(\mathsf{log1p}\left(4 \cdot u\right) - \log \color{blue}{\left(\mathsf{fma}\left(u \cdot u, -16, 1\right)\right)}\right) \cdot s \]

8. Applied rewrites 89.3%

   \[\leadsto \left(\mathsf{log1p}\left(4 \cdot u\right) - \color{blue}{\log \left(\mathsf{fma}\left(u \cdot u, -16, 1\right)\right)}\right) \cdot s \]
9. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(\mathsf{log1p}\left(4 \cdot u\right) - \log \left(\mathsf{fma}\left(u \cdot u, -16, 1\right)\right)\right) \cdot s} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{s \cdot \left(\mathsf{log1p}\left(4 \cdot u\right) - \log \left(\mathsf{fma}\left(u \cdot u, -16, 1\right)\right)\right)} \]

   3. lift--.f32 N/A

      \[\leadsto s \cdot \color{blue}{\left(\mathsf{log1p}\left(4 \cdot u\right) - \log \left(\mathsf{fma}\left(u \cdot u, -16, 1\right)\right)\right)} \]

   4. sub-neg N/A

      \[\leadsto s \cdot \color{blue}{\left(\mathsf{log1p}\left(4 \cdot u\right) + \left(\mathsf{neg}\left(\log \left(\mathsf{fma}\left(u \cdot u, -16, 1\right)\right)\right)\right)\right)} \]

   5. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\mathsf{log1p}\left(4 \cdot u\right) \cdot s + \left(\mathsf{neg}\left(\log \left(\mathsf{fma}\left(u \cdot u, -16, 1\right)\right)\right)\right) \cdot s} \]

   6. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{log1p}\left(4 \cdot u\right), s, \left(\mathsf{neg}\left(\log \left(\mathsf{fma}\left(u \cdot u, -16, 1\right)\right)\right)\right) \cdot s\right)} \]

   7. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(4 \cdot u\right), s, \color{blue}{\left(\mathsf{neg}\left(\log \left(\mathsf{fma}\left(u \cdot u, -16, 1\right)\right)\right)\right) \cdot s}\right) \]

   8. lower-neg.f32 89.2

      \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(4 \cdot u\right), s, \color{blue}{\left(-\log \left(\mathsf{fma}\left(u \cdot u, -16, 1\right)\right)\right)} \cdot s\right) \]

   9. lift-log.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(4 \cdot u\right), s, \left(\mathsf{neg}\left(\color{blue}{\log \left(\mathsf{fma}\left(u \cdot u, -16, 1\right)\right)}\right)\right) \cdot s\right) \]

   10. lift-fma.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(4 \cdot u\right), s, \left(\mathsf{neg}\left(\log \color{blue}{\left(\left(u \cdot u\right) \cdot -16 + 1\right)}\right)\right) \cdot s\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(4 \cdot u\right), s, \left(\mathsf{neg}\left(\log \color{blue}{\left(1 + \left(u \cdot u\right) \cdot -16\right)}\right)\right) \cdot s\right) \]

   12. lower-log1p.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(4 \cdot u\right), s, \left(\mathsf{neg}\left(\color{blue}{\mathsf{log1p}\left(\left(u \cdot u\right) \cdot -16\right)}\right)\right) \cdot s\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(4 \cdot u\right), s, \left(\mathsf{neg}\left(\mathsf{log1p}\left(\color{blue}{-16 \cdot \left(u \cdot u\right)}\right)\right)\right) \cdot s\right) \]

   14. lift-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(4 \cdot u\right), s, \left(\mathsf{neg}\left(\mathsf{log1p}\left(-16 \cdot \color{blue}{\left(u \cdot u\right)}\right)\right)\right) \cdot s\right) \]

   15. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(4 \cdot u\right), s, \left(\mathsf{neg}\left(\mathsf{log1p}\left(\color{blue}{\left(-16 \cdot u\right) \cdot u}\right)\right)\right) \cdot s\right) \]

   16. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(4 \cdot u\right), s, \left(\mathsf{neg}\left(\mathsf{log1p}\left(\color{blue}{\left(-16 \cdot u\right) \cdot u}\right)\right)\right) \cdot s\right) \]

   17. lower-*.f32 99.3

       \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(4 \cdot u\right), s, \left(-\mathsf{log1p}\left(\color{blue}{\left(-16 \cdot u\right)} \cdot u\right)\right) \cdot s\right) \]

10. Applied rewrites 99.3%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{log1p}\left(4 \cdot u\right), s, \left(-\mathsf{log1p}\left(\left(-16 \cdot u\right) \cdot u\right)\right) \cdot s\right)} \]

11. Final simplification 99.3%

    \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(u \cdot 4\right), s, \left(-s\right) \cdot \mathsf{log1p}\left(\left(-16 \cdot u\right) \cdot u\right)\right) \]
12. Add Preprocessing

Alternative 2: 99.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \mathsf{log1p}\left(-4 \cdot u\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (- s) (log1p (* -4.0 u))))

C

float code(float s, float u) {
	return -s * log1pf((-4.0f * u));
}

Julia

function code(s, u)
	return Float32(Float32(-s) * log1p(Float32(Float32(-4.0) * u)))
end

Derivation

1. Initial program 61.4%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 61.4

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

   4. lift-log.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. log-rec N/A

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

   7. lower-neg.f32 N/A

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

   8. lift--.f32 N/A

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

   9. sub-neg N/A

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

   10. lower-log1p.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(4 \cdot u\right)\right)}\right)\right) \cdot s \]

   11. lift-*.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(\color{blue}{4 \cdot u}\right)\right)\right)\right) \cdot s \]

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

       \[\leadsto \left(\mathsf{neg}\left(\mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot u}\right)\right)\right) \cdot s \]

   13. lower-*.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(\mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot u}\right)\right)\right) \cdot s \]

   14. metadata-eval 99.4

       \[\leadsto \left(-\mathsf{log1p}\left(\color{blue}{-4} \cdot u\right)\right) \cdot s \]

4. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\left(-\mathsf{log1p}\left(-4 \cdot u\right)\right) \cdot s} \]

5. Final simplification 99.4%

   \[\leadsto \left(-s\right) \cdot \mathsf{log1p}\left(-4 \cdot u\right) \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024230 
(FPCore (s u)
  :name "Disney BSSRDF, sample scattering profile, lower"
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (<= 2.328306437e-10 u) (<= u 0.25)))
  (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))