Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.9% → 96.1%

Time: 7.3s

Alternatives: 3

Speedup: 1.2×

Specification

\[\left(0 \leq s \land s \leq 256\right) \land \left(0.25 \leq u \land u \leq 1\right)\]

Math

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

FPCore

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

C

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

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (3.0e0 * s) * log((1.0e0 / (1.0e0 - ((u - 0.25e0) / 0.75e0))))
end function

Julia

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

MATLAB

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

Initial Program: 95.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (3.0e0 * s) * log((1.0e0 / (1.0e0 - ((u - 0.25e0) / 0.75e0))))
end function

Julia

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

MATLAB

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

Alternative 1: 96.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.7%

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

3. Taylor expanded in u around 0

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

5. Applied rewrites 7.4%

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

6. Taylor expanded in s around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. cancel-sign-sub-inv N/A

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

   4. metadata-eval N/A

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

   5. associate-*l* N/A

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

   6. log-rec N/A

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

   7. neg-mul-1 N/A

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

   8. associate-*r* N/A

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

   9. metadata-eval N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. metadata-eval N/A

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

   15. cancel-sign-sub-inv N/A

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

   16. lower-log.f32 N/A

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

8. Applied rewrites 3.7%

   \[\leadsto \color{blue}{\left(-3 \cdot s\right) \cdot \log \left(\mathsf{fma}\left(u, -1.3333333333333333, 1.3333333333333333\right)\right)} \]
9. Step-by-step derivation

10. Applied rewrites 96.2%

    \[\leadsto \left(-3 \cdot s\right) \cdot \log \left(-1.3333333333333333 \cdot u + 1.3333333333333333\right) \]

11. Final simplification 96.2%

    \[\leadsto \log \left(1.3333333333333333 + -1.3333333333333333 \cdot u\right) \cdot \left(-3 \cdot s\right) \]
12. Add Preprocessing

Alternative 2: 28.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.7%

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

   1. lift-*.f32 N/A

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

   2. +-lft-identity N/A

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

   3. metadata-eval N/A

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

   4. distribute-lft-in N/A

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

   5. lift-*.f32 N/A

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

   6. lower-fma.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. metadata-eval 7.0

       \[\leadsto \mathsf{fma}\left(s \cdot 3, \color{blue}{0}, \left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right)\right) \]

   11. lift-*.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. associate-*l* N/A

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

   14. *-commutative N/A

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

   15. associate-*r* N/A

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

   16. lower-*.f32 N/A

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

4. Applied rewrites 33.4%

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

6. Applied rewrites 10.5%

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

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 28.1

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

9. Applied rewrites 28.1%

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

Alternative 3: 28.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \left(\log 1.5 \cdot 3\right) \cdot s \end{array} \]

FPCore

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

C

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

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (log(1.5e0) * 3.0e0) * s
end function

Julia

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

MATLAB

function tmp = code(s, u)
	tmp = (log(single(1.5)) * single(3.0)) * s;
end

Derivation

1. Initial program 95.7%

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

   1. lift--.f32 N/A

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

   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 95.7

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

4. Applied rewrites 95.7%

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

6. Applied rewrites 10.4%

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

7. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-log.f32 28.1

      \[\leadsto \left(\color{blue}{\log 1.5} \cdot s\right) \cdot 3 \]

9. Applied rewrites 28.1%

   \[\leadsto \color{blue}{\left(\log 1.5 \cdot s\right) \cdot 3} \]
10. Step-by-step derivation

11. Applied rewrites 28.1%

    \[\leadsto \left(\log 1.5 \cdot 3\right) \cdot \color{blue}{s} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024243 
(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))))))