Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.8% → 95.8%

Time: 8.1s

Alternatives: 7

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: 95.8% 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: 95.8% 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 95.6%

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

3. Final simplification 95.6%

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

Alternative 2: 95.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float s, float u) {
	return logf((1.0f / (1.0f + (-1.3333333333333333f * (u - 0.25f))))) * (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 + ((-1.3333333333333333e0) * (u - 0.25e0))))) * (s * 3.0e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 95.6%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 95.4

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

4. Applied rewrites 95.4%

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

5. Final simplification 95.4%

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

Alternative 3: 36.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(1.5 + u\right) \cdot u\right) \cdot u + \left(\log 0.75 + u\right) \cdot 3\right) \cdot s \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (+ (* (* (+ 1.5 u) u) u) (* (+ (log 0.75) u) 3.0)) s))

C

float code(float s, float u) {
	return ((((1.5f + u) * u) * u) + ((logf(0.75f) + u) * 3.0f)) * s;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = ((((1.5e0 + u) * u) * u) + ((log(0.75e0) + u) * 3.0e0)) * s
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(Float32(Float32(Float32(1.5) + u) * u) * u) + Float32(Float32(log(Float32(0.75)) + u) * Float32(3.0))) * s)
end

MATLAB

function tmp = code(s, u)
	tmp = ((((single(1.5) + u) * u) * u) + ((log(single(0.75)) + u) * single(3.0))) * s;
end

Derivation

1. Initial program 95.6%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 7.3

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

5. Applied rewrites 7.3%

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

6. Taylor expanded in u around 0

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

8. Applied rewrites 14.1%

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

10. Applied rewrites 36.6%

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

11. Final simplification 36.6%

    \[\leadsto \left(\left(\left(1.5 + u\right) \cdot u\right) \cdot u + \left(\log 0.75 + u\right) \cdot 3\right) \cdot s \]
12. Add Preprocessing

Alternative 4: 11.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ -3 \cdot \left(\mathsf{log1p}\left(0.3333333333333333 + -1.3333333333333333 \cdot u\right) \cdot s\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* -3.0 (* (log1p (+ 0.3333333333333333 (* -1.3333333333333333 u))) s)))

C

float code(float s, float u) {
	return -3.0f * (log1pf((0.3333333333333333f + (-1.3333333333333333f * u))) * s);
}

Julia

function code(s, u)
	return Float32(Float32(-3.0) * Float32(log1p(Float32(Float32(0.3333333333333333) + Float32(Float32(-1.3333333333333333) * u))) * s))
end

Derivation

1. Initial program 95.6%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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-rgt-in N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. lower-fma.f32 N/A

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

   8. metadata-eval 7.5

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 7.5

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. associate-*l* N/A

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

   15. *-commutative N/A

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

   16. associate-*r* N/A

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

   17. lower-*.f32 N/A

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

4. Applied rewrites 34.1%

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

   1. lift-fma.f32 N/A

      \[\leadsto \color{blue}{0 \cdot \left(s \cdot 3\right) + \left(-3 \cdot \mathsf{log1p}\left(\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right)\right)\right) \cdot s} \]

   2. mul0-lft N/A

      \[\leadsto \color{blue}{0} + \left(-3 \cdot \mathsf{log1p}\left(\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right)\right)\right) \cdot s \]

   3. +-lft-identity 9.3

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 34.7

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

   10. lift-*.f32 N/A

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

   11. lift--.f32 N/A

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

   12. sub-neg N/A

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

   13. distribute-lft-in N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. lower-fma.f32 N/A

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

   18. metadata-eval 7.4

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

6. Applied rewrites 7.3%

   \[\leadsto \color{blue}{\left(\mathsf{log1p}\left(\mathsf{fma}\left(-1.3333333333333333, u, 0.3333333333333333\right)\right) \cdot s\right) \cdot -3} \]
7. Step-by-step derivation

   1. lift-fma.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-+.f32 34.7

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

   5. *-lft-identity N/A

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

   6. lift-+.f32 N/A

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

   7. distribute-lft-in N/A

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

   8. metadata-eval N/A

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

   9. lower-+.f32 N/A

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

   10. lower-*.f32 34.7

       \[\leadsto \left(\mathsf{log1p}\left(\color{blue}{1 \cdot \left(u \cdot -1.3333333333333333\right)} + 0.3333333333333333\right) \cdot s\right) \cdot -3 \]

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 34.7

       \[\leadsto \left(\mathsf{log1p}\left(1 \cdot \color{blue}{\left(-1.3333333333333333 \cdot u\right)} + 0.3333333333333333\right) \cdot s\right) \cdot -3 \]

8. Applied rewrites 34.7%

   \[\leadsto \left(\mathsf{log1p}\left(\color{blue}{1 \cdot \left(-1.3333333333333333 \cdot u\right) + 0.3333333333333333}\right) \cdot s\right) \cdot -3 \]

9. Final simplification 34.7%

   \[\leadsto -3 \cdot \left(\mathsf{log1p}\left(0.3333333333333333 + -1.3333333333333333 \cdot u\right) \cdot s\right) \]
10. Add Preprocessing

Alternative 5: 26.4% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (((u * u) * 0.5e0) * s) * 3.0e0
end function

Julia

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

MATLAB

function tmp = code(s, u)
	tmp = (((u * u) * single(0.5)) * s) * single(3.0);
end

Derivation

1. Initial program 95.6%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 10.8

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

5. Applied rewrites 10.8%

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

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 26.5%

   \[\leadsto \left(3 \cdot s\right) \cdot \left(\left(u \cdot u\right) \cdot \color{blue}{0.5}\right) \]
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. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 26.5

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

10. Applied rewrites 26.5%

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

Alternative 6: 26.4% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (((u * u) * 0.5e0) * 3.0e0) * s
end function

Julia

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

MATLAB

function tmp = code(s, u)
	tmp = (((u * u) * single(0.5)) * single(3.0)) * s;
end

Derivation

1. Initial program 95.6%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 10.8

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

5. Applied rewrites 10.8%

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

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 26.5%

   \[\leadsto \left(3 \cdot s\right) \cdot \left(\left(u \cdot u\right) \cdot \color{blue}{0.5}\right) \]
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 26.5

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

10. Applied rewrites 26.5%

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

Alternative 7: 26.4% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = ((u * u) * 0.5e0) * (s * 3.0e0)
end function

Julia

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

MATLAB

function tmp = code(s, u)
	tmp = ((u * u) * single(0.5)) * (s * single(3.0));
end

Derivation

1. Initial program 95.6%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 10.8

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

5. Applied rewrites 10.8%

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

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 26.5%

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

9. Final simplification 26.5%

   \[\leadsto \left(\left(u \cdot u\right) \cdot 0.5\right) \cdot \left(s \cdot 3\right) \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024244 
(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))))))