Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.9% → 97.8%

Time: 9.6s

Alternatives: 8

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: 97.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ s \cdot \sqrt{{\left(\mathsf{log1p}\left(\mathsf{fma}\left(u, -1.3333333333333333, 0.3333333333333333\right)\right)\right)}^{2} \cdot 9} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (*
  s
  (sqrt
   (* (pow (log1p (fma u -1.3333333333333333 0.3333333333333333)) 2.0) 9.0))))

C

float code(float s, float u) {
	return s * sqrtf((powf(log1pf(fmaf(u, -1.3333333333333333f, 0.3333333333333333f)), 2.0f) * 9.0f));
}

Julia

function code(s, u)
	return Float32(s * sqrt(Float32((log1p(fma(u, Float32(-1.3333333333333333), Float32(0.3333333333333333))) ^ Float32(2.0)) * Float32(9.0))))
end

Derivation

1. Initial program 95.8%

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

   1. *-commutative 95.8%

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

   2. associate-*l* 95.9%

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

   3. *-commutative 95.9%

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

   4. log-rec 96.8%

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

   5. distribute-lft-neg-out 96.8%

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

   6. distribute-rgt-neg-in 96.8%

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

   7. sub-neg 96.8%

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

   8. log1p-define 98.4%

      \[\leadsto s \cdot \left(\color{blue}{\mathsf{log1p}\left(-\frac{u - 0.25}{0.75}\right)} \cdot \left(-3\right)\right) \]

   9. neg-sub0 98.4%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{0 - \frac{u - 0.25}{0.75}}\right) \cdot \left(-3\right)\right) \]

   10. div-sub 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(0 - \color{blue}{\left(\frac{u}{0.75} - \frac{0.25}{0.75}\right)}\right) \cdot \left(-3\right)\right) \]

   11. associate--r- 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{\left(0 - \frac{u}{0.75}\right) + \frac{0.25}{0.75}}\right) \cdot \left(-3\right)\right) \]

   12. neg-sub0 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{\left(-\frac{u}{0.75}\right)} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   13. distribute-frac-neg 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{\frac{-u}{0.75}} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   14. neg-mul-1 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\frac{\color{blue}{-1 \cdot u}}{0.75} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   15. *-commutative 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\frac{\color{blue}{u \cdot -1}}{0.75} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   16. associate-/l* 97.0%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{u \cdot \frac{-1}{0.75}} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   17. fma-define 98.0%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(u, \frac{-1}{0.75}, \frac{0.25}{0.75}\right)}\right) \cdot \left(-3\right)\right) \]

   18. metadata-eval 98.0%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u, \color{blue}{-1.3333333333333333}, \frac{0.25}{0.75}\right)\right) \cdot \left(-3\right)\right) \]

   19. metadata-eval 98.0%

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

   20. metadata-eval 98.0%

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

3. Simplified 98.0%

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

   1. add-sqr-sqrt 97.6%

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

   2. sqrt-unprod 98.0%

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

   3. swap-sqr 98.1%

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

   4. pow2 98.1%

      \[\leadsto s \cdot \sqrt{\color{blue}{{\left(\mathsf{log1p}\left(\mathsf{fma}\left(u, -1.3333333333333333, 0.3333333333333333\right)\right)\right)}^{2}} \cdot \left(-3 \cdot -3\right)} \]

   5. metadata-eval 98.1%

      \[\leadsto s \cdot \sqrt{{\left(\mathsf{log1p}\left(\mathsf{fma}\left(u, -1.3333333333333333, 0.3333333333333333\right)\right)\right)}^{2} \cdot \color{blue}{9}} \]

6. Applied egg-rr 98.1%

   \[\leadsto s \cdot \color{blue}{\sqrt{{\left(\mathsf{log1p}\left(\mathsf{fma}\left(u, -1.3333333333333333, 0.3333333333333333\right)\right)\right)}^{2} \cdot 9}} \]
7. Add Preprocessing

Alternative 2: 97.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 95.8%

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

   1. *-commutative 95.8%

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

   2. associate-*l* 95.9%

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

   3. *-commutative 95.9%

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

   4. log-rec 96.8%

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

   5. distribute-lft-neg-out 96.8%

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

   6. distribute-rgt-neg-in 96.8%

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

   7. sub-neg 96.8%

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

   8. log1p-define 98.4%

      \[\leadsto s \cdot \left(\color{blue}{\mathsf{log1p}\left(-\frac{u - 0.25}{0.75}\right)} \cdot \left(-3\right)\right) \]

   9. neg-sub0 98.4%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{0 - \frac{u - 0.25}{0.75}}\right) \cdot \left(-3\right)\right) \]

   10. div-sub 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(0 - \color{blue}{\left(\frac{u}{0.75} - \frac{0.25}{0.75}\right)}\right) \cdot \left(-3\right)\right) \]

   11. associate--r- 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{\left(0 - \frac{u}{0.75}\right) + \frac{0.25}{0.75}}\right) \cdot \left(-3\right)\right) \]

   12. neg-sub0 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{\left(-\frac{u}{0.75}\right)} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   13. distribute-frac-neg 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{\frac{-u}{0.75}} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   14. neg-mul-1 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\frac{\color{blue}{-1 \cdot u}}{0.75} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   15. *-commutative 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\frac{\color{blue}{u \cdot -1}}{0.75} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   16. associate-/l* 97.0%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{u \cdot \frac{-1}{0.75}} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   17. fma-define 98.0%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(u, \frac{-1}{0.75}, \frac{0.25}{0.75}\right)}\right) \cdot \left(-3\right)\right) \]

   18. metadata-eval 98.0%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u, \color{blue}{-1.3333333333333333}, \frac{0.25}{0.75}\right)\right) \cdot \left(-3\right)\right) \]

   19. metadata-eval 98.0%

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

   20. metadata-eval 98.0%

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

3. Simplified 98.0%

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

Alternative 3: 96.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(-3 \cdot \mathsf{log1p}\left(u \cdot \left(0.3333333333333333 \cdot \frac{1}{u} - 1.3333333333333333\right)\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (*
  s
  (*
   -3.0
   (log1p (* u (- (* 0.3333333333333333 (/ 1.0 u)) 1.3333333333333333))))))

C

float code(float s, float u) {
	return s * (-3.0f * log1pf((u * ((0.3333333333333333f * (1.0f / u)) - 1.3333333333333333f))));
}

Julia

function code(s, u)
	return Float32(s * Float32(Float32(-3.0) * log1p(Float32(u * Float32(Float32(Float32(0.3333333333333333) * Float32(Float32(1.0) / u)) - Float32(1.3333333333333333))))))
end

Derivation

1. Initial program 95.8%

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

   1. *-commutative 95.8%

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

   2. associate-*l* 95.9%

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

   3. *-commutative 95.9%

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

   4. log-rec 96.8%

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

   5. distribute-lft-neg-out 96.8%

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

   6. distribute-rgt-neg-in 96.8%

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

   7. sub-neg 96.8%

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

   8. log1p-define 98.4%

      \[\leadsto s \cdot \left(\color{blue}{\mathsf{log1p}\left(-\frac{u - 0.25}{0.75}\right)} \cdot \left(-3\right)\right) \]

   9. neg-sub0 98.4%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{0 - \frac{u - 0.25}{0.75}}\right) \cdot \left(-3\right)\right) \]

   10. div-sub 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(0 - \color{blue}{\left(\frac{u}{0.75} - \frac{0.25}{0.75}\right)}\right) \cdot \left(-3\right)\right) \]

   11. associate--r- 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{\left(0 - \frac{u}{0.75}\right) + \frac{0.25}{0.75}}\right) \cdot \left(-3\right)\right) \]

   12. neg-sub0 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{\left(-\frac{u}{0.75}\right)} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   13. distribute-frac-neg 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{\frac{-u}{0.75}} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   14. neg-mul-1 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\frac{\color{blue}{-1 \cdot u}}{0.75} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   15. *-commutative 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\frac{\color{blue}{u \cdot -1}}{0.75} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   16. associate-/l* 97.0%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{u \cdot \frac{-1}{0.75}} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   17. fma-define 98.0%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(u, \frac{-1}{0.75}, \frac{0.25}{0.75}\right)}\right) \cdot \left(-3\right)\right) \]

   18. metadata-eval 98.0%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u, \color{blue}{-1.3333333333333333}, \frac{0.25}{0.75}\right)\right) \cdot \left(-3\right)\right) \]

   19. metadata-eval 98.0%

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

   20. metadata-eval 98.0%

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

3. Simplified 98.0%

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

5. Taylor expanded in u around inf 97.2%

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

6. Final simplification 97.2%

   \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(u \cdot \left(0.3333333333333333 \cdot \frac{1}{u} - 1.3333333333333333\right)\right)\right) \]
7. Add Preprocessing

Alternative 4: 96.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 95.8%

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

   1. *-commutative 95.8%

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

   2. associate-*l* 95.9%

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

   3. *-commutative 95.9%

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

   4. log-rec 96.8%

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

   5. distribute-lft-neg-out 96.8%

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

   6. distribute-rgt-neg-in 96.8%

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

   7. sub-neg 96.8%

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

   8. log1p-define 98.4%

      \[\leadsto s \cdot \left(\color{blue}{\mathsf{log1p}\left(-\frac{u - 0.25}{0.75}\right)} \cdot \left(-3\right)\right) \]

   9. neg-sub0 98.4%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{0 - \frac{u - 0.25}{0.75}}\right) \cdot \left(-3\right)\right) \]

   10. div-sub 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(0 - \color{blue}{\left(\frac{u}{0.75} - \frac{0.25}{0.75}\right)}\right) \cdot \left(-3\right)\right) \]

   11. associate--r- 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{\left(0 - \frac{u}{0.75}\right) + \frac{0.25}{0.75}}\right) \cdot \left(-3\right)\right) \]

   12. neg-sub0 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{\left(-\frac{u}{0.75}\right)} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   13. distribute-frac-neg 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{\frac{-u}{0.75}} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   14. neg-mul-1 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\frac{\color{blue}{-1 \cdot u}}{0.75} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   15. *-commutative 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\frac{\color{blue}{u \cdot -1}}{0.75} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   16. associate-/l* 97.0%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{u \cdot \frac{-1}{0.75}} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   17. fma-define 98.0%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(u, \frac{-1}{0.75}, \frac{0.25}{0.75}\right)}\right) \cdot \left(-3\right)\right) \]

   18. metadata-eval 98.0%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u, \color{blue}{-1.3333333333333333}, \frac{0.25}{0.75}\right)\right) \cdot \left(-3\right)\right) \]

   19. metadata-eval 98.0%

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

   20. metadata-eval 98.0%

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

3. Simplified 98.0%

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

5. Taylor expanded in u around inf 97.2%

   \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{u \cdot \left(0.3333333333333333 \cdot \frac{1}{u} - 1.3333333333333333\right)}\right) \cdot -3\right) \]
6. Step-by-step derivation

   1. sub-neg 97.2%

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

   2. metadata-eval 97.2%

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

   3. +-commutative 97.2%

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

   4. associate-*r/ 97.2%

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

   5. metadata-eval 97.2%

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

7. Simplified 97.2%

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

8. Final simplification 97.2%

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

Alternative 5: 96.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 95.8%

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

   1. *-commutative 95.8%

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

   2. associate-*l* 95.9%

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

   3. *-commutative 95.9%

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

   4. log-rec 96.8%

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

   5. distribute-lft-neg-out 96.8%

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

   6. distribute-rgt-neg-in 96.8%

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

   7. sub-neg 96.8%

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

   8. log1p-define 98.4%

      \[\leadsto s \cdot \left(\color{blue}{\mathsf{log1p}\left(-\frac{u - 0.25}{0.75}\right)} \cdot \left(-3\right)\right) \]

   9. neg-sub0 98.4%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{0 - \frac{u - 0.25}{0.75}}\right) \cdot \left(-3\right)\right) \]

   10. div-sub 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(0 - \color{blue}{\left(\frac{u}{0.75} - \frac{0.25}{0.75}\right)}\right) \cdot \left(-3\right)\right) \]

   11. associate--r- 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{\left(0 - \frac{u}{0.75}\right) + \frac{0.25}{0.75}}\right) \cdot \left(-3\right)\right) \]

   12. neg-sub0 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{\left(-\frac{u}{0.75}\right)} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   13. distribute-frac-neg 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{\frac{-u}{0.75}} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   14. neg-mul-1 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\frac{\color{blue}{-1 \cdot u}}{0.75} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   15. *-commutative 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\frac{\color{blue}{u \cdot -1}}{0.75} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   16. associate-/l* 97.0%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{u \cdot \frac{-1}{0.75}} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   17. fma-define 98.0%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(u, \frac{-1}{0.75}, \frac{0.25}{0.75}\right)}\right) \cdot \left(-3\right)\right) \]

   18. metadata-eval 98.0%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u, \color{blue}{-1.3333333333333333}, \frac{0.25}{0.75}\right)\right) \cdot \left(-3\right)\right) \]

   19. metadata-eval 98.0%

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

   20. metadata-eval 98.0%

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

3. Simplified 98.0%

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

   1. fma-undefine 97.0%

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

6. Applied egg-rr 97.0%

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

7. Final simplification 97.0%

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

Alternative 6: 96.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.8%

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

   1. *-commutative 95.8%

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

   2. associate-*l* 95.9%

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

   3. *-commutative 95.9%

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

   4. log-rec 96.8%

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

   5. distribute-lft-neg-out 96.8%

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

   6. distribute-rgt-neg-in 96.8%

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

   7. sub-neg 96.8%

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

   8. log1p-define 98.4%

      \[\leadsto s \cdot \left(\color{blue}{\mathsf{log1p}\left(-\frac{u - 0.25}{0.75}\right)} \cdot \left(-3\right)\right) \]

   9. neg-sub0 98.4%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{0 - \frac{u - 0.25}{0.75}}\right) \cdot \left(-3\right)\right) \]

   10. div-sub 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(0 - \color{blue}{\left(\frac{u}{0.75} - \frac{0.25}{0.75}\right)}\right) \cdot \left(-3\right)\right) \]

   11. associate--r- 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{\left(0 - \frac{u}{0.75}\right) + \frac{0.25}{0.75}}\right) \cdot \left(-3\right)\right) \]

   12. neg-sub0 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{\left(-\frac{u}{0.75}\right)} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   13. distribute-frac-neg 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{\frac{-u}{0.75}} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   14. neg-mul-1 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\frac{\color{blue}{-1 \cdot u}}{0.75} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   15. *-commutative 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\frac{\color{blue}{u \cdot -1}}{0.75} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   16. associate-/l* 97.0%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{u \cdot \frac{-1}{0.75}} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   17. fma-define 98.0%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(u, \frac{-1}{0.75}, \frac{0.25}{0.75}\right)}\right) \cdot \left(-3\right)\right) \]

   18. metadata-eval 98.0%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u, \color{blue}{-1.3333333333333333}, \frac{0.25}{0.75}\right)\right) \cdot \left(-3\right)\right) \]

   19. metadata-eval 98.0%

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

   20. metadata-eval 98.0%

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

3. Simplified 98.0%

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

5. Taylor expanded in s around 0 96.1%

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

6. Final simplification 96.1%

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

Alternative 7: 28.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.8%

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

   1. *-commutative 95.8%

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

   2. associate-*l* 95.9%

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

   3. *-commutative 95.9%

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

   4. log-rec 96.8%

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

   5. distribute-lft-neg-out 96.8%

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

   6. distribute-rgt-neg-in 96.8%

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

   7. sub-neg 96.8%

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

   8. log1p-define 98.4%

      \[\leadsto s \cdot \left(\color{blue}{\mathsf{log1p}\left(-\frac{u - 0.25}{0.75}\right)} \cdot \left(-3\right)\right) \]

   9. neg-sub0 98.4%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{0 - \frac{u - 0.25}{0.75}}\right) \cdot \left(-3\right)\right) \]

   10. div-sub 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(0 - \color{blue}{\left(\frac{u}{0.75} - \frac{0.25}{0.75}\right)}\right) \cdot \left(-3\right)\right) \]

   11. associate--r- 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{\left(0 - \frac{u}{0.75}\right) + \frac{0.25}{0.75}}\right) \cdot \left(-3\right)\right) \]

   12. neg-sub0 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{\left(-\frac{u}{0.75}\right)} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   13. distribute-frac-neg 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{\frac{-u}{0.75}} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   14. neg-mul-1 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\frac{\color{blue}{-1 \cdot u}}{0.75} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   15. *-commutative 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\frac{\color{blue}{u \cdot -1}}{0.75} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   16. associate-/l* 97.0%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{u \cdot \frac{-1}{0.75}} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   17. fma-define 98.0%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(u, \frac{-1}{0.75}, \frac{0.25}{0.75}\right)}\right) \cdot \left(-3\right)\right) \]

   18. metadata-eval 98.0%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u, \color{blue}{-1.3333333333333333}, \frac{0.25}{0.75}\right)\right) \cdot \left(-3\right)\right) \]

   19. metadata-eval 98.0%

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

   20. metadata-eval 98.0%

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

3. Simplified 98.0%

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

   1. add-sqr-sqrt 97.6%

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

   2. sqrt-unprod 98.0%

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

   3. swap-sqr 98.1%

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

   4. pow2 98.1%

      \[\leadsto s \cdot \sqrt{\color{blue}{{\left(\mathsf{log1p}\left(\mathsf{fma}\left(u, -1.3333333333333333, 0.3333333333333333\right)\right)\right)}^{2}} \cdot \left(-3 \cdot -3\right)} \]

   5. metadata-eval 98.1%

      \[\leadsto s \cdot \sqrt{{\left(\mathsf{log1p}\left(\mathsf{fma}\left(u, -1.3333333333333333, 0.3333333333333333\right)\right)\right)}^{2} \cdot \color{blue}{9}} \]

6. Applied egg-rr 98.1%

   \[\leadsto s \cdot \color{blue}{\sqrt{{\left(\mathsf{log1p}\left(\mathsf{fma}\left(u, -1.3333333333333333, 0.3333333333333333\right)\right)\right)}^{2} \cdot 9}} \]

7. Taylor expanded in u around 0 27.9%

   \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log 1.3333333333333333\right)} \]
8. Add Preprocessing

Alternative 8: 7.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ s \cdot \log 0.421875 \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (log 0.421875)))

C

float code(float s, float u) {
	return s * logf(0.421875f);
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * log(0.421875e0)
end function

Julia

function code(s, u)
	return Float32(s * log(Float32(0.421875)))
end

MATLAB

function tmp = code(s, u)
	tmp = s * log(single(0.421875));
end

Derivation

1. Initial program 95.8%

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

   1. *-commutative 95.8%

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

   2. associate-*l* 95.9%

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

   3. *-commutative 95.9%

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

   4. log-rec 96.8%

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

   5. distribute-lft-neg-out 96.8%

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

   6. distribute-rgt-neg-in 96.8%

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

   7. sub-neg 96.8%

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

   8. log1p-define 98.4%

      \[\leadsto s \cdot \left(\color{blue}{\mathsf{log1p}\left(-\frac{u - 0.25}{0.75}\right)} \cdot \left(-3\right)\right) \]

   9. neg-sub0 98.4%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{0 - \frac{u - 0.25}{0.75}}\right) \cdot \left(-3\right)\right) \]

   10. div-sub 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(0 - \color{blue}{\left(\frac{u}{0.75} - \frac{0.25}{0.75}\right)}\right) \cdot \left(-3\right)\right) \]

   11. associate--r- 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{\left(0 - \frac{u}{0.75}\right) + \frac{0.25}{0.75}}\right) \cdot \left(-3\right)\right) \]

   12. neg-sub0 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{\left(-\frac{u}{0.75}\right)} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   13. distribute-frac-neg 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{\frac{-u}{0.75}} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   14. neg-mul-1 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\frac{\color{blue}{-1 \cdot u}}{0.75} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   15. *-commutative 96.6%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\frac{\color{blue}{u \cdot -1}}{0.75} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   16. associate-/l* 97.0%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{u \cdot \frac{-1}{0.75}} + \frac{0.25}{0.75}\right) \cdot \left(-3\right)\right) \]

   17. fma-define 98.0%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(u, \frac{-1}{0.75}, \frac{0.25}{0.75}\right)}\right) \cdot \left(-3\right)\right) \]

   18. metadata-eval 98.0%

       \[\leadsto s \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u, \color{blue}{-1.3333333333333333}, \frac{0.25}{0.75}\right)\right) \cdot \left(-3\right)\right) \]

   19. metadata-eval 98.0%

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

   20. metadata-eval 98.0%

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

3. Simplified 98.0%

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

5. Taylor expanded in s around 0 96.1%

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

7. Simplified 96.9%

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

8. Taylor expanded in u around 0 7.3%

   \[\leadsto s \cdot \log \color{blue}{0.421875} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(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))))))