Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.8% → 98.3%

Time: 18.2s

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: 95.8% 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: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\frac{\left(--0.25\right) - u}{0.75}\right) \cdot \left(s \cdot -3\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (log1p (/ (- (- -0.25) u) 0.75)) (* s -3.0)))

C

float code(float s, float u) {
	return log1pf(((-(-0.25f) - u) / 0.75f)) * (s * -3.0f);
}

Julia

function code(s, u)
	return Float32(log1p(Float32(Float32(Float32(-Float32(-0.25)) - u) / Float32(0.75))) * Float32(s * 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. log-rec 97.0%

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

   2. distribute-rgt-neg-out 97.0%

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

   3. distribute-lft-neg-out 97.0%

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

   4. *-commutative 97.0%

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

   5. distribute-rgt-neg-in 97.0%

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

   6. metadata-eval 97.0%

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

   7. sub-neg 97.0%

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

   8. log1p-def 98.5%

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

   9. distribute-neg-frac 98.5%

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

   10. sub-neg 98.5%

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

   11. metadata-eval 98.5%

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

3. Simplified 98.5%

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

5. Final simplification 98.5%

   \[\leadsto \mathsf{log1p}\left(\frac{\left(--0.25\right) - u}{0.75}\right) \cdot \left(s \cdot -3\right) \]
6. Add Preprocessing

Alternative 2: 96.1% 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. log-rec 97.0%

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

   2. div-sub 96.1%

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

   3. metadata-eval 96.1%

      \[\leadsto \left(3 \cdot s\right) \cdot \left(-\log \left(1 - \left(\frac{u}{0.75} - \color{blue}{0.3333333333333333}\right)\right)\right) \]

3. Simplified 96.1%

   \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \left(-\log \left(1 - \left(\frac{u}{0.75} - 0.3333333333333333\right)\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around 0 96.3%

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

6. Final simplification 96.3%

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

Alternative 3: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ -3 \cdot \left(s \cdot \mathsf{log1p}\left(1.3333333333333333 \cdot \left(0.25 - u\right)\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* -3.0 (* s (log1p (* 1.3333333333333333 (- 0.25 u))))))

C

float code(float s, float u) {
	return -3.0f * (s * log1pf((1.3333333333333333f * (0.25f - u))));
}

Julia

function code(s, u)
	return Float32(Float32(-3.0) * Float32(s * log1p(Float32(Float32(1.3333333333333333) * Float32(Float32(0.25) - 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. log-rec 97.0%

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

   2. distribute-rgt-neg-out 97.0%

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

   3. distribute-lft-neg-out 97.0%

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

   4. *-commutative 97.0%

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

   5. distribute-rgt-neg-in 97.0%

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

   6. metadata-eval 97.0%

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

   7. sub-neg 97.0%

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

   8. log1p-def 98.5%

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

   9. distribute-neg-frac 98.5%

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

   10. sub-neg 98.5%

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

   11. metadata-eval 98.5%

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

3. Simplified 98.5%

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

5. Taylor expanded in s around 0 96.6%

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

   1. log1p-def 97.9%

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

7. Simplified 97.9%

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

8. Final simplification 97.9%

   \[\leadsto -3 \cdot \left(s \cdot \mathsf{log1p}\left(1.3333333333333333 \cdot \left(0.25 - u\right)\right)\right) \]
9. Add Preprocessing

Alternative 4: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(-3 \cdot \mathsf{log1p}\left(\left(u + -0.25\right) \cdot -1.3333333333333333\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* s (* -3.0 (log1p (* (+ u -0.25) -1.3333333333333333)))))

C

float code(float s, float u) {
	return s * (-3.0f * log1pf(((u + -0.25f) * -1.3333333333333333f)));
}

Julia

function code(s, u)
	return Float32(s * Float32(Float32(-3.0) * log1p(Float32(Float32(u + Float32(-0.25)) * 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 96.9%

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

   5. distribute-lft-neg-out 96.9%

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

   6. distribute-rgt-neg-in 96.9%

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

3. Simplified 97.1%

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

   1. expm1-log1p-u 96.6%

      \[\leadsto s \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\mathsf{fma}\left(u, -1.3333333333333333, 1.3333333333333333\right)\right) \cdot -3\right)\right)} \]

   2. expm1-udef 96.2%

      \[\leadsto s \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log \left(\mathsf{fma}\left(u, -1.3333333333333333, 1.3333333333333333\right)\right) \cdot -3\right)} - 1\right)} \]

6. Applied egg-rr 96.1%

   \[\leadsto s \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log \left({\left(\mathsf{fma}\left(u, -1.3333333333333333, 1.3333333333333333\right)\right)}^{-3}\right)\right)} - 1\right)} \]
7. Step-by-step derivation

   1. expm1-def 96.2%

      \[\leadsto s \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left({\left(\mathsf{fma}\left(u, -1.3333333333333333, 1.3333333333333333\right)\right)}^{-3}\right)\right)\right)} \]

   2. expm1-log1p 96.7%

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

   3. log-pow 97.1%

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

   4. log1p-expm1 96.6%

      \[\leadsto s \cdot \left(-3 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\mathsf{fma}\left(u, -1.3333333333333333, 1.3333333333333333\right)\right)\right)\right)}\right) \]

   5. expm1-def 96.6%

      \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{e^{\log \left(\mathsf{fma}\left(u, -1.3333333333333333, 1.3333333333333333\right)\right)} - 1}\right)\right) \]

   6. rem-exp-log 96.6%

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

   7. fma-def 96.4%

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

   8. associate--l+ 95.4%

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

   9. metadata-eval 97.1%

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

   10. +-commutative 97.1%

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

   11. metadata-eval 97.1%

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

   12. *-commutative 97.1%

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

   13. metadata-eval 97.1%

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

   14. distribute-lft-neg-in 97.1%

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

   15. distribute-rgt-neg-in 97.1%

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

   16. distribute-lft-in 98.0%

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

   17. sub-neg 98.0%

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

   18. log1p-def 96.7%

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

   19. log1p-def 98.0%

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

   20. sub-neg 98.0%

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

   21. distribute-lft-in 97.1%

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

   22. metadata-eval 97.1%

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

   23. distribute-rgt-neg-in 97.1%

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

   24. *-commutative 97.1%

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

   25. distribute-rgt-neg-in 97.1%

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

   26. metadata-eval 97.1%

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

8. Simplified 98.0%

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

   1. fma-udef 97.1%

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

   2. metadata-eval 97.1%

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

   3. metadata-eval 97.1%

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

   4. distribute-rgt-in 98.0%

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

   5. sub-neg 98.0%

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

   6. *-commutative 98.0%

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

   7. sub-neg 98.0%

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

   8. metadata-eval 98.0%

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

10. Applied egg-rr 98.0%

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

11. Final simplification 98.0%

    \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\left(u + -0.25\right) \cdot -1.3333333333333333\right)\right) \]
12. Add Preprocessing

Alternative 5: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(1.3333333333333333 \cdot \left(0.25 - u\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* s -3.0) (log1p (* 1.3333333333333333 (- 0.25 u)))))

C

float code(float s, float u) {
	return (s * -3.0f) * log1pf((1.3333333333333333f * (0.25f - u)));
}

Julia

function code(s, u)
	return Float32(Float32(s * Float32(-3.0)) * log1p(Float32(Float32(1.3333333333333333) * Float32(Float32(0.25) - 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. log-rec 97.0%

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

   2. distribute-rgt-neg-out 97.0%

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

   3. distribute-lft-neg-out 97.0%

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

   4. *-commutative 97.0%

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

   5. distribute-rgt-neg-in 97.0%

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

   6. metadata-eval 97.0%

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

   7. sub-neg 97.0%

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

   8. log1p-def 98.5%

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

   9. distribute-neg-frac 98.5%

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

   10. sub-neg 98.5%

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

   11. metadata-eval 98.5%

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

3. Simplified 98.5%

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

5. Taylor expanded in s around 0 96.6%

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

   1. log1p-def 97.9%

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

7. Simplified 97.9%

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

   1. add-sqr-sqrt 97.4%

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

   2. pow2 97.4%

      \[\leadsto \color{blue}{{\left(\sqrt{-3 \cdot \left(s \cdot \mathsf{log1p}\left(1.3333333333333333 \cdot \left(0.25 - u\right)\right)\right)}\right)}^{2}} \]

   3. associate-*r* 97.4%

      \[\leadsto {\left(\sqrt{\color{blue}{\left(-3 \cdot s\right) \cdot \mathsf{log1p}\left(1.3333333333333333 \cdot \left(0.25 - u\right)\right)}}\right)}^{2} \]

9. Applied egg-rr 97.4%

   \[\leadsto \color{blue}{{\left(\sqrt{\left(-3 \cdot s\right) \cdot \mathsf{log1p}\left(1.3333333333333333 \cdot \left(0.25 - u\right)\right)}\right)}^{2}} \]
10. Step-by-step derivation

    1. unpow2 97.4%

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

    2. add-sqr-sqrt 98.1%

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

11. Applied egg-rr 98.1%

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

12. Final simplification 98.1%

    \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(1.3333333333333333 \cdot \left(0.25 - u\right)\right) \]
13. Add Preprocessing

Alternative 6: 29.6% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(s, u)
	tmp = (s * single(-3.0)) * (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. log-rec 97.0%

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

   2. distribute-rgt-neg-out 97.0%

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

   3. distribute-lft-neg-out 97.0%

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

   4. *-commutative 97.0%

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

   5. distribute-rgt-neg-in 97.0%

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

   6. metadata-eval 97.0%

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

   7. sub-neg 97.0%

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

   8. log1p-def 98.5%

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

   9. neg-mul-1 98.5%

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

   10. associate-*r/ 98.5%

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

   11. associate-/l* 98.2%

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

   12. associate-/r/ 98.1%

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

   13. sub-neg 98.1%

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

   14. distribute-lft-in 97.2%

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

   15. metadata-eval 97.2%

       \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\color{blue}{-1.3333333333333333} \cdot u + \frac{-1}{0.75} \cdot \left(-0.25\right)\right) \]

   16. metadata-eval 97.2%

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

   17. metadata-eval 97.2%

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

   18. metadata-eval 97.2%

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

3. Simplified 97.2%

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

5. Taylor expanded in u around inf 19.8%

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

   1. *-commutative 19.8%

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

7. Simplified 19.8%

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

8. Taylor expanded in u around 0 29.7%

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

   1. *-commutative 29.7%

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

10. Simplified 29.7%

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

11. Final simplification 29.7%

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

Alternative 7: 29.6% accurate, 22.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(s, u)
	tmp = single(4.0) * (s * 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. log-rec 97.0%

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

   2. distribute-rgt-neg-out 97.0%

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

   3. distribute-lft-neg-out 97.0%

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

   4. *-commutative 97.0%

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

   5. distribute-rgt-neg-in 97.0%

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

   6. metadata-eval 97.0%

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

   7. sub-neg 97.0%

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

   8. log1p-def 98.5%

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

   9. neg-mul-1 98.5%

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

   10. associate-*r/ 98.5%

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

   11. associate-/l* 98.2%

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

   12. associate-/r/ 98.1%

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

   13. sub-neg 98.1%

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

   14. distribute-lft-in 97.2%

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

   15. metadata-eval 97.2%

       \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\color{blue}{-1.3333333333333333} \cdot u + \frac{-1}{0.75} \cdot \left(-0.25\right)\right) \]

   16. metadata-eval 97.2%

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

   17. metadata-eval 97.2%

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

   18. metadata-eval 97.2%

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

3. Simplified 97.2%

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

5. Taylor expanded in u around inf 19.8%

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

   1. *-commutative 19.8%

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

7. Simplified 19.8%

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

8. Taylor expanded in u around 0 29.7%

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

9. Final simplification 29.7%

   \[\leadsto 4 \cdot \left(s \cdot u\right) \]
10. Add Preprocessing

Alternative 8: 29.6% accurate, 22.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(s, u)
	tmp = u * (s * single(4.0));
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. log-rec 97.0%

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

   2. distribute-rgt-neg-out 97.0%

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

   3. distribute-lft-neg-out 97.0%

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

   4. *-commutative 97.0%

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

   5. distribute-rgt-neg-in 97.0%

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

   6. metadata-eval 97.0%

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

   7. sub-neg 97.0%

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

   8. log1p-def 98.5%

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

   9. neg-mul-1 98.5%

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

   10. associate-*r/ 98.5%

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

   11. associate-/l* 98.2%

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

   12. associate-/r/ 98.1%

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

   13. sub-neg 98.1%

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

   14. distribute-lft-in 97.2%

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

   15. metadata-eval 97.2%

       \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\color{blue}{-1.3333333333333333} \cdot u + \frac{-1}{0.75} \cdot \left(-0.25\right)\right) \]

   16. metadata-eval 97.2%

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

   17. metadata-eval 97.2%

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

   18. metadata-eval 97.2%

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

3. Simplified 97.2%

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

5. Taylor expanded in u around inf 19.8%

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

   1. *-commutative 19.8%

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

7. Simplified 19.8%

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

8. Taylor expanded in u around 0 29.7%

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

   1. associate-*r* 29.7%

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

10. Simplified 29.7%

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

11. Final simplification 29.7%

    \[\leadsto u \cdot \left(s \cdot 4\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024020 
(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))))))