Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.8% → 98.3%

Time: 10.4s

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

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

FPCore

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

C

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

Julia

function code(s, u)
	return Float32(s * Float32(log1p(Float32(Float32(Float32(0.25) - u) / Float32(0.75))) * Float32(-3.0)))
end

Derivation

1. Initial program 96.2%

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

   1. associate-*l* 96.3%

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

   2. *-commutative 96.3%

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

   3. associate-*l* 96.3%

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

   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-def 98.4%

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

   9. distribute-neg-frac 98.4%

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

   10. sub-neg 98.4%

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

   11. +-commutative 98.4%

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

   12. distribute-neg-in 98.4%

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

   13. metadata-eval 98.4%

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

   14. metadata-eval 98.4%

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

   15. unsub-neg 98.4%

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

   16. metadata-eval 98.4%

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

3. Simplified 98.4%

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

4. Final simplification 98.4%

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

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 96.2%

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

   1. associate-*l* 96.3%

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

   2. *-commutative 96.3%

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

   3. associate-*l* 96.3%

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

   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. div-sub 95.9%

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

   8. associate--r- 95.8%

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

   9. sub-neg 95.8%

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

   10. +-commutative 95.8%

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

   11. associate-+l+ 95.1%

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

   12. distribute-neg-frac 95.1%

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

   13. neg-mul-1 95.1%

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

   14. associate-/l* 94.7%

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

   15. associate-/r/ 96.5%

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

   16. metadata-eval 96.5%

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

   17. metadata-eval 96.5%

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

   18. metadata-eval 96.5%

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

   19. metadata-eval 96.5%

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

3. Simplified 96.5%

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

4. Taylor expanded in u around inf 96.5%

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

   1. +-commutative 96.5%

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

   2. *-commutative 96.5%

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

   3. fma-udef 97.0%

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

6. Simplified 97.0%

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

7. Taylor expanded in s around 0 96.3%

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

8. Final simplification 96.3%

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

Alternative 3: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, u)
	return Float32(Float32(-3.0) * Float32(s * log1p(Float32(Float32(Float32(0.25) - u) * Float32(1.3333333333333333)))))
end

Derivation

1. Initial program 96.2%

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

   1. associate-*l* 96.3%

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

   2. *-commutative 96.3%

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

   3. associate-*l* 96.3%

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

   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-def 98.4%

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

   9. distribute-neg-frac 98.4%

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

   10. sub-neg 98.4%

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

   11. +-commutative 98.4%

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

   12. distribute-neg-in 98.4%

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

   13. metadata-eval 98.4%

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

   14. metadata-eval 98.4%

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

   15. unsub-neg 98.4%

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

   16. metadata-eval 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in s around 0 96.4%

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

   1. log1p-def 97.9%

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

6. Simplified 97.9%

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

7. Final simplification 97.9%

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

Alternative 4: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, u)
	return Float32(s * Float32(Float32(-3.0) * log1p(Float32(Float32(Float32(0.25) - u) * Float32(1.3333333333333333)))))
end

Derivation

1. Initial program 96.2%

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

   1. associate-*l* 96.3%

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

   2. *-commutative 96.3%

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

   3. associate-*l* 96.3%

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

   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-def 98.4%

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

   9. distribute-neg-frac 98.4%

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

   10. sub-neg 98.4%

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

   11. +-commutative 98.4%

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

   12. distribute-neg-in 98.4%

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

   13. metadata-eval 98.4%

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

   14. metadata-eval 98.4%

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

   15. unsub-neg 98.4%

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

   16. metadata-eval 98.4%

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

3. Simplified 98.4%

   \[\leadsto \color{blue}{s \cdot \left(\mathsf{log1p}\left(\frac{0.25 - u}{0.75}\right) \cdot -3\right)} \]
4. Step-by-step derivation

   1. expm1-log1p-u 71.6%

      \[\leadsto s \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\frac{0.25 - u}{0.75}\right)\right)\right)} \cdot -3\right) \]

   2. expm1-udef 70.2%

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

   3. div-inv 70.1%

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

   4. metadata-eval 70.1%

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

5. Applied egg-rr 70.1%

   \[\leadsto s \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{log1p}\left(\left(0.25 - u\right) \cdot 1.3333333333333333\right)\right)} - 1\right)} \cdot -3\right) \]
6. Step-by-step derivation

   1. expm1-def 71.4%

      \[\leadsto s \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\left(0.25 - u\right) \cdot 1.3333333333333333\right)\right)\right)} \cdot -3\right) \]

   2. expm1-log1p 98.1%

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

   3. *-commutative 98.1%

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

7. Simplified 98.1%

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

8. Final simplification 98.1%

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

Alternative 5: 19.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 96.2%

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

   1. associate-*l* 96.3%

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

   2. *-commutative 96.3%

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

   3. associate-*l* 96.3%

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

   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-def 98.4%

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

   9. distribute-neg-frac 98.4%

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

   10. sub-neg 98.4%

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

   11. +-commutative 98.4%

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

   12. distribute-neg-in 98.4%

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

   13. metadata-eval 98.4%

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

   14. metadata-eval 98.4%

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

   15. unsub-neg 98.4%

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

   16. metadata-eval 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in s around 0 96.4%

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

   1. log1p-def 97.9%

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

6. Simplified 97.9%

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

7. Taylor expanded in u around inf 19.4%

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

   1. *-commutative 19.4%

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

9. Simplified 19.4%

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

10. Final simplification 19.4%

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

Alternative 6: 25.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 96.2%

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

2. Taylor expanded in u around 0 25.9%

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

   1. distribute-lft-out 25.9%

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

   2. distribute-lft-out 25.9%

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

   3. +-commutative 25.9%

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

4. Simplified 25.9%

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

5. Final simplification 25.9%

   \[\leadsto 3 \cdot \left(s \cdot \left(u + \log 0.75\right)\right) \]

Alternative 7: 25.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 96.2%

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

2. Taylor expanded in u around 0 25.9%

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

3. Final simplification 25.9%

   \[\leadsto \left(s \cdot 3\right) \cdot \left(u + \log 0.75\right) \]

Alternative 8: 7.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ s \cdot \mathsf{log1p}\left(-0.578125\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (log1p -0.578125)))

C

float code(float s, float u) {
	return s * log1pf(-0.578125f);
}

Julia

function code(s, u)
	return Float32(s * log1p(Float32(-0.578125)))
end

Derivation

1. Initial program 96.2%

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

   1. associate-*l* 96.3%

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

   2. log-rec 96.6%

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

   3. sub-neg 96.6%

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

   4. metadata-eval 96.6%

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

3. Simplified 96.6%

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

4. Taylor expanded in s around 0 96.4%

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

   1. *-commutative 96.4%

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

   2. associate-*l* 96.6%

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

   3. sub-neg 96.6%

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

   4. metadata-eval 96.6%

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

   5. cancel-sign-sub-inv 96.6%

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

   6. log1p-def 98.1%

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

   7. metadata-eval 98.1%

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

   8. +-commutative 98.1%

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

   9. distribute-rgt-in 96.9%

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

   10. metadata-eval 96.9%

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

6. Simplified 96.9%

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

7. Taylor expanded in u around 0 7.1%

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

   1. log1p-expm1-u 7.1%

      \[\leadsto s \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(0.3333333333333333\right) \cdot -3\right)\right)} \]

   2. expm1-udef 7.1%

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

   3. log1p-udef 7.1%

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

   4. metadata-eval 7.1%

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

   5. exp-to-pow 7.1%

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

   6. metadata-eval 7.1%

      \[\leadsto s \cdot \mathsf{log1p}\left(\color{blue}{0.421875} - 1\right) \]

   7. metadata-eval 7.1%

      \[\leadsto s \cdot \mathsf{log1p}\left(\color{blue}{-0.578125}\right) \]

9. Applied egg-rr 7.1%

   \[\leadsto s \cdot \color{blue}{\mathsf{log1p}\left(-0.578125\right)} \]

10. Final simplification 7.1%

    \[\leadsto s \cdot \mathsf{log1p}\left(-0.578125\right) \]

Reproduce

herbie shell --seed 2023256 
(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))))))