Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.9% → 98.3%

Time: 10.5s

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: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 96.1%

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

   1. log-rec 96.7%

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

   2. sub-neg 96.7%

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

   3. log1p-def 98.3%

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

   4. distribute-neg-frac 98.3%

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

   5. sub-neg 98.3%

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

   6. metadata-eval 98.3%

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

3. Simplified 98.3%

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

4. Final simplification 98.3%

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

Alternative 2: 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 96.1%

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

   1. associate-*l* 96.0%

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

   2. *-commutative 96.0%

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

   3. associate-*l* 96.0%

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

   4. log-rec 96.6%

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

   5. distribute-lft-neg-out 96.6%

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

   6. distribute-rgt-neg-in 96.6%

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

   7. sub-neg 96.6%

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

   8. log1p-def 98.2%

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

   9. distribute-neg-frac 98.2%

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

   10. sub-neg 98.2%

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

   11. +-commutative 98.2%

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

   12. distribute-neg-in 98.2%

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

   13. metadata-eval 98.2%

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

   14. metadata-eval 98.2%

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

   15. unsub-neg 98.2%

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

   16. metadata-eval 98.2%

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

3. Simplified 98.2%

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

   1. add-exp-log 91.2%

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

   2. *-commutative 91.2%

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

   3. div-sub 90.8%

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

   4. metadata-eval 90.8%

      \[\leadsto e^{\log \left(s \cdot \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{0.3333333333333333} - \frac{u}{0.75}\right)\right)\right)} \]

   5. div-inv 91.0%

      \[\leadsto e^{\log \left(s \cdot \left(-3 \cdot \mathsf{log1p}\left(0.3333333333333333 - \color{blue}{u \cdot \frac{1}{0.75}}\right)\right)\right)} \]

   6. metadata-eval 91.0%

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

5. Applied egg-rr 91.0%

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

6. 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. Final simplification 96.1%

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

Alternative 3: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 96.1%

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

   1. associate-*l* 96.0%

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

   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 96.5%

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

   1. *-un-lft-identity 96.5%

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

   2. cancel-sign-sub-inv 96.5%

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

   3. metadata-eval 96.5%

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

6. Applied egg-rr 96.5%

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

   1. *-lft-identity 96.5%

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

   2. log1p-def 97.8%

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

   3. sub-neg 97.8%

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

   4. metadata-eval 97.8%

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

8. Simplified 97.8%

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

9. Final simplification 97.8%

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

Alternative 4: 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 96.1%

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

   1. associate-*l* 96.0%

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

   2. *-commutative 96.0%

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

   3. associate-*l* 96.0%

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

   4. log-rec 96.6%

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

   5. distribute-lft-neg-out 96.6%

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

   6. distribute-rgt-neg-in 96.6%

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

   7. sub-neg 96.6%

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

   8. log1p-def 98.2%

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

   9. distribute-neg-frac 98.2%

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

   10. sub-neg 98.2%

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

   11. +-commutative 98.2%

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

   12. distribute-neg-in 98.2%

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

   13. metadata-eval 98.2%

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

   14. metadata-eval 98.2%

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

   15. unsub-neg 98.2%

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

   16. metadata-eval 98.2%

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

3. Simplified 98.2%

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

4. Final simplification 98.2%

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

Alternative 5: 18.8% 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 96.1%

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

   1. associate-*l* 96.0%

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

   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 96.5%

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

   1. *-un-lft-identity 96.5%

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

   2. cancel-sign-sub-inv 96.5%

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

   3. metadata-eval 96.5%

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

6. Applied egg-rr 96.5%

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

   1. *-lft-identity 96.5%

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

   2. log1p-def 97.8%

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

   3. sub-neg 97.8%

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

   4. metadata-eval 97.8%

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

8. Simplified 97.8%

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

9. Taylor expanded in u around inf 19.3%

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

    1. *-commutative 19.3%

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

11. Simplified 19.3%

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

12. Final simplification 19.3%

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

Alternative 6: 25.8% 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 96.1%

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

2. Taylor expanded in u around 0 26.0%

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

   1. distribute-lft-out 26.0%

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

   2. distribute-lft-out 26.0%

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

   3. +-commutative 26.0%

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

4. Simplified 26.0%

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

5. Final simplification 26.0%

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

Alternative 7: 25.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(3 \cdot u + \log 0.421875\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (+ (* 3.0 u) (log 0.421875))))

C

float code(float s, float u) {
	return s * ((3.0f * u) + logf(0.421875f));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * ((3.0e0 * u) + log(0.421875e0))
end function

Julia

function code(s, u)
	return Float32(s * Float32(Float32(Float32(3.0) * u) + log(Float32(0.421875))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * ((single(3.0) * u) + log(single(0.421875)));
end

Derivation

1. Initial program 96.1%

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

   1. associate-*l* 96.0%

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

   2. *-commutative 96.0%

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

   3. associate-*l* 96.0%

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

   4. log-rec 96.6%

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

   5. distribute-lft-neg-out 96.6%

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

   6. distribute-rgt-neg-in 96.6%

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

   7. sub-neg 96.6%

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

   8. log1p-def 98.2%

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

   9. distribute-neg-frac 98.2%

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

   10. sub-neg 98.2%

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

   11. +-commutative 98.2%

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

   12. distribute-neg-in 98.2%

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

   13. metadata-eval 98.2%

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

   14. metadata-eval 98.2%

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

   15. unsub-neg 98.2%

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

   16. metadata-eval 98.2%

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

3. Simplified 98.2%

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

   1. add-log-exp 97.5%

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

   2. log1p-udef 96.6%

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

   3. exp-to-pow 96.6%

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

   4. div-sub 95.6%

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

   5. metadata-eval 95.6%

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

   6. div-inv 95.5%

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

   7. metadata-eval 95.5%

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

5. Applied egg-rr 95.5%

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

6. Taylor expanded in u around 0 26.0%

   \[\leadsto s \cdot \color{blue}{\left(3 \cdot u + \log 0.421875\right)} \]

7. Final simplification 26.0%

   \[\leadsto s \cdot \left(3 \cdot u + \log 0.421875\right) \]

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 96.1%

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

   1. associate-*l* 96.0%

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

   2. *-commutative 96.0%

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

   3. associate-*l* 96.0%

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

   4. log-rec 96.6%

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

   5. distribute-lft-neg-out 96.6%

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

   6. distribute-rgt-neg-in 96.6%

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

   7. sub-neg 96.6%

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

   8. log1p-def 98.2%

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

   9. distribute-neg-frac 98.2%

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

   10. sub-neg 98.2%

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

   11. +-commutative 98.2%

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

   12. distribute-neg-in 98.2%

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

   13. metadata-eval 98.2%

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

   14. metadata-eval 98.2%

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

   15. unsub-neg 98.2%

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

   16. metadata-eval 98.2%

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

3. Simplified 98.2%

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

   1. add-log-exp 97.5%

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

   2. log1p-udef 96.6%

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

   3. exp-to-pow 96.6%

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

   4. div-sub 95.6%

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

   5. metadata-eval 95.6%

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

   6. div-inv 95.5%

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

   7. metadata-eval 95.5%

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

5. Applied egg-rr 95.5%

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

6. Taylor expanded in u around 0 7.2%

   \[\leadsto s \cdot \log \color{blue}{0.421875} \]

7. Final simplification 7.2%

   \[\leadsto s \cdot \log 0.421875 \]

Reproduce

herbie shell --seed 2023242 
(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))))))