Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.8% → 98.3%

Time: 16.0s

Alternatives: 10

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, 0.3× speedup

Math

\[\begin{array}{l} \\ \left(s \cdot -3\right) \cdot \mathsf{log1p}\left({\left(0.25 - u\right)}^{2} \cdot -1.7777777777777777\right) - \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (-
  (* (* s -3.0) (log1p (* (pow (- 0.25 u) 2.0) -1.7777777777777777)))
  (* (* s -3.0) (log1p (fma u 1.3333333333333333 -0.3333333333333333)))))

C

float code(float s, float u) {
	return ((s * -3.0f) * log1pf((powf((0.25f - u), 2.0f) * -1.7777777777777777f))) - ((s * -3.0f) * log1pf(fmaf(u, 1.3333333333333333f, -0.3333333333333333f)));
}

Julia

function code(s, u)
	return Float32(Float32(Float32(s * Float32(-3.0)) * log1p(Float32((Float32(Float32(0.25) - u) ^ Float32(2.0)) * Float32(-1.7777777777777777)))) - Float32(Float32(s * Float32(-3.0)) * log1p(fma(u, Float32(1.3333333333333333), Float32(-0.3333333333333333)))))
end

Derivation

1. Initial program 96.4%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 98.3%

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

   9. distribute-neg-frac 98.3%

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

   10. sub-neg 98.3%

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

   11. metadata-eval 98.3%

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

3. Simplified 98.3%

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

4. Taylor expanded in s around 0 96.6%

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

   1. flip-+ 95.9%

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

   2. log-div 95.8%

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

   3. metadata-eval 95.8%

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

   4. swap-sqr 95.6%

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

   5. metadata-eval 96.3%

      \[\leadsto -3 \cdot \left(s \cdot \left(\log \left(1 - \color{blue}{1.7777777777777777} \cdot \left(\left(0.25 - u\right) \cdot \left(0.25 - u\right)\right)\right) - \log \left(1 - 1.3333333333333333 \cdot \left(0.25 - u\right)\right)\right)\right) \]

   6. pow2 96.3%

      \[\leadsto -3 \cdot \left(s \cdot \left(\log \left(1 - 1.7777777777777777 \cdot \color{blue}{{\left(0.25 - u\right)}^{2}}\right) - \log \left(1 - 1.3333333333333333 \cdot \left(0.25 - u\right)\right)\right)\right) \]

   7. cancel-sign-sub-inv 96.3%

      \[\leadsto -3 \cdot \left(s \cdot \left(\log \left(1 - 1.7777777777777777 \cdot {\left(0.25 - u\right)}^{2}\right) - \log \color{blue}{\left(1 + \left(-1.3333333333333333\right) \cdot \left(0.25 - u\right)\right)}\right)\right) \]

   8. metadata-eval 96.3%

      \[\leadsto -3 \cdot \left(s \cdot \left(\log \left(1 - 1.7777777777777777 \cdot {\left(0.25 - u\right)}^{2}\right) - \log \left(1 + \color{blue}{-1.3333333333333333} \cdot \left(0.25 - u\right)\right)\right)\right) \]

6. Applied egg-rr 96.3%

   \[\leadsto -3 \cdot \left(s \cdot \color{blue}{\left(\log \left(1 - 1.7777777777777777 \cdot {\left(0.25 - u\right)}^{2}\right) - \log \left(1 + -1.3333333333333333 \cdot \left(0.25 - u\right)\right)\right)}\right) \]
7. Step-by-step derivation

   1. sub-neg 96.3%

      \[\leadsto -3 \cdot \left(s \cdot \left(\log \color{blue}{\left(1 + \left(-1.7777777777777777 \cdot {\left(0.25 - u\right)}^{2}\right)\right)} - \log \left(1 + -1.3333333333333333 \cdot \left(0.25 - u\right)\right)\right)\right) \]

   2. log1p-def 96.8%

      \[\leadsto -3 \cdot \left(s \cdot \left(\color{blue}{\mathsf{log1p}\left(-1.7777777777777777 \cdot {\left(0.25 - u\right)}^{2}\right)} - \log \left(1 + -1.3333333333333333 \cdot \left(0.25 - u\right)\right)\right)\right) \]

   3. *-commutative 96.8%

      \[\leadsto -3 \cdot \left(s \cdot \left(\mathsf{log1p}\left(-\color{blue}{{\left(0.25 - u\right)}^{2} \cdot 1.7777777777777777}\right) - \log \left(1 + -1.3333333333333333 \cdot \left(0.25 - u\right)\right)\right)\right) \]

   4. distribute-rgt-neg-in 96.8%

      \[\leadsto -3 \cdot \left(s \cdot \left(\mathsf{log1p}\left(\color{blue}{{\left(0.25 - u\right)}^{2} \cdot \left(-1.7777777777777777\right)}\right) - \log \left(1 + -1.3333333333333333 \cdot \left(0.25 - u\right)\right)\right)\right) \]

   5. metadata-eval 96.8%

      \[\leadsto -3 \cdot \left(s \cdot \left(\mathsf{log1p}\left({\left(0.25 - u\right)}^{2} \cdot \color{blue}{-1.7777777777777777}\right) - \log \left(1 + -1.3333333333333333 \cdot \left(0.25 - u\right)\right)\right)\right) \]

   6. log1p-def 98.1%

      \[\leadsto -3 \cdot \left(s \cdot \left(\mathsf{log1p}\left({\left(0.25 - u\right)}^{2} \cdot -1.7777777777777777\right) - \color{blue}{\mathsf{log1p}\left(-1.3333333333333333 \cdot \left(0.25 - u\right)\right)}\right)\right) \]

   7. sub-neg 98.1%

      \[\leadsto -3 \cdot \left(s \cdot \left(\mathsf{log1p}\left({\left(0.25 - u\right)}^{2} \cdot -1.7777777777777777\right) - \mathsf{log1p}\left(-1.3333333333333333 \cdot \color{blue}{\left(0.25 + \left(-u\right)\right)}\right)\right)\right) \]

   8. mul-1-neg 98.1%

      \[\leadsto -3 \cdot \left(s \cdot \left(\mathsf{log1p}\left({\left(0.25 - u\right)}^{2} \cdot -1.7777777777777777\right) - \mathsf{log1p}\left(-1.3333333333333333 \cdot \left(0.25 + \color{blue}{-1 \cdot u}\right)\right)\right)\right) \]

   9. distribute-lft-in 97.7%

      \[\leadsto -3 \cdot \left(s \cdot \left(\mathsf{log1p}\left({\left(0.25 - u\right)}^{2} \cdot -1.7777777777777777\right) - \mathsf{log1p}\left(\color{blue}{-1.3333333333333333 \cdot 0.25 + -1.3333333333333333 \cdot \left(-1 \cdot u\right)}\right)\right)\right) \]

   10. metadata-eval 97.7%

       \[\leadsto -3 \cdot \left(s \cdot \left(\mathsf{log1p}\left({\left(0.25 - u\right)}^{2} \cdot -1.7777777777777777\right) - \mathsf{log1p}\left(\color{blue}{-0.3333333333333333} + -1.3333333333333333 \cdot \left(-1 \cdot u\right)\right)\right)\right) \]

   11. mul-1-neg 97.7%

       \[\leadsto -3 \cdot \left(s \cdot \left(\mathsf{log1p}\left({\left(0.25 - u\right)}^{2} \cdot -1.7777777777777777\right) - \mathsf{log1p}\left(-0.3333333333333333 + -1.3333333333333333 \cdot \color{blue}{\left(-u\right)}\right)\right)\right) \]

   12. distribute-rgt-neg-in 97.7%

       \[\leadsto -3 \cdot \left(s \cdot \left(\mathsf{log1p}\left({\left(0.25 - u\right)}^{2} \cdot -1.7777777777777777\right) - \mathsf{log1p}\left(-0.3333333333333333 + \color{blue}{\left(--1.3333333333333333 \cdot u\right)}\right)\right)\right) \]

   13. distribute-lft-neg-in 97.7%

       \[\leadsto -3 \cdot \left(s \cdot \left(\mathsf{log1p}\left({\left(0.25 - u\right)}^{2} \cdot -1.7777777777777777\right) - \mathsf{log1p}\left(-0.3333333333333333 + \color{blue}{\left(--1.3333333333333333\right) \cdot u}\right)\right)\right) \]

   14. metadata-eval 97.7%

       \[\leadsto -3 \cdot \left(s \cdot \left(\mathsf{log1p}\left({\left(0.25 - u\right)}^{2} \cdot -1.7777777777777777\right) - \mathsf{log1p}\left(-0.3333333333333333 + \color{blue}{1.3333333333333333} \cdot u\right)\right)\right) \]

8. Simplified 97.7%

   \[\leadsto -3 \cdot \left(s \cdot \color{blue}{\left(\mathsf{log1p}\left({\left(0.25 - u\right)}^{2} \cdot -1.7777777777777777\right) - \mathsf{log1p}\left(-0.3333333333333333 + 1.3333333333333333 \cdot u\right)\right)}\right) \]
9. Step-by-step derivation

   1. associate-*r* 97.7%

      \[\leadsto \color{blue}{\left(-3 \cdot s\right) \cdot \left(\mathsf{log1p}\left({\left(0.25 - u\right)}^{2} \cdot -1.7777777777777777\right) - \mathsf{log1p}\left(-0.3333333333333333 + 1.3333333333333333 \cdot u\right)\right)} \]

   2. sub-neg 97.7%

      \[\leadsto \left(-3 \cdot s\right) \cdot \color{blue}{\left(\mathsf{log1p}\left({\left(0.25 - u\right)}^{2} \cdot -1.7777777777777777\right) + \left(-\mathsf{log1p}\left(-0.3333333333333333 + 1.3333333333333333 \cdot u\right)\right)\right)} \]

   3. distribute-lft-in 97.8%

      \[\leadsto \color{blue}{\left(-3 \cdot s\right) \cdot \mathsf{log1p}\left({\left(0.25 - u\right)}^{2} \cdot -1.7777777777777777\right) + \left(-3 \cdot s\right) \cdot \left(-\mathsf{log1p}\left(-0.3333333333333333 + 1.3333333333333333 \cdot u\right)\right)} \]

   4. *-commutative 97.8%

      \[\leadsto \color{blue}{\left(s \cdot -3\right)} \cdot \mathsf{log1p}\left({\left(0.25 - u\right)}^{2} \cdot -1.7777777777777777\right) + \left(-3 \cdot s\right) \cdot \left(-\mathsf{log1p}\left(-0.3333333333333333 + 1.3333333333333333 \cdot u\right)\right) \]

   5. *-commutative 97.8%

      \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left({\left(0.25 - u\right)}^{2} \cdot -1.7777777777777777\right) + \color{blue}{\left(s \cdot -3\right)} \cdot \left(-\mathsf{log1p}\left(-0.3333333333333333 + 1.3333333333333333 \cdot u\right)\right) \]

   6. +-commutative 97.8%

      \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left({\left(0.25 - u\right)}^{2} \cdot -1.7777777777777777\right) + \left(s \cdot -3\right) \cdot \left(-\mathsf{log1p}\left(\color{blue}{1.3333333333333333 \cdot u + -0.3333333333333333}\right)\right) \]

   7. *-commutative 97.8%

      \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left({\left(0.25 - u\right)}^{2} \cdot -1.7777777777777777\right) + \left(s \cdot -3\right) \cdot \left(-\mathsf{log1p}\left(\color{blue}{u \cdot 1.3333333333333333} + -0.3333333333333333\right)\right) \]

   8. fma-def 98.4%

      \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left({\left(0.25 - u\right)}^{2} \cdot -1.7777777777777777\right) + \left(s \cdot -3\right) \cdot \left(-\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)}\right)\right) \]

10. Applied egg-rr 98.4%

    \[\leadsto \color{blue}{\left(s \cdot -3\right) \cdot \mathsf{log1p}\left({\left(0.25 - u\right)}^{2} \cdot -1.7777777777777777\right) + \left(s \cdot -3\right) \cdot \left(-\mathsf{log1p}\left(\mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)\right)\right)} \]

11. Final simplification 98.4%

    \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left({\left(0.25 - u\right)}^{2} \cdot -1.7777777777777777\right) - \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)\right) \]

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 96.4%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 98.3%

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

   9. distribute-neg-frac 98.3%

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

   10. sub-neg 98.3%

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

   11. metadata-eval 98.3%

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

3. Simplified 98.3%

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

4. Final simplification 98.3%

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

Alternative 3: 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 96.4%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 95.9%

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

   3. metadata-eval 95.9%

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

3. Simplified 95.9%

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

4. Taylor expanded in s around 0 96.4%

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

5. Final simplification 96.4%

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

Alternative 4: 97.9% 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 96.4%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 98.3%

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

   9. distribute-neg-frac 98.3%

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

   10. sub-neg 98.3%

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

   11. metadata-eval 98.3%

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

3. Simplified 98.3%

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

4. Taylor expanded in s around 0 96.6%

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

   1. log1p-def 97.8%

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

   2. sub-neg 97.8%

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

   3. distribute-lft-in 96.9%

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

   4. metadata-eval 96.9%

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

   5. distribute-rgt-neg-in 96.9%

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

   6. *-commutative 96.9%

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

   7. distribute-rgt-neg-in 96.9%

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

   8. metadata-eval 96.9%

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

   9. +-commutative 96.9%

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

   10. fma-udef 97.8%

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

6. Simplified 97.8%

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

   1. fma-udef 96.9%

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

   2. +-commutative 96.9%

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

   3. metadata-eval 96.9%

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

   4. *-commutative 96.9%

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

   5. metadata-eval 96.9%

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

   6. distribute-lft-neg-in 96.9%

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

   7. distribute-rgt-neg-in 96.9%

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

   8. distribute-lft-in 97.8%

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

   9. sub-neg 97.8%

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

   10. *-commutative 97.8%

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

8. Applied egg-rr 97.8%

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

9. Final simplification 97.8%

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

Alternative 5: 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 96.4%

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

2. Taylor expanded in s around 0 96.1%

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

   1. *-commutative 96.1%

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

   2. associate-*l* 96.1%

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

   3. log-rec 96.7%

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

   4. sub-neg 96.7%

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

   5. log1p-def 97.9%

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

   6. sub-neg 97.9%

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

   7. metadata-eval 97.9%

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

   8. distribute-lft-neg-in 97.9%

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

   9. metadata-eval 97.9%

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

   10. distribute-rgt-in 96.8%

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

   11. metadata-eval 96.8%

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

   12. fma-udef 97.9%

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

4. Simplified 97.9%

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

   1. fma-udef 96.9%

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

   2. +-commutative 96.9%

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

   3. metadata-eval 96.9%

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

   4. *-commutative 96.9%

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

   5. metadata-eval 96.9%

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

   6. distribute-lft-neg-in 96.9%

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

   7. distribute-rgt-neg-in 96.9%

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

   8. distribute-lft-in 97.8%

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

   9. sub-neg 97.8%

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

   10. *-commutative 97.8%

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

6. Applied egg-rr 97.9%

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

   1. distribute-lft-neg-out 97.9%

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

   2. neg-sub0 97.9%

      \[\leadsto s \cdot \color{blue}{\left(0 - \mathsf{log1p}\left(\left(0.25 - u\right) \cdot 1.3333333333333333\right) \cdot 3\right)} \]

8. Applied egg-rr 97.9%

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

   1. neg-sub0 97.9%

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

   2. distribute-rgt-neg-in 97.9%

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

   3. *-commutative 97.9%

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

   4. metadata-eval 97.9%

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

10. Simplified 97.9%

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

11. Final simplification 97.9%

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

Alternative 6: 97.9% accurate, 1.0× speedup

Math

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

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(Float32(s * Float32(-3.0)) * log1p(Float32(Float32(Float32(0.25) - u) * Float32(1.3333333333333333))))
end

Derivation

1. Initial program 96.4%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 98.3%

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

   9. distribute-neg-frac 98.3%

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

   10. sub-neg 98.3%

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

   11. metadata-eval 98.3%

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

3. Simplified 98.3%

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

4. Taylor expanded in s around 0 96.6%

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

   1. associate-*r* 96.7%

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

   2. log1p-def 98.0%

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

6. Simplified 98.0%

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

7. Final simplification 98.0%

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

Alternative 7: 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 96.4%

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

2. Taylor expanded in u around 0 26.0%

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

   1. distribute-lft-out 26.0%

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

   2. distribute-lft-out 26.0%

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

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 8: 25.8% 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 96.4%

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

2. Taylor expanded in u around 0 26.0%

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

3. Final simplification 26.0%

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

Alternative 9: 7.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 96.4%

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

2. Taylor expanded in u around 0 7.4%

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

3. Final simplification 7.4%

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

Alternative 10: 10.4% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ \left(s \cdot 3\right) \cdot 0 \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (* s 3.0) 0.0))

C

float code(float s, float u) {
	return (s * 3.0f) * 0.0f;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (s * 3.0e0) * 0.0e0
end function

Julia

function code(s, u)
	return Float32(Float32(s * Float32(3.0)) * Float32(0.0))
end

MATLAB

function tmp = code(s, u)
	tmp = (s * single(3.0)) * single(0.0);
end

Derivation

1. Initial program 96.4%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 95.9%

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

   3. metadata-eval 95.9%

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

3. Simplified 95.9%

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

5. Applied egg-rr 4.9%

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

   1. *-inverses 10.5%

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

7. Simplified 10.5%

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

8. Final simplification 10.5%

   \[\leadsto \left(s \cdot 3\right) \cdot 0 \]

Reproduce

herbie shell --seed 2023334 
(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))))))