Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.8% → 98.3%

Time: 14.5s

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: 95.8% 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

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

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 96.2%

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

   1. 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. neg-mul-1 96.7%

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

   3. associate-*r* 96.7%

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

   4. *-commutative 96.7%

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

   5. associate-*l* 96.7%

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

   6. metadata-eval 96.7%

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

   7. sub-neg 96.7%

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

   8. log1p-def 98.1%

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

   9. distribute-neg-frac 98.1%

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

   10. sub-neg 98.1%

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

   11. metadata-eval 98.1%

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

3. Simplified 98.1%

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

4. Final simplification 98.1%

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

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. *-commutative 96.2%

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

   2. associate-*l* 96.1%

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

   3. log-rec 96.8%

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

   4. neg-mul-1 96.8%

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

   5. associate-*r* 96.8%

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

   6. metadata-eval 96.8%

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

   7. metadata-eval 96.8%

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

   8. *-commutative 96.8%

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

3. Simplified 98.1%

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

4. Taylor expanded in s around 0 96.2%

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

5. Final simplification 96.2%

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

Alternative 3: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 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. 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. neg-mul-1 96.7%

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

   3. associate-*r* 96.7%

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

   4. *-commutative 96.7%

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

   5. associate-*l* 96.7%

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

   6. metadata-eval 96.7%

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

   7. sub-neg 96.7%

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

   8. log1p-def 98.1%

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

   9. distribute-neg-frac 98.1%

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

   10. sub-neg 98.1%

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

   11. metadata-eval 98.1%

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

3. Simplified 98.1%

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

4. Taylor expanded in s around 0 96.7%

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

   1. log1p-def 98.0%

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

6. Simplified 98.0%

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

7. Final simplification 98.0%

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

Alternative 4: 18.7% 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. 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. neg-mul-1 96.7%

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

   3. associate-*r* 96.7%

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

   4. *-commutative 96.7%

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

   5. associate-*l* 96.7%

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

   6. metadata-eval 96.7%

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

   7. sub-neg 96.7%

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

   8. log1p-def 98.1%

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

   9. distribute-neg-frac 98.1%

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

   10. sub-neg 98.1%

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

   11. metadata-eval 98.1%

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

3. Simplified 98.1%

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

4. Taylor expanded in s around 0 96.7%

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

   1. log1p-def 98.0%

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

6. Simplified 98.0%

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

7. Taylor expanded in u around inf 19.5%

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

   1. *-commutative 19.5%

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

9. Simplified 19.5%

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

10. Final simplification 19.5%

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

Alternative 5: 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 96.2%

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

2. Taylor expanded in u around 0 26.4%

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

   1. distribute-lft-out 26.4%

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

   2. distribute-lft-out 26.4%

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

4. Simplified 26.4%

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

5. Final simplification 26.4%

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

Alternative 6: 25.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(s, u)
	tmp = s * (log(single(0.421875)) + (u * single(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. *-commutative 96.2%

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

   2. associate-*l* 96.1%

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

   3. log-rec 96.8%

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

   4. neg-mul-1 96.8%

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

   5. associate-*r* 96.8%

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

   6. metadata-eval 96.8%

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

   7. sub-neg 96.8%

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

   8. log1p-def 98.4%

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

   9. neg-mul-1 98.4%

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

   10. associate-*r/ 98.4%

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

   11. associate-/l* 97.9%

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

   12. associate-/r/ 98.1%

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

   13. sub-neg 98.1%

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

   14. distribute-lft-in 97.0%

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

   15. metadata-eval 97.0%

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

   16. metadata-eval 97.0%

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

   17. metadata-eval 97.0%

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

   18. metadata-eval 97.0%

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

3. Simplified 97.0%

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

   1. *-commutative 97.0%

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

   2. fma-udef 98.1%

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

   3. expm1-log1p-u 97.8%

      \[\leadsto s \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-3 \cdot \mathsf{log1p}\left(\mathsf{fma}\left(u, -1.3333333333333333, 0.3333333333333333\right)\right)\right)\right)} \]

5. Applied egg-rr 97.8%

   \[\leadsto s \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-3 \cdot \mathsf{log1p}\left(\mathsf{fma}\left(u, -1.3333333333333333, 0.3333333333333333\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. expm1-log1p 98.1%

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

   2. add-log-exp 97.5%

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

   3. *-commutative 97.5%

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

   4. log1p-udef 96.5%

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

   5. fma-udef 95.6%

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

   6. *-commutative 95.6%

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

   7. associate-+r+ 95.6%

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

   8. +-commutative 95.6%

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

   9. *-commutative 95.6%

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

   10. fma-udef 96.0%

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

   11. exp-to-pow 96.2%

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

   12. fma-udef 95.6%

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

   13. *-commutative 95.6%

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

   14. associate-+l+ 96.1%

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

   15. metadata-eval 96.1%

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

   16. *-commutative 96.1%

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

   17. fma-def 96.6%

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

7. Applied egg-rr 96.6%

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

8. Taylor expanded in u around 0 26.4%

   \[\leadsto s \cdot \color{blue}{\left(\log 0.421875 + 3 \cdot u\right)} \]
9. Step-by-step derivation

   1. *-commutative 26.4%

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

10. Simplified 26.4%

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

11. Final simplification 26.4%

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

Alternative 7: 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. *-commutative 96.2%

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

   2. associate-*l* 96.1%

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

   3. log-rec 96.8%

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

   4. neg-mul-1 96.8%

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

   5. associate-*r* 96.8%

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

   6. metadata-eval 96.8%

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

   7. sub-neg 96.8%

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

   8. log1p-def 98.4%

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

   9. neg-mul-1 98.4%

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

   10. associate-*r/ 98.4%

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

   11. associate-/l* 97.9%

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

   12. associate-/r/ 98.1%

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

   13. sub-neg 98.1%

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

   14. distribute-lft-in 97.0%

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

   15. metadata-eval 97.0%

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

   16. metadata-eval 97.0%

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

   17. metadata-eval 97.0%

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

   18. metadata-eval 97.0%

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

3. Simplified 97.0%

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

   1. *-commutative 97.0%

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

   2. fma-udef 98.1%

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

   3. log1p-expm1-u 98.1%

      \[\leadsto s \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-3 \cdot \mathsf{log1p}\left(\mathsf{fma}\left(u, -1.3333333333333333, 0.3333333333333333\right)\right)\right)\right)} \]

5. Applied egg-rr 98.1%

   \[\leadsto s \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-3 \cdot \mathsf{log1p}\left(\mathsf{fma}\left(u, -1.3333333333333333, 0.3333333333333333\right)\right)\right)\right)} \]

6. Taylor expanded in u around 0 7.7%

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

7. Final simplification 7.7%

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

Alternative 8: 10.5% 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 96.2%

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

   1. 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. neg-mul-1 96.7%

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

   4. metadata-eval 96.7%

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

   5. cancel-sign-sub-inv 96.7%

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

   6. *-lft-identity 96.7%

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

   7. sub-neg 96.7%

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

   8. metadata-eval 96.7%

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

3. Simplified 96.7%

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

5. Applied egg-rr 10.8%

   \[\leadsto \left(3 \cdot s\right) \cdot \left(-\log \color{blue}{\left(\frac{\sqrt{u \cdot 1.3333333333333333 + 0.6666666666666666}}{\sqrt{u \cdot 1.3333333333333333 + 0.6666666666666666}}\right)}\right) \]
6. Step-by-step derivation

   1. *-inverses 10.8%

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

7. Simplified 10.8%

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

8. Final simplification 10.8%

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

Reproduce

herbie shell --seed 2023326 
(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))))))