Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.8% → 98.3%

Time: 9.7s

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

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

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(u + Float32(-0.25)) / Float32(-0.75))))
end

Derivation

1. Initial program 95.7%

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

   1. log-rec 96.8%

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

   2. distribute-rgt-neg-out 96.8%

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

   3. distribute-lft-neg-out 96.8%

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

   4. *-commutative 96.8%

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

   5. distribute-rgt-neg-in 96.8%

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

   6. metadata-eval 96.8%

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

   7. sub-neg 96.8%

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

   8. log1p-define 98.1%

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

   9. distribute-neg-frac2 98.1%

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

   10. sub-neg 98.1%

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

   11. metadata-eval 98.1%

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

   12. metadata-eval 98.1%

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

3. Simplified 98.1%

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

Alternative 2: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(-3 \cdot \mathsf{log1p}\left(u \cdot -1.3333333333333333 + 0.3333333333333333\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* s (* -3.0 (log1p (+ (* u -1.3333333333333333) 0.3333333333333333)))))

C

float code(float s, float u) {
	return s * (-3.0f * log1pf(((u * -1.3333333333333333f) + 0.3333333333333333f)));
}

Julia

function code(s, u)
	return Float32(s * Float32(Float32(-3.0) * log1p(Float32(Float32(u * Float32(-1.3333333333333333)) + Float32(0.3333333333333333)))))
end

Derivation

1. Initial program 95.7%

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

   1. log-rec 96.8%

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

   2. distribute-rgt-neg-out 96.8%

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

   3. distribute-lft-neg-out 96.8%

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

   4. *-commutative 96.8%

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

   5. distribute-rgt-neg-in 96.8%

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

   6. metadata-eval 96.8%

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

   7. sub-neg 96.8%

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

   8. log1p-define 98.1%

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

   9. distribute-neg-frac2 98.1%

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

   10. sub-neg 98.1%

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

   11. metadata-eval 98.1%

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

   12. metadata-eval 98.1%

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

3. Simplified 98.1%

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

5. Taylor expanded in s around 0 96.4%

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

   1. associate-*r* 96.4%

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

   2. *-commutative 96.4%

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

   3. log1p-define 97.7%

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

   4. sub-neg 97.7%

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

   5. metadata-eval 97.7%

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

   6. distribute-rgt-in 96.8%

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

   7. metadata-eval 96.8%

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

   8. fma-undefine 97.7%

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

   9. associate-*r* 97.9%

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

7. Simplified 97.9%

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

   1. fma-undefine 96.8%

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

9. Applied egg-rr 96.8%

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

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 95.7%

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

   1. associate-*l* 95.6%

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

   2. log-rec 96.7%

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

   3. div-sub 95.6%

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

   4. metadata-eval 95.6%

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

3. Simplified 95.6%

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

5. Taylor expanded in s around 0 96.2%

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

6. Final simplification 96.2%

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

Alternative 4: 25.7% 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 95.7%

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

   1. log-rec 96.8%

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

   2. distribute-rgt-neg-out 96.8%

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

   3. distribute-lft-neg-out 96.8%

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

   4. *-commutative 96.8%

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

   5. distribute-rgt-neg-in 96.8%

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

   6. metadata-eval 96.8%

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

   7. sub-neg 96.8%

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

   8. log1p-define 98.1%

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

   9. distribute-neg-frac2 98.1%

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

   10. sub-neg 98.1%

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

   11. metadata-eval 98.1%

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

   12. metadata-eval 98.1%

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

3. Simplified 98.1%

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

5. Taylor expanded in s around 0 96.4%

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

   1. associate-*r* 96.4%

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

   2. *-commutative 96.4%

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

   3. log1p-define 97.7%

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

   4. sub-neg 97.7%

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

   5. metadata-eval 97.7%

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

   6. distribute-rgt-in 96.8%

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

   7. metadata-eval 96.8%

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

   8. fma-undefine 97.7%

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

   9. associate-*r* 97.9%

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

7. Simplified 97.9%

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

   1. pow1 97.9%

      \[\leadsto s \cdot \color{blue}{{\left(-3 \cdot \mathsf{log1p}\left(\mathsf{fma}\left(u, -1.3333333333333333, 0.3333333333333333\right)\right)\right)}^{1}} \]

   2. *-commutative 97.9%

      \[\leadsto s \cdot {\color{blue}{\left(\mathsf{log1p}\left(\mathsf{fma}\left(u, -1.3333333333333333, 0.3333333333333333\right)\right) \cdot -3\right)}}^{1} \]

9. Applied egg-rr 97.9%

   \[\leadsto s \cdot \color{blue}{{\left(\mathsf{log1p}\left(\mathsf{fma}\left(u, -1.3333333333333333, 0.3333333333333333\right)\right) \cdot -3\right)}^{1}} \]
10. Step-by-step derivation

    1. unpow1 97.9%

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

    2. log1p-expm1 97.9%

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

    3. log1p-define 97.1%

       \[\leadsto s \cdot \color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(u, -1.3333333333333333, 0.3333333333333333\right)\right) \cdot -3\right)\right)} \]

    4. rem-exp-log 97.1%

       \[\leadsto s \cdot \log \color{blue}{\left(e^{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(u, -1.3333333333333333, 0.3333333333333333\right)\right) \cdot -3\right)\right)}\right)} \]

    5. log1p-define 97.1%

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

    6. log1p-expm1 97.2%

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

    7. log1p-undefine 96.3%

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

    8. +-commutative 96.3%

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

    9. exp-to-pow 96.4%

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

    10. fma-define 95.6%

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

    11. associate-+l+ 96.1%

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

    12. metadata-eval 96.1%

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

    13. fma-define 97.0%

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

11. Simplified 97.0%

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

12. Taylor expanded in u around 0 25.7%

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

    1. *-commutative 25.7%

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

14. Simplified 25.7%

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

Alternative 5: 25.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.7%

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

3. Taylor expanded in u around 0 25.7%

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

   1. distribute-lft-out 25.7%

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

   2. distribute-lft-out 25.7%

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

5. Simplified 25.7%

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

Alternative 6: 7.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 95.7%

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

3. Taylor expanded in u around 0 7.1%

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

   1. log1p-expm1-u 7.1%

      \[\leadsto 3 \cdot \left(s \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log 0.75\right)\right)}\right) \]

   2. expm1-undefine 7.1%

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

   3. rem-exp-log 7.1%

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

   4. metadata-eval 7.1%

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

5. Applied egg-rr 7.1%

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

Alternative 7: 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 95.7%

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

   1. log-rec 96.8%

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

   2. distribute-rgt-neg-out 96.8%

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

   3. distribute-lft-neg-out 96.8%

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

   4. *-commutative 96.8%

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

   5. distribute-rgt-neg-in 96.8%

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

   6. metadata-eval 96.8%

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

   7. sub-neg 96.8%

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

   8. log1p-define 98.1%

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

   9. distribute-neg-frac2 98.1%

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

   10. sub-neg 98.1%

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

   11. metadata-eval 98.1%

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

   12. metadata-eval 98.1%

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

3. Simplified 98.1%

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

5. Taylor expanded in s around 0 96.4%

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

   1. associate-*r* 96.4%

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

   2. *-commutative 96.4%

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

   3. log1p-define 97.7%

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

   4. sub-neg 97.7%

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

   5. metadata-eval 97.7%

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

   6. distribute-rgt-in 96.8%

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

   7. metadata-eval 96.8%

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

   8. fma-undefine 97.7%

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

   9. associate-*r* 97.9%

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

7. Simplified 97.9%

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

   1. pow1 97.9%

      \[\leadsto s \cdot \color{blue}{{\left(-3 \cdot \mathsf{log1p}\left(\mathsf{fma}\left(u, -1.3333333333333333, 0.3333333333333333\right)\right)\right)}^{1}} \]

   2. *-commutative 97.9%

      \[\leadsto s \cdot {\color{blue}{\left(\mathsf{log1p}\left(\mathsf{fma}\left(u, -1.3333333333333333, 0.3333333333333333\right)\right) \cdot -3\right)}}^{1} \]

9. Applied egg-rr 97.9%

   \[\leadsto s \cdot \color{blue}{{\left(\mathsf{log1p}\left(\mathsf{fma}\left(u, -1.3333333333333333, 0.3333333333333333\right)\right) \cdot -3\right)}^{1}} \]
10. Step-by-step derivation

    1. unpow1 97.9%

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

    2. log1p-expm1 97.9%

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

    3. log1p-define 97.1%

       \[\leadsto s \cdot \color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(u, -1.3333333333333333, 0.3333333333333333\right)\right) \cdot -3\right)\right)} \]

    4. rem-exp-log 97.1%

       \[\leadsto s \cdot \log \color{blue}{\left(e^{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(u, -1.3333333333333333, 0.3333333333333333\right)\right) \cdot -3\right)\right)}\right)} \]

    5. log1p-define 97.1%

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

    6. log1p-expm1 97.2%

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

    7. log1p-undefine 96.3%

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

    8. +-commutative 96.3%

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

    9. exp-to-pow 96.4%

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

    10. fma-define 95.6%

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

    11. associate-+l+ 96.1%

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

    12. metadata-eval 96.1%

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

    13. fma-define 97.0%

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

11. Simplified 97.0%

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

12. Taylor expanded in u around 0 7.1%

    \[\leadsto s \cdot \log \color{blue}{0.421875} \]
13. Add Preprocessing

Alternative 8: 10.5% accurate, 22.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.7%

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

   1. associate-*l* 95.6%

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

   2. log-rec 96.7%

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

   3. div-sub 95.6%

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

   4. metadata-eval 95.6%

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

3. Simplified 95.6%

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

6. Applied egg-rr 5.5%

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

   1. *-inverses 10.3%

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

8. Simplified 10.3%

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

9. Final simplification 10.3%

   \[\leadsto 3 \cdot \left(s \cdot 0\right) \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024135 
(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))))))