Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.8% → 98.2%

Time: 9.4s

Alternatives: 6

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: 98.2% 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 95.3%

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

   1. log-rec 96.4%

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

   2. distribute-rgt-neg-out 96.4%

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

   3. distribute-lft-neg-out 96.4%

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

   4. *-commutative 96.4%

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

   5. distribute-rgt-neg-in 96.4%

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

   6. metadata-eval 96.4%

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

   7. sub-neg 96.4%

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

   8. log1p-define 98.4%

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

   9. distribute-neg-frac2 98.4%

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

   10. sub-neg 98.4%

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

   11. metadata-eval 98.4%

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

   12. metadata-eval 98.4%

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

3. Simplified 98.4%

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

Alternative 2: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 95.3%

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

   1. log-rec 96.4%

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

   2. distribute-rgt-neg-out 96.4%

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

   3. distribute-lft-neg-out 96.4%

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

   4. *-commutative 96.4%

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

   5. distribute-rgt-neg-in 96.4%

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

   6. metadata-eval 96.4%

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

   7. sub-neg 96.4%

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

   8. log1p-define 98.4%

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

   9. neg-sub0 98.4%

      \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\color{blue}{0 - \frac{u - 0.25}{0.75}}\right) \]

   10. div-sub 96.3%

       \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(0 - \color{blue}{\left(\frac{u}{0.75} - \frac{0.25}{0.75}\right)}\right) \]

   11. associate--r- 96.3%

       \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\color{blue}{\left(0 - \frac{u}{0.75}\right) + \frac{0.25}{0.75}}\right) \]

   12. neg-sub0 96.3%

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

   13. neg-mul-1 96.3%

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

   14. associate-/l* 96.3%

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

   15. *-commutative 96.3%

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

   16. associate-/l* 96.7%

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

   17. fma-define 98.0%

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

   18. metadata-eval 98.0%

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

   19. metadata-eval 98.0%

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

3. Simplified 98.0%

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

   1. fma-undefine 96.7%

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

   2. metadata-eval 96.7%

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

   3. metadata-eval 96.7%

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

   4. distribute-rgt-in 98.0%

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

   5. sub-neg 98.0%

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

   6. metadata-eval 98.0%

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

   7. distribute-lft-neg-in 98.0%

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

   8. *-commutative 98.0%

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

   9. distribute-rgt-neg-in 98.0%

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

   10. sub-neg 98.0%

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

   11. metadata-eval 98.0%

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

   12. metadata-eval 98.0%

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

6. Applied egg-rr 98.0%

   \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\color{blue}{\left(u + -0.25\right) \cdot -1.3333333333333333}\right) \]
7. 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 95.3%

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

   1. associate-*l* 95.3%

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

   2. log-rec 96.4%

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

   3. div-sub 95.4%

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

   4. metadata-eval 95.4%

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

3. Simplified 95.4%

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

5. Taylor expanded in s around 0 95.8%

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

6. Final simplification 95.8%

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

Alternative 4: 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 95.3%

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

3. Taylor expanded in u around 0 25.1%

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

   1. associate-*r* 25.1%

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

   2. associate-*r* 25.1%

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

   3. distribute-lft-out 25.1%

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

   4. *-commutative 25.1%

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

5. Simplified 25.1%

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

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 95.3%

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

3. Taylor expanded in s around 0 95.3%

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

4. Taylor expanded in u around 0 25.1%

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

Alternative 6: 10.6% accurate, 113.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (s u) :precision binary32 0.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.3%

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

   1. associate-*l* 95.3%

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

   2. log-rec 96.4%

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

   3. div-sub 95.4%

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

   4. metadata-eval 95.4%

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

3. Simplified 95.4%

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

6. Applied egg-rr 6.4%

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

   1. *-inverses 11.0%

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

8. Simplified 11.0%

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

9. Taylor expanded in s around 0 11.0%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024131 
(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))))))