Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.9% → 96.2%

Time: 6.7s

Alternatives: 3

Speedup: 1.2×

Specification

\[\left(0 \leq s \land s \leq 256\right) \land \left(0.25 \leq u \land u \leq 1\right)\]

Math

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

FPCore

(FPCore (s u)
 :precision binary32
 (* (* 3.0 s) (log (/ 1.0 (- 1.0 (/ (- u 0.25) 0.75))))))

C

float code(float s, float u) {
	return (3.0f * s) * logf((1.0f / (1.0f - ((u - 0.25f) / 0.75f))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (3.0e0 * s) * log((1.0e0 / (1.0e0 - ((u - 0.25e0) / 0.75e0))))
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(3.0) * s) * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(u - Float32(0.25)) / Float32(0.75))))))
end

MATLAB

function tmp = code(s, u)
	tmp = (single(3.0) * s) * log((single(1.0) / (single(1.0) - ((u - single(0.25)) / single(0.75)))));
end

Initial Program: 95.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (s u)
 :precision binary32
 (* (* 3.0 s) (log (/ 1.0 (- 1.0 (/ (- u 0.25) 0.75))))))

C

float code(float s, float u) {
	return (3.0f * s) * logf((1.0f / (1.0f - ((u - 0.25f) / 0.75f))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (3.0e0 * s) * log((1.0e0 / (1.0e0 - ((u - 0.25e0) / 0.75e0))))
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(3.0) * s) * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(u - Float32(0.25)) / Float32(0.75))))))
end

MATLAB

function tmp = code(s, u)
	tmp = (single(3.0) * s) * log((single(1.0) / (single(1.0) - ((u - single(0.25)) / single(0.75)))));
end

Alternative 1: 96.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = ((-3.0e0) * s) * log((1.3333333333333333e0 + ((-1.3333333333333333e0) * u)))
end function

Julia

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

MATLAB

function tmp = code(s, u)
	tmp = (single(-3.0) * s) * log((single(1.3333333333333333) + (single(-1.3333333333333333) * u)));
end

Derivation

1. Initial program 95.8%

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

3. Taylor expanded in s around 0

   \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \left(\frac{1}{1 - \frac{4}{3} \cdot \left(u - \frac{1}{4}\right)}\right)\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. log-rec N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\log \left(1 - \frac{4}{3} \cdot \left(u - \frac{1}{4}\right)\right)\right)\right)} \cdot \left(3 \cdot s\right) \]

   4. distribute-lft-neg-out N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\log \left(1 - \frac{4}{3} \cdot \left(u - \frac{1}{4}\right)\right) \cdot \left(3 \cdot s\right)\right)} \]

   5. distribute-rgt-neg-in N/A

      \[\leadsto \color{blue}{\log \left(1 - \frac{4}{3} \cdot \left(u - \frac{1}{4}\right)\right) \cdot \left(\mathsf{neg}\left(3 \cdot s\right)\right)} \]

   6. distribute-lft-neg-in N/A

      \[\leadsto \log \left(1 - \frac{4}{3} \cdot \left(u - \frac{1}{4}\right)\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(3\right)\right) \cdot s\right)} \]

   7. metadata-eval N/A

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

   8. lower-*.f32 N/A

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

5. Applied rewrites 7.2%

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

7. Applied rewrites 96.3%

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

8. Final simplification 96.3%

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

Alternative 2: 26.4% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(u \cdot u\right) \cdot 0.5\right) \cdot 3\right) \cdot s \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (* (* (* u u) 0.5) 3.0) s))

C

float code(float s, float u) {
	return (((u * u) * 0.5f) * 3.0f) * s;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (((u * u) * 0.5e0) * 3.0e0) * s
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(Float32(u * u) * Float32(0.5)) * Float32(3.0)) * s)
end

MATLAB

function tmp = code(s, u)
	tmp = (((u * u) * single(0.5)) * single(3.0)) * s;
end

Derivation

1. Initial program 95.8%

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

3. Taylor expanded in u around 0

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} + u \cdot \left(1 + \frac{1}{2} \cdot u\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(u \cdot \left(1 + \frac{1}{2} \cdot u\right) + \log \frac{3}{4}\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\left(1 + \frac{1}{2} \cdot u\right) \cdot u} + \log \frac{3}{4}\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\mathsf{fma}\left(1 + \frac{1}{2} \cdot u, u, \log \frac{3}{4}\right)} \]

   4. +-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot u + 1}, u, \log \frac{3}{4}\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, u, 1\right)}, u, \log \frac{3}{4}\right) \]

   6. lower-log.f32 10.8

      \[\leadsto \left(3 \cdot s\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, u, 1\right), u, \color{blue}{\log 0.75}\right) \]

5. Applied rewrites 10.8%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, u, 1\right), u, \log 0.75\right)} \]

6. Taylor expanded in u around inf

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

8. Applied rewrites 26.5%

   \[\leadsto \left(3 \cdot s\right) \cdot \left(\left(u \cdot u\right) \cdot \color{blue}{0.5}\right) \]
9. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \left(\left(u \cdot u\right) \cdot \frac{1}{2}\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(u \cdot u\right) \cdot \frac{1}{2}\right) \cdot \left(3 \cdot s\right)} \]

   3. lift-*.f32 N/A

      \[\leadsto \left(\left(u \cdot u\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(3 \cdot s\right)} \]

   4. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\left(u \cdot u\right) \cdot \frac{1}{2}\right) \cdot 3\right) \cdot s} \]

   5. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\left(\left(u \cdot u\right) \cdot \frac{1}{2}\right) \cdot 3\right) \cdot s} \]

   6. lower-*.f32 26.5

      \[\leadsto \color{blue}{\left(\left(\left(u \cdot u\right) \cdot 0.5\right) \cdot 3\right)} \cdot s \]

10. Applied rewrites 26.5%

    \[\leadsto \color{blue}{\left(\left(\left(u \cdot u\right) \cdot 0.5\right) \cdot 3\right) \cdot s} \]
11. Add Preprocessing

Alternative 3: 26.4% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \left(s \cdot 3\right) \cdot \left(\left(u \cdot u\right) \cdot 0.5\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (* s 3.0) (* (* u u) 0.5)))

C

float code(float s, float u) {
	return (s * 3.0f) * ((u * u) * 0.5f);
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (s * 3.0e0) * ((u * u) * 0.5e0)
end function

Julia

function code(s, u)
	return Float32(Float32(s * Float32(3.0)) * Float32(Float32(u * u) * Float32(0.5)))
end

MATLAB

function tmp = code(s, u)
	tmp = (s * single(3.0)) * ((u * u) * single(0.5));
end

Derivation

1. Initial program 95.8%

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

3. Taylor expanded in u around 0

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} + u \cdot \left(1 + \frac{1}{2} \cdot u\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(u \cdot \left(1 + \frac{1}{2} \cdot u\right) + \log \frac{3}{4}\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\left(1 + \frac{1}{2} \cdot u\right) \cdot u} + \log \frac{3}{4}\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\mathsf{fma}\left(1 + \frac{1}{2} \cdot u, u, \log \frac{3}{4}\right)} \]

   4. +-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot u + 1}, u, \log \frac{3}{4}\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, u, 1\right)}, u, \log \frac{3}{4}\right) \]

   6. lower-log.f32 10.8

      \[\leadsto \left(3 \cdot s\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, u, 1\right), u, \color{blue}{\log 0.75}\right) \]

5. Applied rewrites 10.8%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, u, 1\right), u, \log 0.75\right)} \]

6. Taylor expanded in u around inf

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

8. Applied rewrites 26.5%

   \[\leadsto \left(3 \cdot s\right) \cdot \left(\left(u \cdot u\right) \cdot \color{blue}{0.5}\right) \]

9. Final simplification 26.5%

   \[\leadsto \left(s \cdot 3\right) \cdot \left(\left(u \cdot u\right) \cdot 0.5\right) \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024255 
(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))))))