Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.9% → 98.3%

Time: 12.2s

Alternatives: 9

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: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(3 \cdot s, \mathsf{log1p}\left(\mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)\right), \mathsf{log1p}\left(\mathsf{fma}\left(1.7777777777777777, u, -0.4444444444444444\right) \cdot \left(0.25 - u\right)\right) \cdot \left(3 \cdot \left(-s\right)\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (fma
  (* 3.0 s)
  (log1p (fma u 1.3333333333333333 -0.3333333333333333))
  (*
   (log1p (* (fma 1.7777777777777777 u -0.4444444444444444) (- 0.25 u)))
   (* 3.0 (- s)))))

C

float code(float s, float u) {
	return fmaf((3.0f * s), log1pf(fmaf(u, 1.3333333333333333f, -0.3333333333333333f)), (log1pf((fmaf(1.7777777777777777f, u, -0.4444444444444444f) * (0.25f - u))) * (3.0f * -s)));
}

Julia

function code(s, u)
	return fma(Float32(Float32(3.0) * s), log1p(fma(u, Float32(1.3333333333333333), Float32(-0.3333333333333333))), Float32(log1p(Float32(fma(Float32(1.7777777777777777), u, Float32(-0.4444444444444444)) * Float32(Float32(0.25) - u))) * Float32(Float32(3.0) * Float32(-s))))
end

Derivation

1. Initial program 95.5%

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

4. Applied rewrites 98.3%

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

6. Applied rewrites 98.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(3 \cdot s, \mathsf{log1p}\left(\mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)\right), \left(3 \cdot s\right) \cdot \left(-\mathsf{log1p}\left(\mathsf{fma}\left(1.7777777777777777, u, -0.4444444444444444\right) \cdot \left(0.25 - u\right)\right)\right)\right)} \]

7. Final simplification 98.5%

   \[\leadsto \mathsf{fma}\left(3 \cdot s, \mathsf{log1p}\left(\mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)\right), \mathsf{log1p}\left(\mathsf{fma}\left(1.7777777777777777, u, -0.4444444444444444\right) \cdot \left(0.25 - u\right)\right) \cdot \left(3 \cdot \left(-s\right)\right)\right) \]
8. Add Preprocessing

Alternative 2: 98.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(s \cdot \mathsf{log1p}\left(\mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)\right), 3, \left(s \cdot \mathsf{log1p}\left(\left(0.25 - u\right) \cdot \mathsf{fma}\left(u, 1.7777777777777777, -0.4444444444444444\right)\right)\right) \cdot -3\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (fma
  (* s (log1p (fma u 1.3333333333333333 -0.3333333333333333)))
  3.0
  (*
   (* s (log1p (* (- 0.25 u) (fma u 1.7777777777777777 -0.4444444444444444))))
   -3.0)))

C

float code(float s, float u) {
	return fmaf((s * log1pf(fmaf(u, 1.3333333333333333f, -0.3333333333333333f))), 3.0f, ((s * log1pf(((0.25f - u) * fmaf(u, 1.7777777777777777f, -0.4444444444444444f)))) * -3.0f));
}

Julia

function code(s, u)
	return fma(Float32(s * log1p(fma(u, Float32(1.3333333333333333), Float32(-0.3333333333333333)))), Float32(3.0), Float32(Float32(s * log1p(Float32(Float32(Float32(0.25) - u) * fma(u, Float32(1.7777777777777777), Float32(-0.4444444444444444))))) * Float32(-3.0)))
end

Derivation

1. Initial program 95.5%

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

4. Applied rewrites 98.3%

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

6. Applied rewrites 98.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(3 \cdot s, \mathsf{log1p}\left(\mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)\right), \left(3 \cdot s\right) \cdot \left(-\mathsf{log1p}\left(\mathsf{fma}\left(1.7777777777777777, u, -0.4444444444444444\right) \cdot \left(0.25 - u\right)\right)\right)\right)} \]

8. Applied rewrites 98.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(s \cdot \mathsf{log1p}\left(\mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)\right), 3, -3 \cdot \left(s \cdot \mathsf{log1p}\left(\left(0.25 - u\right) \cdot \mathsf{fma}\left(u, 1.7777777777777777, -0.4444444444444444\right)\right)\right)\right)} \]
9. Step-by-step derivation

   1. lift--.f32 N/A

      \[\leadsto \mathsf{fma}\left(s \cdot \mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{4}{3}, \frac{-1}{3}\right)\right), 3, \mathsf{neg}\left(3 \cdot \left(s \cdot \log \left(1 + \color{blue}{\left(\frac{1}{4} - u\right)} \cdot \left(u \cdot \frac{16}{9} + \frac{-4}{9}\right)\right)\right)\right)\right) \]

   2. lift-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(s \cdot \mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{4}{3}, \frac{-1}{3}\right)\right), 3, \mathsf{neg}\left(3 \cdot \left(s \cdot \log \left(1 + \left(\frac{1}{4} - u\right) \cdot \color{blue}{\mathsf{fma}\left(u, \frac{16}{9}, \frac{-4}{9}\right)}\right)\right)\right)\right) \]

   3. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(s \cdot \mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{4}{3}, \frac{-1}{3}\right)\right), 3, \mathsf{neg}\left(3 \cdot \left(s \cdot \log \left(1 + \color{blue}{\left(\frac{1}{4} - u\right) \cdot \mathsf{fma}\left(u, \frac{16}{9}, \frac{-4}{9}\right)}\right)\right)\right)\right) \]

   4. lift-log1p.f32 N/A

      \[\leadsto \mathsf{fma}\left(s \cdot \mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{4}{3}, \frac{-1}{3}\right)\right), 3, \mathsf{neg}\left(3 \cdot \left(s \cdot \color{blue}{\mathsf{log1p}\left(\left(\frac{1}{4} - u\right) \cdot \mathsf{fma}\left(u, \frac{16}{9}, \frac{-4}{9}\right)\right)}\right)\right)\right) \]

   5. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(s \cdot \mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{4}{3}, \frac{-1}{3}\right)\right), 3, \mathsf{neg}\left(3 \cdot \color{blue}{\left(s \cdot \mathsf{log1p}\left(\left(\frac{1}{4} - u\right) \cdot \mathsf{fma}\left(u, \frac{16}{9}, \frac{-4}{9}\right)\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{fma}\left(s \cdot \mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{4}{3}, \frac{-1}{3}\right)\right), 3, \color{blue}{\left(\mathsf{neg}\left(3\right)\right) \cdot \left(s \cdot \mathsf{log1p}\left(\left(\frac{1}{4} - u\right) \cdot \mathsf{fma}\left(u, \frac{16}{9}, \frac{-4}{9}\right)\right)\right)}\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(s \cdot \mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{4}{3}, \frac{-1}{3}\right)\right), 3, \color{blue}{\left(s \cdot \mathsf{log1p}\left(\left(\frac{1}{4} - u\right) \cdot \mathsf{fma}\left(u, \frac{16}{9}, \frac{-4}{9}\right)\right)\right) \cdot \left(\mathsf{neg}\left(3\right)\right)}\right) \]

   8. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(s \cdot \mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{4}{3}, \frac{-1}{3}\right)\right), 3, \color{blue}{\left(s \cdot \mathsf{log1p}\left(\left(\frac{1}{4} - u\right) \cdot \mathsf{fma}\left(u, \frac{16}{9}, \frac{-4}{9}\right)\right)\right) \cdot \left(\mathsf{neg}\left(3\right)\right)}\right) \]

   9. metadata-eval 98.4

      \[\leadsto \mathsf{fma}\left(s \cdot \mathsf{log1p}\left(\mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)\right), 3, \left(s \cdot \mathsf{log1p}\left(\left(0.25 - u\right) \cdot \mathsf{fma}\left(u, 1.7777777777777777, -0.4444444444444444\right)\right)\right) \cdot \color{blue}{-3}\right) \]

10. Applied rewrites 98.4%

    \[\leadsto \mathsf{fma}\left(s \cdot \mathsf{log1p}\left(\mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)\right), 3, \color{blue}{\left(s \cdot \mathsf{log1p}\left(\left(0.25 - u\right) \cdot \mathsf{fma}\left(u, 1.7777777777777777, -0.4444444444444444\right)\right)\right) \cdot -3}\right) \]
11. Add Preprocessing

Alternative 3: 98.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ 3 \cdot \mathsf{fma}\left(s, -\mathsf{log1p}\left(\left(0.25 - u\right) \cdot \mathsf{fma}\left(u, 1.7777777777777777, -0.4444444444444444\right)\right), s \cdot \mathsf{log1p}\left(\mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (*
  3.0
  (fma
   s
   (- (log1p (* (- 0.25 u) (fma u 1.7777777777777777 -0.4444444444444444))))
   (* s (log1p (fma u 1.3333333333333333 -0.3333333333333333))))))

C

float code(float s, float u) {
	return 3.0f * fmaf(s, -log1pf(((0.25f - u) * fmaf(u, 1.7777777777777777f, -0.4444444444444444f))), (s * log1pf(fmaf(u, 1.3333333333333333f, -0.3333333333333333f))));
}

Julia

function code(s, u)
	return Float32(Float32(3.0) * fma(s, Float32(-log1p(Float32(Float32(Float32(0.25) - u) * fma(u, Float32(1.7777777777777777), Float32(-0.4444444444444444))))), Float32(s * log1p(fma(u, Float32(1.3333333333333333), Float32(-0.3333333333333333))))))
end

Derivation

1. Initial program 95.5%

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

4. Applied rewrites 98.3%

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

6. Applied rewrites 98.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(3 \cdot s, \mathsf{log1p}\left(\mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)\right), \left(3 \cdot s\right) \cdot \left(-\mathsf{log1p}\left(\mathsf{fma}\left(1.7777777777777777, u, -0.4444444444444444\right) \cdot \left(0.25 - u\right)\right)\right)\right)} \]

8. Applied rewrites 98.4%

   \[\leadsto \color{blue}{3 \cdot \mathsf{fma}\left(s, -\mathsf{log1p}\left(\left(0.25 - u\right) \cdot \mathsf{fma}\left(u, 1.7777777777777777, -0.4444444444444444\right)\right), s \cdot \mathsf{log1p}\left(\mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)\right)\right)} \]
9. Add Preprocessing

Alternative 4: 98.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(1.7777777777777777, u, -0.4444444444444444\right) \cdot \left(0.25 - u\right)\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (*
  (* 3.0 s)
  (-
   (log1p (fma u 1.3333333333333333 -0.3333333333333333))
   (log1p (* (fma 1.7777777777777777 u -0.4444444444444444) (- 0.25 u))))))

C

float code(float s, float u) {
	return (3.0f * s) * (log1pf(fmaf(u, 1.3333333333333333f, -0.3333333333333333f)) - log1pf((fmaf(1.7777777777777777f, u, -0.4444444444444444f) * (0.25f - u))));
}

Julia

function code(s, u)
	return Float32(Float32(Float32(3.0) * s) * Float32(log1p(fma(u, Float32(1.3333333333333333), Float32(-0.3333333333333333))) - log1p(Float32(fma(Float32(1.7777777777777777), u, Float32(-0.4444444444444444)) * Float32(Float32(0.25) - u)))))
end

Derivation

1. Initial program 95.5%

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

4. Applied rewrites 98.3%

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

   1. lift-fma.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \left(1 + \color{blue}{\mathsf{fma}\left(u, \frac{4}{3}, \frac{-1}{3}\right)}\right) - \log \left(1 + \left(\frac{1}{4} - u\right) \cdot \left(\frac{16}{9} \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   2. lift-log1p.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{4}{3}, \frac{-1}{3}\right)\right)} - \log \left(1 + \left(\frac{1}{4} - u\right) \cdot \left(\frac{16}{9} \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   3. lift--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{4}{3}, \frac{-1}{3}\right)\right) - \log \left(1 + \color{blue}{\left(\frac{1}{4} - u\right)} \cdot \left(\frac{16}{9} \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   4. lift-+.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{4}{3}, \frac{-1}{3}\right)\right) - \log \left(1 + \left(\frac{1}{4} - u\right) \cdot \left(\frac{16}{9} \cdot \color{blue}{\left(u + \frac{-1}{4}\right)}\right)\right)\right) \]

   5. lift-*.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{4}{3}, \frac{-1}{3}\right)\right) - \log \left(1 + \left(\frac{1}{4} - u\right) \cdot \color{blue}{\left(\frac{16}{9} \cdot \left(u + \frac{-1}{4}\right)\right)}\right)\right) \]

   6. lift-*.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{4}{3}, \frac{-1}{3}\right)\right) - \log \left(1 + \color{blue}{\left(\frac{1}{4} - u\right) \cdot \left(\frac{16}{9} \cdot \left(u + \frac{-1}{4}\right)\right)}\right)\right) \]

   7. lift-log1p.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{4}{3}, \frac{-1}{3}\right)\right) - \color{blue}{\mathsf{log1p}\left(\left(\frac{1}{4} - u\right) \cdot \left(\frac{16}{9} \cdot \left(u + \frac{-1}{4}\right)\right)\right)}\right) \]

   8. lift--.f32 98.3

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

   9. lift-*.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{4}{3}, \frac{-1}{3}\right)\right) - \mathsf{log1p}\left(\color{blue}{\left(\frac{1}{4} - u\right) \cdot \left(\frac{16}{9} \cdot \left(u + \frac{-1}{4}\right)\right)}\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{4}{3}, \frac{-1}{3}\right)\right) - \mathsf{log1p}\left(\color{blue}{\left(\frac{16}{9} \cdot \left(u + \frac{-1}{4}\right)\right) \cdot \left(\frac{1}{4} - u\right)}\right)\right) \]

   11. lower-*.f32 98.3

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

   12. lift-*.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{4}{3}, \frac{-1}{3}\right)\right) - \mathsf{log1p}\left(\color{blue}{\left(\frac{16}{9} \cdot \left(u + \frac{-1}{4}\right)\right)} \cdot \left(\frac{1}{4} - u\right)\right)\right) \]

   13. lift-+.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{4}{3}, \frac{-1}{3}\right)\right) - \mathsf{log1p}\left(\left(\frac{16}{9} \cdot \color{blue}{\left(u + \frac{-1}{4}\right)}\right) \cdot \left(\frac{1}{4} - u\right)\right)\right) \]

   14. distribute-lft-in N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{4}{3}, \frac{-1}{3}\right)\right) - \mathsf{log1p}\left(\color{blue}{\left(\frac{16}{9} \cdot u + \frac{16}{9} \cdot \frac{-1}{4}\right)} \cdot \left(\frac{1}{4} - u\right)\right)\right) \]

   15. lower-fma.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{4}{3}, \frac{-1}{3}\right)\right) - \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\frac{16}{9}, u, \frac{16}{9} \cdot \frac{-1}{4}\right)} \cdot \left(\frac{1}{4} - u\right)\right)\right) \]

   16. metadata-eval 98.3

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(1.7777777777777777, u, \color{blue}{-0.4444444444444444}\right) \cdot \left(0.25 - u\right)\right)\right) \]

6. Applied rewrites 98.3%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\mathsf{log1p}\left(\mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(1.7777777777777777, u, -0.4444444444444444\right) \cdot \left(0.25 - u\right)\right)\right)} \]
7. Add Preprocessing

Alternative 5: 98.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u, \mathsf{fma}\left(u, -1.7777777777777777, 0.8888888888888888\right), -0.1111111111111111\right)\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (*
  (* 3.0 s)
  (-
   (log1p (fma u 1.3333333333333333 -0.3333333333333333))
   (log1p
    (fma
     u
     (fma u -1.7777777777777777 0.8888888888888888)
     -0.1111111111111111)))))

C

float code(float s, float u) {
	return (3.0f * s) * (log1pf(fmaf(u, 1.3333333333333333f, -0.3333333333333333f)) - log1pf(fmaf(u, fmaf(u, -1.7777777777777777f, 0.8888888888888888f), -0.1111111111111111f)));
}

Julia

function code(s, u)
	return Float32(Float32(Float32(3.0) * s) * Float32(log1p(fma(u, Float32(1.3333333333333333), Float32(-0.3333333333333333))) - log1p(fma(u, fma(u, Float32(-1.7777777777777777), Float32(0.8888888888888888)), Float32(-0.1111111111111111)))))
end

Derivation

1. Initial program 95.5%

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in u around 0

   \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{4}{3}, \frac{-1}{3}\right)\right) - \mathsf{log1p}\left(\color{blue}{u \cdot \left(\frac{8}{9} + \frac{-16}{9} \cdot u\right) - \frac{1}{9}}\right)\right) \]
6. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{4}{3}, \frac{-1}{3}\right)\right) - \mathsf{log1p}\left(\color{blue}{u \cdot \left(\frac{8}{9} + \frac{-16}{9} \cdot u\right) + \left(\mathsf{neg}\left(\frac{1}{9}\right)\right)}\right)\right) \]

   2. lower-fma.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{4}{3}, \frac{-1}{3}\right)\right) - \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(u, \frac{8}{9} + \frac{-16}{9} \cdot u, \mathsf{neg}\left(\frac{1}{9}\right)\right)}\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{4}{3}, \frac{-1}{3}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u, \color{blue}{\frac{-16}{9} \cdot u + \frac{8}{9}}, \mathsf{neg}\left(\frac{1}{9}\right)\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{4}{3}, \frac{-1}{3}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u, \color{blue}{u \cdot \frac{-16}{9}} + \frac{8}{9}, \mathsf{neg}\left(\frac{1}{9}\right)\right)\right)\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{4}{3}, \frac{-1}{3}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left(u, \frac{-16}{9}, \frac{8}{9}\right)}, \mathsf{neg}\left(\frac{1}{9}\right)\right)\right)\right) \]

   6. metadata-eval 98.0

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u, \mathsf{fma}\left(u, -1.7777777777777777, 0.8888888888888888\right), \color{blue}{-0.1111111111111111}\right)\right)\right) \]

7. Applied rewrites 98.0%

   \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)\right) - \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, -1.7777777777777777, 0.8888888888888888\right), -0.1111111111111111\right)}\right)\right) \]
8. Add Preprocessing

Alternative 6: 97.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 95.5%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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. log-rec N/A

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

   3. distribute-rgt-neg-out N/A

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

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

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

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

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

   6. metadata-eval N/A

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

   7. lower-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. sub-neg N/A

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

   11. lower-log1p.f32 N/A

       \[\leadsto \left(s \cdot -3\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(\frac{4}{3} \cdot \left(u - \frac{1}{4}\right)\right)\right)} \]

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

       \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{4}{3}\right)\right) \cdot \left(u - \frac{1}{4}\right)}\right) \]

   13. metadata-eval N/A

       \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\color{blue}{\frac{-4}{3}} \cdot \left(u - \frac{1}{4}\right)\right) \]

   14. sub-neg N/A

       \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\frac{-4}{3} \cdot \color{blue}{\left(u + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)}\right) \]

   15. distribute-lft-in N/A

       \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\color{blue}{\frac{-4}{3} \cdot u + \frac{-4}{3} \cdot \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}\right) \]

   16. metadata-eval N/A

       \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\frac{-4}{3} \cdot u + \frac{-4}{3} \cdot \color{blue}{\frac{-1}{4}}\right) \]

   17. metadata-eval N/A

       \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\frac{-4}{3} \cdot u + \color{blue}{\frac{1}{3}}\right) \]

   18. lower-fma.f32 97.9

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

5. Applied rewrites 97.9%

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

Alternative 7: 96.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 95.5%

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

4. Applied rewrites 97.6%

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

5. Taylor expanded in s around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. cancel-sign-sub-inv N/A

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

   6. metadata-eval N/A

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

   7. lower-log.f32 N/A

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f32 96.9

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

7. Applied rewrites 96.9%

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

Alternative 8: 30.0% accurate, 12.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.5%

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

3. Taylor expanded in u around 0

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

   1. distribute-lft-out N/A

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

   2. lower-*.f32 N/A

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

   3. distribute-lft-out N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-+.f32 N/A

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

   7. lower-log.f32 25.6

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

5. Applied rewrites 25.6%

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

6. Taylor expanded in u around inf

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 30.0

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

8. Applied rewrites 30.0%

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

9. Final simplification 30.0%

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

Alternative 9: 30.0% accurate, 12.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.5%

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

3. Taylor expanded in u around 0

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

   1. distribute-lft-out N/A

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

   2. lower-*.f32 N/A

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

   3. distribute-lft-out N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-+.f32 N/A

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

   7. lower-log.f32 25.6

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

5. Applied rewrites 25.6%

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

6. Taylor expanded in u around inf

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

   1. lower-*.f32 30.0

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

8. Applied rewrites 30.0%

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

Reproduce

herbie shell --seed 2024216 
(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))))))