Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.9% → 98.4%

Time: 12.6s

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: 98.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

4. Applied rewrites 98.4%

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

   1. lift-*.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-log1p.f32 N/A

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

   4. lift--.f32 N/A

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

   5. lift-+.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-log1p.f32 N/A

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

   9. sub-neg N/A

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

   10. distribute-lft-in N/A

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

   11. lower-fma.f32 N/A

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

   12. lower-*.f32 N/A

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

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 98.5%

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

9. Final simplification 98.5%

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

Alternative 2: 98.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

4. Applied rewrites 98.4%

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

   1. lift-*.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-log1p.f32 N/A

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

   4. lift--.f32 N/A

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

   5. lift-+.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-log1p.f32 N/A

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

   9. sub-neg N/A

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

   10. distribute-lft-in N/A

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

   11. lower-fma.f32 N/A

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

   12. lower-*.f32 N/A

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

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. Taylor expanded in s around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-log1p.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower--.f32 N/A

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

   9. sub-neg N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. metadata-eval 98.5

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

9. Applied rewrites 98.5%

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

10. Final simplification 98.5%

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

Alternative 3: 98.3% 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 95.7%

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

4. Applied rewrites 98.4%

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

   1. lift-*.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-log1p.f32 N/A

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

   4. lift--.f32 N/A

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

   5. lift-+.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-log1p.f32 N/A

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

   9. sub-neg N/A

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

   10. distribute-lft-in N/A

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

   11. lower-fma.f32 N/A

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

   12. lower-*.f32 N/A

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

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

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

FPCore

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

C

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

Julia

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

4. Applied rewrites 98.4%

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

5. Final simplification 98.4%

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

Alternative 5: 98.1% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (s u)
 :precision binary32
 (*
  (* s 3.0)
  (-
   (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 (s * 3.0f) * (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(s * Float32(3.0)) * 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 95.7%

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

4. Applied rewrites 98.4%

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

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 98.3

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

7. Applied rewrites 98.3%

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

8. Final simplification 98.3%

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

Alternative 6: 97.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, u)
	return Float32(s * Float32(Float32(-3.0) * log1p(Float32(-fma(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. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative N/A

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

   2. lift--.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift--.f32 N/A

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

   5. /-rgt-identity N/A

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

   6. clear-num N/A

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

   7. lift-/.f32 N/A

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

   8. lift-log.f32 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

4. Applied rewrites 98.0%

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

5. Final simplification 98.0%

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

Alternative 7: 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 95.7%

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

3. Taylor expanded in 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 98.0

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

5. Applied rewrites 98.0%

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

Alternative 8: 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 95.7%

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

   1. div-sub N/A

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

   2. sub-neg N/A

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

   3. associate--r+ N/A

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

   4. lower--.f32 N/A

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

   5. lower--.f32 N/A

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

   6. div-inv N/A

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

   7. lower-*.f32 N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval 95.2

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

4. Applied rewrites 95.2%

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

5. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 96.0

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

7. Applied rewrites 96.0%

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

   1. lift-*.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-log.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 96.0

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

   10. lift-log.f32 N/A

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

   11. lift-/.f32 N/A

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

   12. log-rec N/A

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

   13. lower-neg.f32 N/A

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

   14. lower-log.f32 96.9

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

9. Applied rewrites 96.9%

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

10. Taylor expanded in s around 0

    \[\leadsto \color{blue}{-3 \cdot \left(s \cdot \log \left(\frac{4}{3} + \frac{-4}{3} \cdot u\right)\right)} \]
11. 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. lower-log.f32 N/A

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

    6. +-commutative N/A

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

    7. *-commutative N/A

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

    8. lower-fma.f32 97.1

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

12. Applied rewrites 97.1%

    \[\leadsto \color{blue}{\left(s \cdot -3\right) \cdot \log \left(\mathsf{fma}\left(u, -1.3333333333333333, 1.3333333333333333\right)\right)} \]
13. 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 95.7%

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

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{3 \cdot \left(s \cdot u\right) + 3 \cdot \left(s \cdot \log \frac{3}{4}\right)} \]
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 26.0

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

5. Applied rewrites 26.0%

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

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 2024219 
(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))))))