Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.9% → 98.5%

Time: 13.4s

Alternatives: 10

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: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(3 \cdot \mathsf{log1p}\left(\left(u + -0.25\right) \cdot \mathsf{fma}\left(u, 1.7777777777777777, 0.8888888888888888\right)\right), s, \mathsf{log1p}\left(\left(u + -0.25\right) \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, -2.3703703703703702, 1.1851851851851851\right), -0.14814814814814814\right)\right) \cdot \left(3 \cdot \left(-s\right)\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (fma
  (* 3.0 (log1p (* (+ u -0.25) (fma u 1.7777777777777777 0.8888888888888888))))
  s
  (*
   (log1p
    (*
     (+ u -0.25)
     (fma
      u
      (fma u -2.3703703703703702 1.1851851851851851)
      -0.14814814814814814)))
   (* 3.0 (- s)))))

C

float code(float s, float u) {
	return fmaf((3.0f * log1pf(((u + -0.25f) * fmaf(u, 1.7777777777777777f, 0.8888888888888888f)))), s, (log1pf(((u + -0.25f) * fmaf(u, fmaf(u, -2.3703703703703702f, 1.1851851851851851f), -0.14814814814814814f))) * (3.0f * -s)));
}

Julia

function code(s, u)
	return fma(Float32(Float32(3.0) * log1p(Float32(Float32(u + Float32(-0.25)) * fma(u, Float32(1.7777777777777777), Float32(0.8888888888888888))))), s, Float32(log1p(Float32(Float32(u + Float32(-0.25)) * fma(u, fma(u, Float32(-2.3703703703703702), Float32(1.1851851851851851)), Float32(-0.14814814814814814)))) * Float32(Float32(3.0) * Float32(-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

4. Applied egg-rr 98.2%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\mathsf{log1p}\left(\left(u + -0.25\right) \cdot \mathsf{fma}\left(1.7777777777777777, u + -0.25, 1.3333333333333333\right)\right) - \mathsf{log1p}\left(\left(\left(u + -0.25\right) \cdot \left(\left(u + -0.25\right) \cdot \left(u + -0.25\right)\right)\right) \cdot -2.3703703703703702\right)\right)} \]

6. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(3 \cdot s, \mathsf{log1p}\left(\left(u + -0.25\right) \cdot \mathsf{fma}\left(1.7777777777777777, u, 0.8888888888888888\right)\right), \left(3 \cdot s\right) \cdot \left(-\mathsf{log1p}\left(\left(u + -0.25\right) \cdot \left(\left(u + -0.25\right) \cdot \left(\left(u + -0.25\right) \cdot -2.3703703703703702\right)\right)\right)\right)\right)} \]

8. Applied egg-rr 98.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{log1p}\left(\left(u + -0.25\right) \cdot \mathsf{fma}\left(u, 1.7777777777777777, 0.8888888888888888\right)\right) \cdot 3, s, -\left(3 \cdot s\right) \cdot \mathsf{log1p}\left(\left(u + -0.25\right) \cdot \left(\left(u + -0.25\right) \cdot \mathsf{fma}\left(u, -2.3703703703703702, 0.5925925925925926\right)\right)\right)\right)} \]

9. Taylor expanded in u around 0

   \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \mathsf{fma}\left(u, \frac{16}{9}, \frac{8}{9}\right)\right) \cdot 3, s, \mathsf{neg}\left(\left(3 \cdot s\right) \cdot \mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \color{blue}{\left(u \cdot \left(\frac{32}{27} + \frac{-64}{27} \cdot u\right) - \frac{4}{27}\right)}\right)\right)\right) \]
10. Step-by-step derivation

    1. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \mathsf{fma}\left(u, \frac{16}{9}, \frac{8}{9}\right)\right) \cdot 3, s, \mathsf{neg}\left(\left(3 \cdot s\right) \cdot \mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \color{blue}{\left(u \cdot \left(\frac{32}{27} + \frac{-64}{27} \cdot u\right) + \left(\mathsf{neg}\left(\frac{4}{27}\right)\right)\right)}\right)\right)\right) \]

    2. lower-fma.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \mathsf{fma}\left(u, \frac{16}{9}, \frac{8}{9}\right)\right) \cdot 3, s, \mathsf{neg}\left(\left(3 \cdot s\right) \cdot \mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \color{blue}{\mathsf{fma}\left(u, \frac{32}{27} + \frac{-64}{27} \cdot u, \mathsf{neg}\left(\frac{4}{27}\right)\right)}\right)\right)\right) \]

    3. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \mathsf{fma}\left(u, \frac{16}{9}, \frac{8}{9}\right)\right) \cdot 3, s, \mathsf{neg}\left(\left(3 \cdot s\right) \cdot \mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \mathsf{fma}\left(u, \color{blue}{\frac{-64}{27} \cdot u + \frac{32}{27}}, \mathsf{neg}\left(\frac{4}{27}\right)\right)\right)\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \mathsf{fma}\left(u, \frac{16}{9}, \frac{8}{9}\right)\right) \cdot 3, s, \mathsf{neg}\left(\left(3 \cdot s\right) \cdot \mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \mathsf{fma}\left(u, \color{blue}{u \cdot \frac{-64}{27}} + \frac{32}{27}, \mathsf{neg}\left(\frac{4}{27}\right)\right)\right)\right)\right) \]

    5. lower-fma.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \mathsf{fma}\left(u, \frac{16}{9}, \frac{8}{9}\right)\right) \cdot 3, s, \mathsf{neg}\left(\left(3 \cdot s\right) \cdot \mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left(u, \frac{-64}{27}, \frac{32}{27}\right)}, \mathsf{neg}\left(\frac{4}{27}\right)\right)\right)\right)\right) \]

    6. metadata-eval 98.6

       \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(\left(u + -0.25\right) \cdot \mathsf{fma}\left(u, 1.7777777777777777, 0.8888888888888888\right)\right) \cdot 3, s, -\left(3 \cdot s\right) \cdot \mathsf{log1p}\left(\left(u + -0.25\right) \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, -2.3703703703703702, 1.1851851851851851\right), \color{blue}{-0.14814814814814814}\right)\right)\right) \]

11. Simplified 98.6%

    \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(\left(u + -0.25\right) \cdot \mathsf{fma}\left(u, 1.7777777777777777, 0.8888888888888888\right)\right) \cdot 3, s, -\left(3 \cdot s\right) \cdot \mathsf{log1p}\left(\left(u + -0.25\right) \cdot \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, -2.3703703703703702, 1.1851851851851851\right), -0.14814814814814814\right)}\right)\right) \]

12. Final simplification 98.6%

    \[\leadsto \mathsf{fma}\left(3 \cdot \mathsf{log1p}\left(\left(u + -0.25\right) \cdot \mathsf{fma}\left(u, 1.7777777777777777, 0.8888888888888888\right)\right), s, \mathsf{log1p}\left(\left(u + -0.25\right) \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, -2.3703703703703702, 1.1851851851851851\right), -0.14814814814814814\right)\right) \cdot \left(3 \cdot \left(-s\right)\right)\right) \]
13. Add Preprocessing

Alternative 2: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u \cdot \left(u + -0.25\right), 1.7777777777777777, \mathsf{fma}\left(u, 0.8888888888888888, -0.2222222222222222\right)\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u, -2.3703703703703702, 0.5925925925925926\right) \cdot \left(\left(u + -0.25\right) \cdot \left(u + -0.25\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (*
  (* 3.0 s)
  (-
   (log1p
    (fma
     (* u (+ u -0.25))
     1.7777777777777777
     (fma u 0.8888888888888888 -0.2222222222222222)))
   (log1p
    (*
     (fma u -2.3703703703703702 0.5925925925925926)
     (* (+ u -0.25) (+ u -0.25)))))))

C

float code(float s, float u) {
	return (3.0f * s) * (log1pf(fmaf((u * (u + -0.25f)), 1.7777777777777777f, fmaf(u, 0.8888888888888888f, -0.2222222222222222f))) - log1pf((fmaf(u, -2.3703703703703702f, 0.5925925925925926f) * ((u + -0.25f) * (u + -0.25f)))));
}

Julia

function code(s, u)
	return Float32(Float32(Float32(3.0) * s) * Float32(log1p(fma(Float32(u * Float32(u + Float32(-0.25))), Float32(1.7777777777777777), fma(u, Float32(0.8888888888888888), Float32(-0.2222222222222222)))) - log1p(Float32(fma(u, Float32(-2.3703703703703702), Float32(0.5925925925925926)) * Float32(Float32(u + Float32(-0.25)) * Float32(u + Float32(-0.25)))))))
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

4. Applied egg-rr 98.2%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\mathsf{log1p}\left(\left(u + -0.25\right) \cdot \mathsf{fma}\left(1.7777777777777777, u + -0.25, 1.3333333333333333\right)\right) - \mathsf{log1p}\left(\left(\left(u + -0.25\right) \cdot \left(\left(u + -0.25\right) \cdot \left(u + -0.25\right)\right)\right) \cdot -2.3703703703703702\right)\right)} \]
5. Step-by-step derivation

   1. distribute-lft-in N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\color{blue}{\left(\frac{16}{9} \cdot u + \frac{16}{9} \cdot \frac{-1}{4}\right)} + \frac{4}{3}\right)\right) - \mathsf{log1p}\left(\left(\left(u + \frac{-1}{4}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right) \cdot \frac{-64}{27}\right)\right) \]

   2. associate-+l+ N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \color{blue}{\left(\frac{16}{9} \cdot u + \left(\frac{16}{9} \cdot \frac{-1}{4} + \frac{4}{3}\right)\right)}\right) - \mathsf{log1p}\left(\left(\left(u + \frac{-1}{4}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right) \cdot \frac{-64}{27}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\color{blue}{u \cdot \frac{16}{9}} + \left(\frac{16}{9} \cdot \frac{-1}{4} + \frac{4}{3}\right)\right)\right) - \mathsf{log1p}\left(\left(\left(u + \frac{-1}{4}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right) \cdot \frac{-64}{27}\right)\right) \]

   4. lower-+.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \color{blue}{\left(u \cdot \frac{16}{9} + \left(\frac{16}{9} \cdot \frac{-1}{4} + \frac{4}{3}\right)\right)}\right) - \mathsf{log1p}\left(\left(\left(u + \frac{-1}{4}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right) \cdot \frac{-64}{27}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\color{blue}{\frac{16}{9} \cdot u} + \left(\frac{16}{9} \cdot \frac{-1}{4} + \frac{4}{3}\right)\right)\right) - \mathsf{log1p}\left(\left(\left(u + \frac{-1}{4}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right) \cdot \frac{-64}{27}\right)\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\color{blue}{\frac{16}{9} \cdot u} + \left(\frac{16}{9} \cdot \frac{-1}{4} + \frac{4}{3}\right)\right)\right) - \mathsf{log1p}\left(\left(\left(u + \frac{-1}{4}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right) \cdot \frac{-64}{27}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\frac{16}{9} \cdot u + \left(\color{blue}{\frac{-4}{9}} + \frac{4}{3}\right)\right)\right) - \mathsf{log1p}\left(\left(\left(u + \frac{-1}{4}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right) \cdot \frac{-64}{27}\right)\right) \]

   8. metadata-eval 98.3

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + -0.25\right) \cdot \left(1.7777777777777777 \cdot u + \color{blue}{0.8888888888888888}\right)\right) - \mathsf{log1p}\left(\left(\left(u + -0.25\right) \cdot \left(\left(u + -0.25\right) \cdot \left(u + -0.25\right)\right)\right) \cdot -2.3703703703703702\right)\right) \]

6. Applied egg-rr 98.3%

   \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + -0.25\right) \cdot \color{blue}{\left(1.7777777777777777 \cdot u + 0.8888888888888888\right)}\right) - \mathsf{log1p}\left(\left(\left(u + -0.25\right) \cdot \left(\left(u + -0.25\right) \cdot \left(u + -0.25\right)\right)\right) \cdot -2.3703703703703702\right)\right) \]
7. Step-by-step derivation

   1. lift-+.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\frac{16}{9} \cdot u + \frac{8}{9}\right)\right) - \mathsf{log1p}\left(\left(\color{blue}{\left(u + \frac{-1}{4}\right)} \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right) \cdot \frac{-64}{27}\right)\right) \]

   2. lift-+.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\frac{16}{9} \cdot u + \frac{8}{9}\right)\right) - \mathsf{log1p}\left(\left(\left(u + \frac{-1}{4}\right) \cdot \left(\color{blue}{\left(u + \frac{-1}{4}\right)} \cdot \left(u + \frac{-1}{4}\right)\right)\right) \cdot \frac{-64}{27}\right)\right) \]

   3. lift-+.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\frac{16}{9} \cdot u + \frac{8}{9}\right)\right) - \mathsf{log1p}\left(\left(\left(u + \frac{-1}{4}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \color{blue}{\left(u + \frac{-1}{4}\right)}\right)\right) \cdot \frac{-64}{27}\right)\right) \]

   4. lift-*.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\frac{16}{9} \cdot u + \frac{8}{9}\right)\right) - \mathsf{log1p}\left(\left(\left(u + \frac{-1}{4}\right) \cdot \color{blue}{\left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)}\right) \cdot \frac{-64}{27}\right)\right) \]

   5. lift-*.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\frac{16}{9} \cdot u + \frac{8}{9}\right)\right) - \mathsf{log1p}\left(\color{blue}{\left(\left(u + \frac{-1}{4}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)} \cdot \frac{-64}{27}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\frac{16}{9} \cdot u + \frac{8}{9}\right)\right) - \mathsf{log1p}\left(\color{blue}{\frac{-64}{27} \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)}\right)\right) \]

   7. lift-*.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\frac{16}{9} \cdot u + \frac{8}{9}\right)\right) - \mathsf{log1p}\left(\frac{-64}{27} \cdot \color{blue}{\left(\left(u + \frac{-1}{4}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)}\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\frac{16}{9} \cdot u + \frac{8}{9}\right)\right) - \mathsf{log1p}\left(\color{blue}{\left(\frac{-64}{27} \cdot \left(u + \frac{-1}{4}\right)\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)}\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\frac{16}{9} \cdot u + \frac{8}{9}\right)\right) - \mathsf{log1p}\left(\color{blue}{\left(\left(u + \frac{-1}{4}\right) \cdot \frac{-64}{27}\right)} \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   10. lift-*.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\frac{16}{9} \cdot u + \frac{8}{9}\right)\right) - \mathsf{log1p}\left(\color{blue}{\left(\left(u + \frac{-1}{4}\right) \cdot \frac{-64}{27}\right)} \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   11. lower-*.f32 98.3

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + -0.25\right) \cdot \left(1.7777777777777777 \cdot u + 0.8888888888888888\right)\right) - \mathsf{log1p}\left(\color{blue}{\left(\left(u + -0.25\right) \cdot -2.3703703703703702\right) \cdot \left(\left(u + -0.25\right) \cdot \left(u + -0.25\right)\right)}\right)\right) \]

   12. lift-*.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\frac{16}{9} \cdot u + \frac{8}{9}\right)\right) - \mathsf{log1p}\left(\color{blue}{\left(\left(u + \frac{-1}{4}\right) \cdot \frac{-64}{27}\right)} \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\frac{16}{9} \cdot u + \frac{8}{9}\right)\right) - \mathsf{log1p}\left(\color{blue}{\left(\frac{-64}{27} \cdot \left(u + \frac{-1}{4}\right)\right)} \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   14. lift-+.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\frac{16}{9} \cdot u + \frac{8}{9}\right)\right) - \mathsf{log1p}\left(\left(\frac{-64}{27} \cdot \color{blue}{\left(u + \frac{-1}{4}\right)}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   15. distribute-rgt-in N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\frac{16}{9} \cdot u + \frac{8}{9}\right)\right) - \mathsf{log1p}\left(\color{blue}{\left(u \cdot \frac{-64}{27} + \frac{-1}{4} \cdot \frac{-64}{27}\right)} \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\frac{16}{9} \cdot u + \frac{8}{9}\right)\right) - \mathsf{log1p}\left(\left(u \cdot \frac{-64}{27} + \color{blue}{\frac{16}{27}}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\frac{16}{9} \cdot u + \frac{8}{9}\right)\right) - \mathsf{log1p}\left(\left(u \cdot \frac{-64}{27} + \color{blue}{\frac{-64}{27} \cdot \frac{-1}{4}}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   18. lower-fma.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\frac{16}{9} \cdot u + \frac{8}{9}\right)\right) - \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(u, \frac{-64}{27}, \frac{-64}{27} \cdot \frac{-1}{4}\right)} \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   19. metadata-eval 98.3

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + -0.25\right) \cdot \left(1.7777777777777777 \cdot u + 0.8888888888888888\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u, -2.3703703703703702, \color{blue}{0.5925925925925926}\right) \cdot \left(\left(u + -0.25\right) \cdot \left(u + -0.25\right)\right)\right)\right) \]

8. Applied egg-rr 98.3%

   \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + -0.25\right) \cdot \left(1.7777777777777777 \cdot u + 0.8888888888888888\right)\right) - \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(u, -2.3703703703703702, 0.5925925925925926\right) \cdot \left(\left(u + -0.25\right) \cdot \left(u + -0.25\right)\right)}\right)\right) \]
9. Step-by-step derivation

   1. lift-+.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\color{blue}{\left(u + \frac{-1}{4}\right)} \cdot \left(\frac{16}{9} \cdot u + \frac{8}{9}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{-64}{27}, \frac{16}{27}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   2. lift-*.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\color{blue}{\frac{16}{9} \cdot u} + \frac{8}{9}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{-64}{27}, \frac{16}{27}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   3. distribute-lft-in N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\color{blue}{\left(u + \frac{-1}{4}\right) \cdot \left(\frac{16}{9} \cdot u\right) + \left(u + \frac{-1}{4}\right) \cdot \frac{8}{9}}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{-64}{27}, \frac{16}{27}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   4. lift-*.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \color{blue}{\left(\frac{16}{9} \cdot u\right)} + \left(u + \frac{-1}{4}\right) \cdot \frac{8}{9}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{-64}{27}, \frac{16}{27}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \color{blue}{\left(u \cdot \frac{16}{9}\right)} + \left(u + \frac{-1}{4}\right) \cdot \frac{8}{9}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{-64}{27}, \frac{16}{27}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\color{blue}{\left(\left(u + \frac{-1}{4}\right) \cdot u\right) \cdot \frac{16}{9}} + \left(u + \frac{-1}{4}\right) \cdot \frac{8}{9}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{-64}{27}, \frac{16}{27}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(\left(u + \frac{-1}{4}\right) \cdot u\right) \cdot \frac{16}{9} + \color{blue}{\frac{8}{9} \cdot \left(u + \frac{-1}{4}\right)}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{-64}{27}, \frac{16}{27}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   8. lower-fma.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\left(u + \frac{-1}{4}\right) \cdot u, \frac{16}{9}, \frac{8}{9} \cdot \left(u + \frac{-1}{4}\right)\right)}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{-64}{27}, \frac{16}{27}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(\color{blue}{u \cdot \left(u + \frac{-1}{4}\right)}, \frac{16}{9}, \frac{8}{9} \cdot \left(u + \frac{-1}{4}\right)\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{-64}{27}, \frac{16}{27}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   10. lower-*.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(\color{blue}{u \cdot \left(u + \frac{-1}{4}\right)}, \frac{16}{9}, \frac{8}{9} \cdot \left(u + \frac{-1}{4}\right)\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{-64}{27}, \frac{16}{27}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   11. lift-+.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u \cdot \left(u + \frac{-1}{4}\right), \frac{16}{9}, \frac{8}{9} \cdot \color{blue}{\left(u + \frac{-1}{4}\right)}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{-64}{27}, \frac{16}{27}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   12. distribute-rgt-in N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u \cdot \left(u + \frac{-1}{4}\right), \frac{16}{9}, \color{blue}{u \cdot \frac{8}{9} + \frac{-1}{4} \cdot \frac{8}{9}}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{-64}{27}, \frac{16}{27}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u \cdot \left(u + \frac{-1}{4}\right), \frac{16}{9}, u \cdot \frac{8}{9} + \color{blue}{\frac{-2}{9}}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{-64}{27}, \frac{16}{27}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u \cdot \left(u + \frac{-1}{4}\right), \frac{16}{9}, u \cdot \frac{8}{9} + \color{blue}{\frac{8}{9} \cdot \frac{-1}{4}}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{-64}{27}, \frac{16}{27}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   15. lower-fma.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u \cdot \left(u + \frac{-1}{4}\right), \frac{16}{9}, \color{blue}{\mathsf{fma}\left(u, \frac{8}{9}, \frac{8}{9} \cdot \frac{-1}{4}\right)}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{-64}{27}, \frac{16}{27}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   16. metadata-eval 98.5

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u \cdot \left(u + -0.25\right), 1.7777777777777777, \mathsf{fma}\left(u, 0.8888888888888888, \color{blue}{-0.2222222222222222}\right)\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u, -2.3703703703703702, 0.5925925925925926\right) \cdot \left(\left(u + -0.25\right) \cdot \left(u + -0.25\right)\right)\right)\right) \]

10. Applied egg-rr 98.5%

    \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(u \cdot \left(u + -0.25\right), 1.7777777777777777, \mathsf{fma}\left(u, 0.8888888888888888, -0.2222222222222222\right)\right)}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u, -2.3703703703703702, 0.5925925925925926\right) \cdot \left(\left(u + -0.25\right) \cdot \left(u + -0.25\right)\right)\right)\right) \]
11. Add Preprocessing

Alternative 3: 98.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + -0.25\right) \cdot \mathsf{fma}\left(u, 1.7777777777777777, 0.8888888888888888\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u, -2.3703703703703702, 0.5925925925925926\right) \cdot \left(\left(u + -0.25\right) \cdot \left(u + -0.25\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (*
  (* 3.0 s)
  (-
   (log1p (* (+ u -0.25) (fma u 1.7777777777777777 0.8888888888888888)))
   (log1p
    (*
     (fma u -2.3703703703703702 0.5925925925925926)
     (* (+ u -0.25) (+ u -0.25)))))))

C

float code(float s, float u) {
	return (3.0f * s) * (log1pf(((u + -0.25f) * fmaf(u, 1.7777777777777777f, 0.8888888888888888f))) - log1pf((fmaf(u, -2.3703703703703702f, 0.5925925925925926f) * ((u + -0.25f) * (u + -0.25f)))));
}

Julia

function code(s, u)
	return Float32(Float32(Float32(3.0) * s) * Float32(log1p(Float32(Float32(u + Float32(-0.25)) * fma(u, Float32(1.7777777777777777), Float32(0.8888888888888888)))) - log1p(Float32(fma(u, Float32(-2.3703703703703702), Float32(0.5925925925925926)) * Float32(Float32(u + Float32(-0.25)) * Float32(u + Float32(-0.25)))))))
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

4. Applied egg-rr 98.2%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\mathsf{log1p}\left(\left(u + -0.25\right) \cdot \mathsf{fma}\left(1.7777777777777777, u + -0.25, 1.3333333333333333\right)\right) - \mathsf{log1p}\left(\left(\left(u + -0.25\right) \cdot \left(\left(u + -0.25\right) \cdot \left(u + -0.25\right)\right)\right) \cdot -2.3703703703703702\right)\right)} \]
5. Step-by-step derivation

   1. distribute-lft-in N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\color{blue}{\left(\frac{16}{9} \cdot u + \frac{16}{9} \cdot \frac{-1}{4}\right)} + \frac{4}{3}\right)\right) - \mathsf{log1p}\left(\left(\left(u + \frac{-1}{4}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right) \cdot \frac{-64}{27}\right)\right) \]

   2. associate-+l+ N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \color{blue}{\left(\frac{16}{9} \cdot u + \left(\frac{16}{9} \cdot \frac{-1}{4} + \frac{4}{3}\right)\right)}\right) - \mathsf{log1p}\left(\left(\left(u + \frac{-1}{4}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right) \cdot \frac{-64}{27}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\color{blue}{u \cdot \frac{16}{9}} + \left(\frac{16}{9} \cdot \frac{-1}{4} + \frac{4}{3}\right)\right)\right) - \mathsf{log1p}\left(\left(\left(u + \frac{-1}{4}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right) \cdot \frac{-64}{27}\right)\right) \]

   4. lower-+.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \color{blue}{\left(u \cdot \frac{16}{9} + \left(\frac{16}{9} \cdot \frac{-1}{4} + \frac{4}{3}\right)\right)}\right) - \mathsf{log1p}\left(\left(\left(u + \frac{-1}{4}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right) \cdot \frac{-64}{27}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\color{blue}{\frac{16}{9} \cdot u} + \left(\frac{16}{9} \cdot \frac{-1}{4} + \frac{4}{3}\right)\right)\right) - \mathsf{log1p}\left(\left(\left(u + \frac{-1}{4}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right) \cdot \frac{-64}{27}\right)\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\color{blue}{\frac{16}{9} \cdot u} + \left(\frac{16}{9} \cdot \frac{-1}{4} + \frac{4}{3}\right)\right)\right) - \mathsf{log1p}\left(\left(\left(u + \frac{-1}{4}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right) \cdot \frac{-64}{27}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\frac{16}{9} \cdot u + \left(\color{blue}{\frac{-4}{9}} + \frac{4}{3}\right)\right)\right) - \mathsf{log1p}\left(\left(\left(u + \frac{-1}{4}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right) \cdot \frac{-64}{27}\right)\right) \]

   8. metadata-eval 98.3

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + -0.25\right) \cdot \left(1.7777777777777777 \cdot u + \color{blue}{0.8888888888888888}\right)\right) - \mathsf{log1p}\left(\left(\left(u + -0.25\right) \cdot \left(\left(u + -0.25\right) \cdot \left(u + -0.25\right)\right)\right) \cdot -2.3703703703703702\right)\right) \]

6. Applied egg-rr 98.3%

   \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + -0.25\right) \cdot \color{blue}{\left(1.7777777777777777 \cdot u + 0.8888888888888888\right)}\right) - \mathsf{log1p}\left(\left(\left(u + -0.25\right) \cdot \left(\left(u + -0.25\right) \cdot \left(u + -0.25\right)\right)\right) \cdot -2.3703703703703702\right)\right) \]
7. Step-by-step derivation

   1. lift-+.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\frac{16}{9} \cdot u + \frac{8}{9}\right)\right) - \mathsf{log1p}\left(\left(\color{blue}{\left(u + \frac{-1}{4}\right)} \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right) \cdot \frac{-64}{27}\right)\right) \]

   2. lift-+.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\frac{16}{9} \cdot u + \frac{8}{9}\right)\right) - \mathsf{log1p}\left(\left(\left(u + \frac{-1}{4}\right) \cdot \left(\color{blue}{\left(u + \frac{-1}{4}\right)} \cdot \left(u + \frac{-1}{4}\right)\right)\right) \cdot \frac{-64}{27}\right)\right) \]

   3. lift-+.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\frac{16}{9} \cdot u + \frac{8}{9}\right)\right) - \mathsf{log1p}\left(\left(\left(u + \frac{-1}{4}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \color{blue}{\left(u + \frac{-1}{4}\right)}\right)\right) \cdot \frac{-64}{27}\right)\right) \]

   4. lift-*.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\frac{16}{9} \cdot u + \frac{8}{9}\right)\right) - \mathsf{log1p}\left(\left(\left(u + \frac{-1}{4}\right) \cdot \color{blue}{\left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)}\right) \cdot \frac{-64}{27}\right)\right) \]

   5. lift-*.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\frac{16}{9} \cdot u + \frac{8}{9}\right)\right) - \mathsf{log1p}\left(\color{blue}{\left(\left(u + \frac{-1}{4}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)} \cdot \frac{-64}{27}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\frac{16}{9} \cdot u + \frac{8}{9}\right)\right) - \mathsf{log1p}\left(\color{blue}{\frac{-64}{27} \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)}\right)\right) \]

   7. lift-*.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\frac{16}{9} \cdot u + \frac{8}{9}\right)\right) - \mathsf{log1p}\left(\frac{-64}{27} \cdot \color{blue}{\left(\left(u + \frac{-1}{4}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)}\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\frac{16}{9} \cdot u + \frac{8}{9}\right)\right) - \mathsf{log1p}\left(\color{blue}{\left(\frac{-64}{27} \cdot \left(u + \frac{-1}{4}\right)\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)}\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\frac{16}{9} \cdot u + \frac{8}{9}\right)\right) - \mathsf{log1p}\left(\color{blue}{\left(\left(u + \frac{-1}{4}\right) \cdot \frac{-64}{27}\right)} \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   10. lift-*.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\frac{16}{9} \cdot u + \frac{8}{9}\right)\right) - \mathsf{log1p}\left(\color{blue}{\left(\left(u + \frac{-1}{4}\right) \cdot \frac{-64}{27}\right)} \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   11. lower-*.f32 98.3

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + -0.25\right) \cdot \left(1.7777777777777777 \cdot u + 0.8888888888888888\right)\right) - \mathsf{log1p}\left(\color{blue}{\left(\left(u + -0.25\right) \cdot -2.3703703703703702\right) \cdot \left(\left(u + -0.25\right) \cdot \left(u + -0.25\right)\right)}\right)\right) \]

   12. lift-*.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\frac{16}{9} \cdot u + \frac{8}{9}\right)\right) - \mathsf{log1p}\left(\color{blue}{\left(\left(u + \frac{-1}{4}\right) \cdot \frac{-64}{27}\right)} \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\frac{16}{9} \cdot u + \frac{8}{9}\right)\right) - \mathsf{log1p}\left(\color{blue}{\left(\frac{-64}{27} \cdot \left(u + \frac{-1}{4}\right)\right)} \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   14. lift-+.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\frac{16}{9} \cdot u + \frac{8}{9}\right)\right) - \mathsf{log1p}\left(\left(\frac{-64}{27} \cdot \color{blue}{\left(u + \frac{-1}{4}\right)}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   15. distribute-rgt-in N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\frac{16}{9} \cdot u + \frac{8}{9}\right)\right) - \mathsf{log1p}\left(\color{blue}{\left(u \cdot \frac{-64}{27} + \frac{-1}{4} \cdot \frac{-64}{27}\right)} \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\frac{16}{9} \cdot u + \frac{8}{9}\right)\right) - \mathsf{log1p}\left(\left(u \cdot \frac{-64}{27} + \color{blue}{\frac{16}{27}}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\frac{16}{9} \cdot u + \frac{8}{9}\right)\right) - \mathsf{log1p}\left(\left(u \cdot \frac{-64}{27} + \color{blue}{\frac{-64}{27} \cdot \frac{-1}{4}}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   18. lower-fma.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\frac{16}{9} \cdot u + \frac{8}{9}\right)\right) - \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(u, \frac{-64}{27}, \frac{-64}{27} \cdot \frac{-1}{4}\right)} \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   19. metadata-eval 98.3

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + -0.25\right) \cdot \left(1.7777777777777777 \cdot u + 0.8888888888888888\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u, -2.3703703703703702, \color{blue}{0.5925925925925926}\right) \cdot \left(\left(u + -0.25\right) \cdot \left(u + -0.25\right)\right)\right)\right) \]

8. Applied egg-rr 98.3%

   \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + -0.25\right) \cdot \left(1.7777777777777777 \cdot u + 0.8888888888888888\right)\right) - \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(u, -2.3703703703703702, 0.5925925925925926\right) \cdot \left(\left(u + -0.25\right) \cdot \left(u + -0.25\right)\right)}\right)\right) \]
9. Step-by-step derivation

   1. lift-+.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\color{blue}{\left(u + \frac{-1}{4}\right)} \cdot \left(\frac{16}{9} \cdot u + \frac{8}{9}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{-64}{27}, \frac{16}{27}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \left(\color{blue}{u \cdot \frac{16}{9}} + \frac{8}{9}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{-64}{27}, \frac{16}{27}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \color{blue}{\left(\frac{8}{9} + u \cdot \frac{16}{9}\right)}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{-64}{27}, \frac{16}{27}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \color{blue}{\left(u \cdot \frac{16}{9} + \frac{8}{9}\right)}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{-64}{27}, \frac{16}{27}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   5. lift-fma.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + \frac{-1}{4}\right) \cdot \color{blue}{\mathsf{fma}\left(u, \frac{16}{9}, \frac{8}{9}\right)}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{-64}{27}, \frac{16}{27}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(u, \frac{16}{9}, \frac{8}{9}\right) \cdot \left(u + \frac{-1}{4}\right)}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{-64}{27}, \frac{16}{27}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)\right)\right) \]

   7. lower-*.f32 98.4

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(u, 1.7777777777777777, 0.8888888888888888\right) \cdot \left(u + -0.25\right)}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u, -2.3703703703703702, 0.5925925925925926\right) \cdot \left(\left(u + -0.25\right) \cdot \left(u + -0.25\right)\right)\right)\right) \]

10. Applied egg-rr 98.4%

    \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(u, 1.7777777777777777, 0.8888888888888888\right) \cdot \left(u + -0.25\right)}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u, -2.3703703703703702, 0.5925925925925926\right) \cdot \left(\left(u + -0.25\right) \cdot \left(u + -0.25\right)\right)\right)\right) \]

11. Final simplification 98.4%

    \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\left(u + -0.25\right) \cdot \mathsf{fma}\left(u, 1.7777777777777777, 0.8888888888888888\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u, -2.3703703703703702, 0.5925925925925926\right) \cdot \left(\left(u + -0.25\right) \cdot \left(u + -0.25\right)\right)\right)\right) \]
12. Add Preprocessing

Alternative 4: 98.1% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

4. Applied egg-rr 98.2%

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

5. Final simplification 98.2%

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

Alternative 5: 97.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, u)
	return Float32(Float32(3.0) * Float32(Float32(-s) * log1p(Float32(-fma(u, Float32(1.3333333333333333), Float32(-0.3333333333333333))))))
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. Step-by-step derivation

   1. lift--.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift--.f32 N/A

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

   4. /-rgt-identity N/A

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

   5. clear-num N/A

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

   6. lift-/.f32 N/A

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

   7. lift-log.f32 N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

4. Applied egg-rr 98.0%

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

5. Final simplification 98.0%

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

Alternative 6: 97.8% 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 95.8%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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. 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. Simplified 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 95.8%

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

3. Taylor expanded in u around 0

   \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{4}{3} + \frac{-4}{3} \cdot u}}\right) \]
4. 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. lower-fma.f32 95.9

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

5. Simplified 95.9%

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

   1. lift-*.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-log.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 95.9

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

   10. lift-log.f32 N/A

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

   11. lift-/.f32 N/A

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

   12. log-rec N/A

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

   13. lower-neg.f32 N/A

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

   14. lower-log.f32 96.7

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

   15. lift-fma.f32 N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f32 96.7

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

7. Applied egg-rr 96.7%

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

8. Taylor expanded in s around 0

   \[\leadsto \color{blue}{-3 \cdot \left(s \cdot \log \left(\frac{4}{3} + \frac{-4}{3} \cdot u\right)\right)} \]
9. 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 96.7

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

10. Simplified 96.7%

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

Alternative 8: 25.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.8%

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

3. Taylor expanded in u around 0

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. distribute-lft-out N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-+.f32 N/A

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

   7. lower-log.f32 N/A

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

   8. lower-*.f32 25.8

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

5. Simplified 25.8%

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

6. Final simplification 25.8%

   \[\leadsto \left(3 \cdot s\right) \cdot \left(u + \log 0.75\right) \]
7. Add Preprocessing

Alternative 9: 25.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.8%

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

3. Taylor expanded in u around 0

   \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{4}{3} + \frac{-4}{3} \cdot u}}\right) \]
4. 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. lower-fma.f32 95.9

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

5. Simplified 95.9%

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

6. Taylor expanded in u around 0

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

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

   5. lower-+.f32 N/A

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

   6. lower-log.f32 25.8

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

8. Simplified 25.8%

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

Alternative 10: 25.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ s \cdot \log 1.3333333333333333 \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (log 1.3333333333333333)))

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(s, u)
	tmp = s * log(single(1.3333333333333333));
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

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

   1. lower-log.f32 7.2

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

5. Simplified 7.2%

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

6. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 25.5

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

8. Simplified 25.5%

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

   1. lift-log.f32 N/A

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

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

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

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

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

   4. lower-*.f32 N/A

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

   5. lift-log.f32 N/A

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

   6. neg-log N/A

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

   7. metadata-eval N/A

      \[\leadsto s \cdot \log \color{blue}{\frac{4}{3}} \]

   8. lower-log.f32 25.5

      \[\leadsto s \cdot \color{blue}{\log 1.3333333333333333} \]

10. Applied egg-rr 25.5%

    \[\leadsto \color{blue}{s \cdot \log 1.3333333333333333} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(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))))))