Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.9% → 98.5%

Time: 13.2s

Alternatives: 12

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

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

FPCore

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

C

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

Julia

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

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

   1. sub-neg N/A

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

   2. distribute-lft-in N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

6. Applied egg-rr 98.5%

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

7. Final simplification 98.5%

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

Alternative 2: 98.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, u)
	return fma(Float32(s * log1p(Float32(Float32(u + Float32(-0.25)) * fma(u, Float32(1.7777777777777777), Float32(0.8888888888888888))))), Float32(3.0), Float32(Float32(Float32(3.0) * Float32(-s)) * log1p(Float32(Float32(u + Float32(-0.25)) * Float32(Float32(u + Float32(-0.25)) * fma(u, Float32(-2.3703703703703702), Float32(0.5925925925925926)))))))
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. sub-neg N/A

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

   2. distribute-lft-in N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

6. Applied egg-rr 98.5%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

   4. *-lowering-*.f32 N/A

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

   5. accelerator-lowering-log1p.f32 N/A

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

   6. *-lowering-*.f32 N/A

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

   7. +-lowering-+.f32 N/A

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

   8. *-commutative N/A

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

   9. accelerator-lowering-fma.f32 N/A

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

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

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

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

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

8. Applied egg-rr 98.4%

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

9. Final simplification 98.4%

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

Alternative 3: 98.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, u)
	return Float32(s * fma(Float32(-3.0), log1p(Float32(fma(Float32(-2.3703703703703702), u, Float32(0.5925925925925926)) * Float32(Float32(u + Float32(-0.25)) * Float32(u + Float32(-0.25))))), Float32(Float32(3.0) * log1p(Float32(Float32(u + Float32(-0.25)) * fma(u, Float32(1.7777777777777777), Float32(0.8888888888888888)))))))
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. sub-neg N/A

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

   2. distribute-lft-in N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

6. Applied egg-rr 98.5%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

   4. *-lowering-*.f32 N/A

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

   5. accelerator-lowering-log1p.f32 N/A

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

   6. *-lowering-*.f32 N/A

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

   7. +-lowering-+.f32 N/A

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

   8. *-commutative N/A

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

   9. accelerator-lowering-fma.f32 N/A

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

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

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

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

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

8. Applied egg-rr 98.4%

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

9. Taylor expanded in s around 0

   \[\leadsto \color{blue}{s \cdot \left(-3 \cdot \log \left(1 + \left(\frac{16}{27} + \frac{-64}{27} \cdot u\right) \cdot {\left(u - \frac{1}{4}\right)}^{2}\right) + 3 \cdot \log \left(1 + \left(\frac{8}{9} + \frac{16}{9} \cdot u\right) \cdot \left(u - \frac{1}{4}\right)\right)\right)} \]
10. Step-by-step derivation

    1. *-lowering-*.f32 N/A

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

    2. accelerator-lowering-fma.f32 N/A

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

    3. accelerator-lowering-log1p.f32 N/A

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

    4. *-lowering-*.f32 N/A

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

    5. +-commutative N/A

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

    6. accelerator-lowering-fma.f32 N/A

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

    7. unpow2 N/A

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

    8. *-lowering-*.f32 N/A

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

    9. sub-neg N/A

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

    10. +-lowering-+.f32 N/A

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

    11. metadata-eval N/A

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

    12. sub-neg N/A

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

    13. +-lowering-+.f32 N/A

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

    14. metadata-eval N/A

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

    15. *-lowering-*.f32 N/A

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

11. Simplified 98.4%

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

Alternative 4: 98.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, u)
	return Float32(s * Float32(Float32(-3.0) * Float32(log1p(Float32(Float32(u + Float32(-0.25)) * Float32(Float32(u + Float32(-0.25)) * Float32(Float32(u + Float32(-0.25)) * Float32(-2.3703703703703702))))) - log1p(Float32(Float32(u + Float32(-0.25)) * fma(Float32(1.7777777777777777), u, Float32(0.8888888888888888)))))))
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. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-lowering-*.f32 N/A

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

4. Applied egg-rr 97.7%

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

6. Applied egg-rr 98.3%

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

7. Final simplification 98.3%

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

Alternative 5: 98.2% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 95.8%

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

   1. flip-- N/A

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

   2. clear-num N/A

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

   3. /-lowering-/.f32 N/A

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

   4. +-commutative N/A

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

   5. div-sub N/A

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

   6. sub-neg N/A

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

   7. associate-+l+ N/A

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

   8. div-inv N/A

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

   9. accelerator-lowering-fma.f32 N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. sub-neg N/A

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

   16. +-commutative N/A

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

4. Applied egg-rr 95.6%

   \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{\mathsf{fma}\left(u, 1.3333333333333333, 0.6666666666666666\right)}{\mathsf{fma}\left(0.25 - u, 1.7777777777777777 \cdot \left(u + -0.25\right), 1\right)}\right)} \]
5. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\log \left(\frac{u \cdot \frac{4}{3} + \frac{2}{3}}{\left(\frac{1}{4} - u\right) \cdot \left(\frac{16}{9} \cdot \left(u + \frac{-1}{4}\right)\right) + 1}\right) \cdot \left(3 \cdot s\right)} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\log \left(\frac{u \cdot \frac{4}{3} + \frac{2}{3}}{\left(\frac{1}{4} - u\right) \cdot \left(\frac{16}{9} \cdot \left(u + \frac{-1}{4}\right)\right) + 1}\right) \cdot 3\right) \cdot s} \]

   3. log-div N/A

      \[\leadsto \left(\color{blue}{\left(\log \left(u \cdot \frac{4}{3} + \frac{2}{3}\right) - \log \left(\left(\frac{1}{4} - u\right) \cdot \left(\frac{16}{9} \cdot \left(u + \frac{-1}{4}\right)\right) + 1\right)\right)} \cdot 3\right) \cdot s \]

   4. metadata-eval N/A

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

   5. associate-+l+ N/A

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

   6. +-commutative N/A

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

   7. +-commutative N/A

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

   8. *-lowering-*.f32 N/A

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

6. Applied egg-rr 95.5%

   \[\leadsto \color{blue}{\left(\log \left(\frac{\mathsf{fma}\left(u, 1.3333333333333333, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(1.7777777777777777, u, -0.4444444444444444\right), 0.25 - u, 1\right)}\right) \cdot 3\right) \cdot s} \]
7. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \color{blue}{\log \left(\frac{u \cdot \frac{4}{3} + \frac{2}{3}}{\left(\frac{16}{9} \cdot u + \frac{-4}{9}\right) \cdot \left(\frac{1}{4} - u\right) + 1}\right) \cdot \left(3 \cdot s\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \log \left(\frac{u \cdot \frac{4}{3} + \frac{2}{3}}{\left(\frac{16}{9} \cdot u + \frac{-4}{9}\right) \cdot \left(\frac{1}{4} - u\right) + 1}\right)} \]

   3. log-div N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \left(u \cdot \frac{4}{3} + \frac{2}{3}\right) - \log \left(\left(\frac{16}{9} \cdot u + \frac{-4}{9}\right) \cdot \left(\frac{1}{4} - u\right) + 1\right)\right)} \]

   4. metadata-eval N/A

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

   5. associate-+l+ N/A

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

   6. +-commutative N/A

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

   7. sub-neg N/A

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

   8. distribute-lft-in N/A

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

   9. +-lowering-+.f32 N/A

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

8. Applied egg-rr 98.2%

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

9. Final simplification 98.2%

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

Alternative 6: 98.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 95.8%

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

   1. flip-- N/A

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

   2. clear-num N/A

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

   3. /-lowering-/.f32 N/A

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

   4. +-commutative N/A

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

   5. div-sub N/A

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

   6. sub-neg N/A

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

   7. associate-+l+ N/A

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

   8. div-inv N/A

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

   9. accelerator-lowering-fma.f32 N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. sub-neg N/A

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

   16. +-commutative N/A

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

4. Applied egg-rr 95.6%

   \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{\mathsf{fma}\left(u, 1.3333333333333333, 0.6666666666666666\right)}{\mathsf{fma}\left(0.25 - u, 1.7777777777777777 \cdot \left(u + -0.25\right), 1\right)}\right)} \]
5. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\log \left(\frac{u \cdot \frac{4}{3} + \frac{2}{3}}{\left(\frac{1}{4} - u\right) \cdot \left(\frac{16}{9} \cdot \left(u + \frac{-1}{4}\right)\right) + 1}\right) \cdot \left(3 \cdot s\right)} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\log \left(\frac{u \cdot \frac{4}{3} + \frac{2}{3}}{\left(\frac{1}{4} - u\right) \cdot \left(\frac{16}{9} \cdot \left(u + \frac{-1}{4}\right)\right) + 1}\right) \cdot 3\right) \cdot s} \]

   3. log-div N/A

      \[\leadsto \left(\color{blue}{\left(\log \left(u \cdot \frac{4}{3} + \frac{2}{3}\right) - \log \left(\left(\frac{1}{4} - u\right) \cdot \left(\frac{16}{9} \cdot \left(u + \frac{-1}{4}\right)\right) + 1\right)\right)} \cdot 3\right) \cdot s \]

   4. metadata-eval N/A

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

   5. associate-+l+ N/A

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

   6. +-commutative N/A

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

   7. +-commutative N/A

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

   8. *-lowering-*.f32 N/A

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

6. Applied egg-rr 95.5%

   \[\leadsto \color{blue}{\left(\log \left(\frac{\mathsf{fma}\left(u, 1.3333333333333333, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(1.7777777777777777, u, -0.4444444444444444\right), 0.25 - u, 1\right)}\right) \cdot 3\right) \cdot s} \]
7. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \color{blue}{\log \left(\frac{u \cdot \frac{4}{3} + \frac{2}{3}}{\left(\frac{16}{9} \cdot u + \frac{-4}{9}\right) \cdot \left(\frac{1}{4} - u\right) + 1}\right) \cdot \left(3 \cdot s\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \log \left(\frac{u \cdot \frac{4}{3} + \frac{2}{3}}{\left(\frac{16}{9} \cdot u + \frac{-4}{9}\right) \cdot \left(\frac{1}{4} - u\right) + 1}\right)} \]

   3. log-div N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \left(u \cdot \frac{4}{3} + \frac{2}{3}\right) - \log \left(\left(\frac{16}{9} \cdot u + \frac{-4}{9}\right) \cdot \left(\frac{1}{4} - u\right) + 1\right)\right)} \]

   4. metadata-eval N/A

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

   5. associate-+l+ N/A

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

   6. +-commutative N/A

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

   7. sub-neg N/A

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

   8. distribute-lft-in N/A

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

   9. accelerator-lowering-fma.f32 N/A

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

   10. *-lowering-*.f32 N/A

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

   11. accelerator-lowering-log1p.f32 N/A

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

   12. accelerator-lowering-fma.f32 N/A

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

8. Applied egg-rr 98.2%

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

9. Final simplification 98.2%

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

Alternative 7: 98.2% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 95.8%

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

   1. flip-- N/A

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

   2. clear-num N/A

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

   3. /-lowering-/.f32 N/A

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

   4. +-commutative N/A

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

   5. div-sub N/A

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

   6. sub-neg N/A

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

   7. associate-+l+ N/A

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

   8. div-inv N/A

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

   9. accelerator-lowering-fma.f32 N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. sub-neg N/A

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

   16. +-commutative N/A

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

4. Applied egg-rr 95.6%

   \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{\mathsf{fma}\left(u, 1.3333333333333333, 0.6666666666666666\right)}{\mathsf{fma}\left(0.25 - u, 1.7777777777777777 \cdot \left(u + -0.25\right), 1\right)}\right)} \]
5. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\log \left(\frac{u \cdot \frac{4}{3} + \frac{2}{3}}{\left(\frac{1}{4} - u\right) \cdot \left(\frac{16}{9} \cdot \left(u + \frac{-1}{4}\right)\right) + 1}\right) \cdot \left(3 \cdot s\right)} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\log \left(\frac{u \cdot \frac{4}{3} + \frac{2}{3}}{\left(\frac{1}{4} - u\right) \cdot \left(\frac{16}{9} \cdot \left(u + \frac{-1}{4}\right)\right) + 1}\right) \cdot 3\right) \cdot s} \]

   3. log-div N/A

      \[\leadsto \left(\color{blue}{\left(\log \left(u \cdot \frac{4}{3} + \frac{2}{3}\right) - \log \left(\left(\frac{1}{4} - u\right) \cdot \left(\frac{16}{9} \cdot \left(u + \frac{-1}{4}\right)\right) + 1\right)\right)} \cdot 3\right) \cdot s \]

   4. metadata-eval N/A

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

   5. associate-+l+ N/A

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

   6. +-commutative N/A

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

   7. +-commutative N/A

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

   8. *-lowering-*.f32 N/A

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

6. Applied egg-rr 95.5%

   \[\leadsto \color{blue}{\left(\log \left(\frac{\mathsf{fma}\left(u, 1.3333333333333333, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(1.7777777777777777, u, -0.4444444444444444\right), 0.25 - u, 1\right)}\right) \cdot 3\right) \cdot s} \]
7. Step-by-step derivation

   1. log-div N/A

      \[\leadsto \left(\color{blue}{\left(\log \left(u \cdot \frac{4}{3} + \frac{2}{3}\right) - \log \left(\left(\frac{16}{9} \cdot u + \frac{-4}{9}\right) \cdot \left(\frac{1}{4} - u\right) + 1\right)\right)} \cdot 3\right) \cdot s \]

   2. metadata-eval N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. --lowering--.f32 N/A

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

   6. accelerator-lowering-log1p.f32 N/A

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

   7. accelerator-lowering-fma.f32 N/A

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

   8. +-commutative N/A

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

   9. accelerator-lowering-log1p.f32 N/A

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

   10. *-lowering-*.f32 N/A

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

   11. *-commutative N/A

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

   12. accelerator-lowering-fma.f32 N/A

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

   13. --lowering--.f32 98.1

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

8. Applied egg-rr 98.1%

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

9. Final simplification 98.1%

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

Alternative 8: 98.2% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

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) * fma(Float32(1.7777777777777777), u, Float32(-0.4444444444444444))))))
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. flip-- N/A

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

   2. clear-num N/A

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

   3. log-div N/A

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

   4. --lowering--.f32 N/A

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

   5. accelerator-lowering-log1p.f32 N/A

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

   6. div-sub N/A

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

   7. sub-neg N/A

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

   8. div-inv N/A

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

   9. accelerator-lowering-fma.f32 N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. sub-neg N/A

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

   15. accelerator-lowering-log1p.f32 N/A

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

4. Applied egg-rr 98.1%

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

   1. *-commutative N/A

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

   2. *-lowering-*.f32 N/A

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

   3. distribute-lft-in N/A

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

   4. accelerator-lowering-fma.f32 N/A

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

   5. metadata-eval N/A

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

   6. --lowering--.f32 98.1

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

6. Applied egg-rr 98.1%

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

7. Final simplification 98.1%

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

Alternative 9: 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 95.8%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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. *-lowering-*.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. *-lowering-*.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. accelerator-lowering-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. accelerator-lowering-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 10: 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. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-lowering-*.f32 N/A

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

4. Applied egg-rr 97.7%

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

   1. *-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. +-lowering-+.f32 N/A

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

   6. *-lowering-*.f32 N/A

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

   7. metadata-eval 96.7

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

6. Applied egg-rr 96.7%

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

7. Taylor expanded in s around 0

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

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

   5. log-lowering-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. accelerator-lowering-fma.f32 96.7

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

9. Simplified 96.7%

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

Alternative 11: 96.9% accurate, 1.2× speedup

Math

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

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(s * Float32(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. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-lowering-*.f32 N/A

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

4. Applied egg-rr 97.7%

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

   1. *-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. +-lowering-+.f32 N/A

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

   6. *-lowering-*.f32 N/A

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

   7. metadata-eval 96.7

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

6. Applied egg-rr 96.7%

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

   1. log-lowering-log.f32 N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. metadata-eval N/A

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

   5. accelerator-lowering-fma.f32 96.6

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

8. Applied egg-rr 96.6%

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

9. Final simplification 96.6%

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

Alternative 12: 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 95.8%

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

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

   2. *-lowering-*.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. *-lowering-*.f32 N/A

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

   6. +-lowering-+.f32 N/A

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

   7. log-lowering-log.f32 25.7

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

5. Simplified 25.7%

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

6. Taylor expanded in u around inf

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

8. Simplified 30.0%

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

9. Final simplification 30.0%

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

Reproduce

herbie shell --seed 2024199 
(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))))))