Rust f32::atanh

Percentage Accurate: 99.8% → 99.9%

Time: 1.3s

Alternatives: 6

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ 0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))

C

float code(float x) {
	return 0.5f * log1pf(((2.0f * x) / (1.0f - x)));
}

Julia

function code(x)
	return Float32(Float32(0.5) * log1p(Float32(Float32(Float32(2.0) * x) / Float32(Float32(1.0) - x))))
end

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))

C

float code(float x) {
	return 0.5f * log1pf(((2.0f * x) / (1.0f - x)));
}

Julia

function code(x)
	return Float32(Float32(0.5) * log1p(Float32(Float32(Float32(2.0) * x) / Float32(Float32(1.0) - x))))
end

Alternative 1: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \mathsf{log1p}\left(\frac{-2 \cdot \mathsf{fma}\left(x, x, x\right)}{\mathsf{fma}\left(x, x, -1\right)}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (* 0.5 (log1p (/ (* -2.0 (fma x x x)) (fma x x -1.0)))))

C

float code(float x) {
	return 0.5f * log1pf(((-2.0f * fmaf(x, x, x)) / fmaf(x, x, -1.0f)));
}

Julia

function code(x)
	return Float32(Float32(0.5) * log1p(Float32(Float32(Float32(-2.0) * fma(x, x, x)) / fma(x, x, Float32(-1.0)))))
end

Derivation

1. Initial program 99.8%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
2. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\color{blue}{\frac{2 \cdot x}{1 - x}}\right) \]

   2. lift--.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{\color{blue}{1 - x}}\right) \]

   3. flip-- N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{\color{blue}{\frac{1 \cdot 1 - x \cdot x}{1 + x}}}\right) \]

   4. associate-/r/ N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\color{blue}{\frac{2 \cdot x}{1 \cdot 1 - x \cdot x} \cdot \left(1 + x\right)}\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\color{blue}{\frac{2 \cdot x}{1 \cdot 1 - x \cdot x} \cdot \left(1 + x\right)}\right) \]

   6. lower-/.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\color{blue}{\frac{2 \cdot x}{1 \cdot 1 - x \cdot x}} \cdot \left(1 + x\right)\right) \]

   7. lift-*.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{\color{blue}{2 \cdot x}}{1 \cdot 1 - x \cdot x} \cdot \left(1 + x\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{\color{blue}{x \cdot 2}}{1 \cdot 1 - x \cdot x} \cdot \left(1 + x\right)\right) \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{\color{blue}{x \cdot 2}}{1 \cdot 1 - x \cdot x} \cdot \left(1 + x\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{x \cdot 2}{\color{blue}{1} - x \cdot x} \cdot \left(1 + x\right)\right) \]

   11. lower--.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{x \cdot 2}{\color{blue}{1 - x \cdot x}} \cdot \left(1 + x\right)\right) \]

   12. lower-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{x \cdot 2}{1 - \color{blue}{x \cdot x}} \cdot \left(1 + x\right)\right) \]

   13. lower-+.f32 99.8

       \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\frac{x \cdot 2}{1 - x \cdot x} \cdot \color{blue}{\left(1 + x\right)}\right) \]

3. Applied rewrites 99.8%

   \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{\frac{x \cdot 2}{1 - x \cdot x} \cdot \left(1 + x\right)}\right) \]
4. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\color{blue}{\frac{x \cdot 2}{1 - x \cdot x} \cdot \left(1 + x\right)}\right) \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\color{blue}{\frac{x \cdot 2}{1 - x \cdot x}} \cdot \left(1 + x\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\color{blue}{\frac{\left(x \cdot 2\right) \cdot \left(1 + x\right)}{1 - x \cdot x}}\right) \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\color{blue}{\frac{\left(x \cdot 2\right) \cdot \left(1 + x\right)}{1 - x \cdot x}}\right) \]

   5. lower-*.f32 99.7

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\frac{\color{blue}{\left(x \cdot 2\right) \cdot \left(1 + x\right)}}{1 - x \cdot x}\right) \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{\color{blue}{\left(x \cdot 2\right)} \cdot \left(1 + x\right)}{1 - x \cdot x}\right) \]

   7. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{\color{blue}{\left(2 \cdot x\right)} \cdot \left(1 + x\right)}{1 - x \cdot x}\right) \]

   8. lower-*.f32 99.7

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\frac{\color{blue}{\left(2 \cdot x\right)} \cdot \left(1 + x\right)}{1 - x \cdot x}\right) \]

5. Applied rewrites 99.7%

   \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{\frac{\left(2 \cdot x\right) \cdot \left(1 + x\right)}{1 - x \cdot x}}\right) \]
6. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\color{blue}{\frac{\left(2 \cdot x\right) \cdot \left(1 + x\right)}{1 - x \cdot x}}\right) \]

   2. frac-2neg N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(2 \cdot x\right) \cdot \left(1 + x\right)\right)}{\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)}}\right) \]

   3. lower-/.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(2 \cdot x\right) \cdot \left(1 + x\right)\right)}{\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)}}\right) \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(2 \cdot x\right) \cdot \left(1 + x\right)}\right)}{\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)}\right) \]

   5. lift-*.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(2 \cdot x\right)} \cdot \left(1 + x\right)\right)}{\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)}\right) \]

   6. associate-*l* N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{\mathsf{neg}\left(\color{blue}{2 \cdot \left(x \cdot \left(1 + x\right)\right)}\right)}{\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)}\right) \]

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

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \left(x \cdot \left(1 + x\right)\right)}}{\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)}\right) \]

   8. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{\color{blue}{-2} \cdot \left(x \cdot \left(1 + x\right)\right)}{\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)}\right) \]

   9. lift-+.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{-2 \cdot \left(x \cdot \color{blue}{\left(1 + x\right)}\right)}{\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)}\right) \]

   10. +-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{-2 \cdot \left(x \cdot \color{blue}{\left(x + 1\right)}\right)}{\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)}\right) \]

   11. distribute-rgt-in N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{-2 \cdot \color{blue}{\left(x \cdot x + 1 \cdot x\right)}}{\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)}\right) \]

   12. lower-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{\color{blue}{-2 \cdot \left(x \cdot x + 1 \cdot x\right)}}{\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)}\right) \]

   13. *-lft-identity N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{-2 \cdot \left(x \cdot x + \color{blue}{x}\right)}{\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)}\right) \]

   14. lower-fma.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{-2 \cdot \color{blue}{\mathsf{fma}\left(x, x, x\right)}}{\mathsf{neg}\left(\left(1 - x \cdot x\right)\right)}\right) \]

   15. lift--.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{-2 \cdot \mathsf{fma}\left(x, x, x\right)}{\mathsf{neg}\left(\color{blue}{\left(1 - x \cdot x\right)}\right)}\right) \]

   16. sub-negate2 N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{-2 \cdot \mathsf{fma}\left(x, x, x\right)}{\color{blue}{x \cdot x - 1}}\right) \]

   17. sub-negate1 N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{-2 \cdot \mathsf{fma}\left(x, x, x\right)}{\color{blue}{x \cdot x + \left(\mathsf{neg}\left(1\right)\right)}}\right) \]

   18. lift-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{-2 \cdot \mathsf{fma}\left(x, x, x\right)}{\color{blue}{x \cdot x} + \left(\mathsf{neg}\left(1\right)\right)}\right) \]

   19. lower-fma.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{-2 \cdot \mathsf{fma}\left(x, x, x\right)}{\color{blue}{\mathsf{fma}\left(x, x, \mathsf{neg}\left(1\right)\right)}}\right) \]

   20. metadata-eval 99.9

       \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\frac{-2 \cdot \mathsf{fma}\left(x, x, x\right)}{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)}\right) \]

7. Applied rewrites 99.9%

   \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{\frac{-2 \cdot \mathsf{fma}\left(x, x, x\right)}{\mathsf{fma}\left(x, x, -1\right)}}\right) \]
8. Add Preprocessing

Alternative 2: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \mathsf{log1p}\left(\frac{x + x}{1 - x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (* 0.5 (log1p (/ (+ x x) (- 1.0 x)))))

C

float code(float x) {
	return 0.5f * log1pf(((x + x) / (1.0f - x)));
}

Julia

function code(x)
	return Float32(Float32(0.5) * log1p(Float32(Float32(x + x) / Float32(Float32(1.0) - x))))
end

Derivation

1. Initial program 99.8%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
2. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{\color{blue}{2 \cdot x}}{1 - x}\right) \]

   2. count-2-rev N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{\color{blue}{x + x}}{1 - x}\right) \]

   3. lower-+.f32 99.8

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\frac{\color{blue}{x + x}}{1 - x}\right) \]

3. Applied rewrites 99.8%

   \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\frac{\color{blue}{x + x}}{1 - x}\right) \]
4. Add Preprocessing

Alternative 3: 99.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.14285714285714285, x \cdot x, 0.2\right), x \cdot x, 0.3333333333333333\right), x \cdot x, 1\right) \cdot x \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (*
  (fma
   (fma (fma 0.14285714285714285 (* x x) 0.2) (* x x) 0.3333333333333333)
   (* x x)
   1.0)
  x))

C

float code(float x) {
	return fmaf(fmaf(fmaf(0.14285714285714285f, (x * x), 0.2f), (x * x), 0.3333333333333333f), (x * x), 1.0f) * x;
}

Julia

function code(x)
	return Float32(fma(fma(fma(Float32(0.14285714285714285), Float32(x * x), Float32(0.2)), Float32(x * x), Float32(0.3333333333333333)), Float32(x * x), Float32(1.0)) * x)
end

Derivation

1. Initial program 99.8%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
2. Step-by-step derivation

   1. lift-log1p.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\log \left(1 + \frac{2 \cdot x}{1 - x}\right)} \]

   2. flip-+ N/A

      \[\leadsto \frac{1}{2} \cdot \log \color{blue}{\left(\frac{1 \cdot 1 - \frac{2 \cdot x}{1 - x} \cdot \frac{2 \cdot x}{1 - x}}{1 - \frac{2 \cdot x}{1 - x}}\right)} \]

   3. log-div N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\log \left(1 \cdot 1 - \frac{2 \cdot x}{1 - x} \cdot \frac{2 \cdot x}{1 - x}\right) - \log \left(1 - \frac{2 \cdot x}{1 - x}\right)\right)} \]

   4. lower--.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\log \left(1 \cdot 1 - \frac{2 \cdot x}{1 - x} \cdot \frac{2 \cdot x}{1 - x}\right) - \log \left(1 - \frac{2 \cdot x}{1 - x}\right)\right)} \]

3. Applied rewrites 99.1%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\mathsf{log1p}\left(-{\left(\frac{x}{1 - x} \cdot 2\right)}^{2}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right)} \]
4. Step-by-step derivation

   1. lift-pow.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\color{blue}{{\left(\frac{x}{1 - x} \cdot 2\right)}^{2}}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - x} \cdot 2\right) \cdot \left(\frac{x}{1 - x} \cdot 2\right)}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - x} \cdot 2\right)} \cdot \left(\frac{x}{1 - x} \cdot 2\right)\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   4. lift-/.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\left(\color{blue}{\frac{x}{1 - x}} \cdot 2\right) \cdot \left(\frac{x}{1 - x} \cdot 2\right)\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   5. associate-*l/ N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\color{blue}{\frac{x \cdot 2}{1 - x}} \cdot \left(\frac{x}{1 - x} \cdot 2\right)\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{\color{blue}{2 \cdot x}}{1 - x} \cdot \left(\frac{x}{1 - x} \cdot 2\right)\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   7. lift-*.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{\color{blue}{2 \cdot x}}{1 - x} \cdot \left(\frac{x}{1 - x} \cdot 2\right)\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   8. lift-*.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{2 \cdot x}{1 - x} \cdot \color{blue}{\left(\frac{x}{1 - x} \cdot 2\right)}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   9. lift-/.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{2 \cdot x}{1 - x} \cdot \left(\color{blue}{\frac{x}{1 - x}} \cdot 2\right)\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   10. associate-*l/ N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{2 \cdot x}{1 - x} \cdot \color{blue}{\frac{x \cdot 2}{1 - x}}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{2 \cdot x}{1 - x} \cdot \frac{\color{blue}{2 \cdot x}}{1 - x}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   12. lift-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{2 \cdot x}{1 - x} \cdot \frac{\color{blue}{2 \cdot x}}{1 - x}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   13. frac-times N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\color{blue}{\frac{\left(2 \cdot x\right) \cdot \left(2 \cdot x\right)}{\left(1 - x\right) \cdot \left(1 - x\right)}}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   14. lower-/.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\color{blue}{\frac{\left(2 \cdot x\right) \cdot \left(2 \cdot x\right)}{\left(1 - x\right) \cdot \left(1 - x\right)}}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   15. lower-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{\color{blue}{\left(2 \cdot x\right) \cdot \left(2 \cdot x\right)}}{\left(1 - x\right) \cdot \left(1 - x\right)}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   16. lift-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{\color{blue}{\left(2 \cdot x\right)} \cdot \left(2 \cdot x\right)}{\left(1 - x\right) \cdot \left(1 - x\right)}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{\color{blue}{\left(x \cdot 2\right)} \cdot \left(2 \cdot x\right)}{\left(1 - x\right) \cdot \left(1 - x\right)}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   18. lower-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{\color{blue}{\left(x \cdot 2\right)} \cdot \left(2 \cdot x\right)}{\left(1 - x\right) \cdot \left(1 - x\right)}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   19. lift-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{\left(x \cdot 2\right) \cdot \color{blue}{\left(2 \cdot x\right)}}{\left(1 - x\right) \cdot \left(1 - x\right)}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   20. *-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{\left(x \cdot 2\right) \cdot \color{blue}{\left(x \cdot 2\right)}}{\left(1 - x\right) \cdot \left(1 - x\right)}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   21. lower-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{\left(x \cdot 2\right) \cdot \color{blue}{\left(x \cdot 2\right)}}{\left(1 - x\right) \cdot \left(1 - x\right)}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   22. lower-*.f32 99.0

       \[\leadsto 0.5 \cdot \left(\mathsf{log1p}\left(-\frac{\left(x \cdot 2\right) \cdot \left(x \cdot 2\right)}{\color{blue}{\left(1 - x\right) \cdot \left(1 - x\right)}}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

5. Applied rewrites 99.0%

   \[\leadsto 0.5 \cdot \left(\mathsf{log1p}\left(-\color{blue}{\frac{\left(x \cdot 2\right) \cdot \left(x \cdot 2\right)}{\left(1 - x\right) \cdot \left(1 - x\right)}}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right) \cdot \color{blue}{x} \]

   2. lower-*.f32 N/A

      \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right) \cdot \color{blue}{x} \]

8. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.14285714285714285, x \cdot x, 0.2\right), x \cdot x, 0.3333333333333333\right), x \cdot x, 1\right) \cdot x} \]
9. Add Preprocessing

Alternative 4: 98.9% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(0.2, x \cdot x, 0.3333333333333333\right), x \cdot x, 1\right) \cdot x \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (* (fma (fma 0.2 (* x x) 0.3333333333333333) (* x x) 1.0) x))

C

float code(float x) {
	return fmaf(fmaf(0.2f, (x * x), 0.3333333333333333f), (x * x), 1.0f) * x;
}

Julia

function code(x)
	return Float32(fma(fma(Float32(0.2), Float32(x * x), Float32(0.3333333333333333)), Float32(x * x), Float32(1.0)) * x)
end

Derivation

1. Initial program 99.8%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
2. Step-by-step derivation

   1. lift-log1p.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\log \left(1 + \frac{2 \cdot x}{1 - x}\right)} \]

   2. flip-+ N/A

      \[\leadsto \frac{1}{2} \cdot \log \color{blue}{\left(\frac{1 \cdot 1 - \frac{2 \cdot x}{1 - x} \cdot \frac{2 \cdot x}{1 - x}}{1 - \frac{2 \cdot x}{1 - x}}\right)} \]

   3. log-div N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\log \left(1 \cdot 1 - \frac{2 \cdot x}{1 - x} \cdot \frac{2 \cdot x}{1 - x}\right) - \log \left(1 - \frac{2 \cdot x}{1 - x}\right)\right)} \]

   4. lower--.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\log \left(1 \cdot 1 - \frac{2 \cdot x}{1 - x} \cdot \frac{2 \cdot x}{1 - x}\right) - \log \left(1 - \frac{2 \cdot x}{1 - x}\right)\right)} \]

3. Applied rewrites 99.1%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\mathsf{log1p}\left(-{\left(\frac{x}{1 - x} \cdot 2\right)}^{2}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right)} \]
4. Step-by-step derivation

   1. lift-pow.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\color{blue}{{\left(\frac{x}{1 - x} \cdot 2\right)}^{2}}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - x} \cdot 2\right) \cdot \left(\frac{x}{1 - x} \cdot 2\right)}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - x} \cdot 2\right)} \cdot \left(\frac{x}{1 - x} \cdot 2\right)\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   4. lift-/.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\left(\color{blue}{\frac{x}{1 - x}} \cdot 2\right) \cdot \left(\frac{x}{1 - x} \cdot 2\right)\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   5. associate-*l/ N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\color{blue}{\frac{x \cdot 2}{1 - x}} \cdot \left(\frac{x}{1 - x} \cdot 2\right)\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{\color{blue}{2 \cdot x}}{1 - x} \cdot \left(\frac{x}{1 - x} \cdot 2\right)\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   7. lift-*.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{\color{blue}{2 \cdot x}}{1 - x} \cdot \left(\frac{x}{1 - x} \cdot 2\right)\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   8. lift-*.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{2 \cdot x}{1 - x} \cdot \color{blue}{\left(\frac{x}{1 - x} \cdot 2\right)}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   9. lift-/.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{2 \cdot x}{1 - x} \cdot \left(\color{blue}{\frac{x}{1 - x}} \cdot 2\right)\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   10. associate-*l/ N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{2 \cdot x}{1 - x} \cdot \color{blue}{\frac{x \cdot 2}{1 - x}}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{2 \cdot x}{1 - x} \cdot \frac{\color{blue}{2 \cdot x}}{1 - x}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   12. lift-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{2 \cdot x}{1 - x} \cdot \frac{\color{blue}{2 \cdot x}}{1 - x}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   13. frac-times N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\color{blue}{\frac{\left(2 \cdot x\right) \cdot \left(2 \cdot x\right)}{\left(1 - x\right) \cdot \left(1 - x\right)}}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   14. lower-/.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\color{blue}{\frac{\left(2 \cdot x\right) \cdot \left(2 \cdot x\right)}{\left(1 - x\right) \cdot \left(1 - x\right)}}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   15. lower-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{\color{blue}{\left(2 \cdot x\right) \cdot \left(2 \cdot x\right)}}{\left(1 - x\right) \cdot \left(1 - x\right)}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   16. lift-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{\color{blue}{\left(2 \cdot x\right)} \cdot \left(2 \cdot x\right)}{\left(1 - x\right) \cdot \left(1 - x\right)}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{\color{blue}{\left(x \cdot 2\right)} \cdot \left(2 \cdot x\right)}{\left(1 - x\right) \cdot \left(1 - x\right)}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   18. lower-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{\color{blue}{\left(x \cdot 2\right)} \cdot \left(2 \cdot x\right)}{\left(1 - x\right) \cdot \left(1 - x\right)}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   19. lift-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{\left(x \cdot 2\right) \cdot \color{blue}{\left(2 \cdot x\right)}}{\left(1 - x\right) \cdot \left(1 - x\right)}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   20. *-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{\left(x \cdot 2\right) \cdot \color{blue}{\left(x \cdot 2\right)}}{\left(1 - x\right) \cdot \left(1 - x\right)}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   21. lower-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{\left(x \cdot 2\right) \cdot \color{blue}{\left(x \cdot 2\right)}}{\left(1 - x\right) \cdot \left(1 - x\right)}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   22. lower-*.f32 99.0

       \[\leadsto 0.5 \cdot \left(\mathsf{log1p}\left(-\frac{\left(x \cdot 2\right) \cdot \left(x \cdot 2\right)}{\color{blue}{\left(1 - x\right) \cdot \left(1 - x\right)}}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

5. Applied rewrites 99.0%

   \[\leadsto 0.5 \cdot \left(\mathsf{log1p}\left(-\color{blue}{\frac{\left(x \cdot 2\right) \cdot \left(x \cdot 2\right)}{\left(1 - x\right) \cdot \left(1 - x\right)}}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right) \cdot \color{blue}{x} \]

   2. lower-*.f32 N/A

      \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right) \cdot \color{blue}{x} \]

   3. +-commutative N/A

      \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right) + 1\right) \cdot x \]

   4. *-commutative N/A

      \[\leadsto \left(\left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right) \cdot {x}^{2} + 1\right) \cdot x \]

   5. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}, {x}^{2}, 1\right) \cdot x \]

   6. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{5} \cdot {x}^{2} + \frac{1}{3}, {x}^{2}, 1\right) \cdot x \]

   7. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5}, {x}^{2}, \frac{1}{3}\right), {x}^{2}, 1\right) \cdot x \]

   8. pow2 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5}, x \cdot x, \frac{1}{3}\right), {x}^{2}, 1\right) \cdot x \]

   9. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5}, x \cdot x, \frac{1}{3}\right), {x}^{2}, 1\right) \cdot x \]

   10. pow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5}, x \cdot x, \frac{1}{3}\right), x \cdot x, 1\right) \cdot x \]

   11. lower-*.f32 98.9

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.2, x \cdot x, 0.3333333333333333\right), x \cdot x, 1\right) \cdot x \]

8. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.2, x \cdot x, 0.3333333333333333\right), x \cdot x, 1\right) \cdot x} \]
9. Add Preprocessing

Alternative 5: 98.4% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot x, 0.3333333333333333, 1\right) \cdot x \end{array} \]

FPCore

(FPCore (x) :precision binary32 (* (fma (* x x) 0.3333333333333333 1.0) x))

C

float code(float x) {
	return fmaf((x * x), 0.3333333333333333f, 1.0f) * x;
}

Julia

function code(x)
	return Float32(fma(Float32(x * x), Float32(0.3333333333333333), Float32(1.0)) * x)
end

Derivation

1. Initial program 99.8%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
2. Step-by-step derivation

   1. lift-log1p.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\log \left(1 + \frac{2 \cdot x}{1 - x}\right)} \]

   2. flip-+ N/A

      \[\leadsto \frac{1}{2} \cdot \log \color{blue}{\left(\frac{1 \cdot 1 - \frac{2 \cdot x}{1 - x} \cdot \frac{2 \cdot x}{1 - x}}{1 - \frac{2 \cdot x}{1 - x}}\right)} \]

   3. log-div N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\log \left(1 \cdot 1 - \frac{2 \cdot x}{1 - x} \cdot \frac{2 \cdot x}{1 - x}\right) - \log \left(1 - \frac{2 \cdot x}{1 - x}\right)\right)} \]

   4. lower--.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\log \left(1 \cdot 1 - \frac{2 \cdot x}{1 - x} \cdot \frac{2 \cdot x}{1 - x}\right) - \log \left(1 - \frac{2 \cdot x}{1 - x}\right)\right)} \]

3. Applied rewrites 99.1%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\mathsf{log1p}\left(-{\left(\frac{x}{1 - x} \cdot 2\right)}^{2}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right)} \]
4. Step-by-step derivation

   1. lift-pow.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\color{blue}{{\left(\frac{x}{1 - x} \cdot 2\right)}^{2}}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - x} \cdot 2\right) \cdot \left(\frac{x}{1 - x} \cdot 2\right)}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - x} \cdot 2\right)} \cdot \left(\frac{x}{1 - x} \cdot 2\right)\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   4. lift-/.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\left(\color{blue}{\frac{x}{1 - x}} \cdot 2\right) \cdot \left(\frac{x}{1 - x} \cdot 2\right)\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   5. associate-*l/ N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\color{blue}{\frac{x \cdot 2}{1 - x}} \cdot \left(\frac{x}{1 - x} \cdot 2\right)\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{\color{blue}{2 \cdot x}}{1 - x} \cdot \left(\frac{x}{1 - x} \cdot 2\right)\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   7. lift-*.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{\color{blue}{2 \cdot x}}{1 - x} \cdot \left(\frac{x}{1 - x} \cdot 2\right)\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   8. lift-*.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{2 \cdot x}{1 - x} \cdot \color{blue}{\left(\frac{x}{1 - x} \cdot 2\right)}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   9. lift-/.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{2 \cdot x}{1 - x} \cdot \left(\color{blue}{\frac{x}{1 - x}} \cdot 2\right)\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   10. associate-*l/ N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{2 \cdot x}{1 - x} \cdot \color{blue}{\frac{x \cdot 2}{1 - x}}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{2 \cdot x}{1 - x} \cdot \frac{\color{blue}{2 \cdot x}}{1 - x}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   12. lift-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{2 \cdot x}{1 - x} \cdot \frac{\color{blue}{2 \cdot x}}{1 - x}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   13. frac-times N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\color{blue}{\frac{\left(2 \cdot x\right) \cdot \left(2 \cdot x\right)}{\left(1 - x\right) \cdot \left(1 - x\right)}}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   14. lower-/.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\color{blue}{\frac{\left(2 \cdot x\right) \cdot \left(2 \cdot x\right)}{\left(1 - x\right) \cdot \left(1 - x\right)}}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   15. lower-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{\color{blue}{\left(2 \cdot x\right) \cdot \left(2 \cdot x\right)}}{\left(1 - x\right) \cdot \left(1 - x\right)}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   16. lift-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{\color{blue}{\left(2 \cdot x\right)} \cdot \left(2 \cdot x\right)}{\left(1 - x\right) \cdot \left(1 - x\right)}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{\color{blue}{\left(x \cdot 2\right)} \cdot \left(2 \cdot x\right)}{\left(1 - x\right) \cdot \left(1 - x\right)}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   18. lower-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{\color{blue}{\left(x \cdot 2\right)} \cdot \left(2 \cdot x\right)}{\left(1 - x\right) \cdot \left(1 - x\right)}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   19. lift-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{\left(x \cdot 2\right) \cdot \color{blue}{\left(2 \cdot x\right)}}{\left(1 - x\right) \cdot \left(1 - x\right)}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   20. *-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{\left(x \cdot 2\right) \cdot \color{blue}{\left(x \cdot 2\right)}}{\left(1 - x\right) \cdot \left(1 - x\right)}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   21. lower-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{\left(x \cdot 2\right) \cdot \color{blue}{\left(x \cdot 2\right)}}{\left(1 - x\right) \cdot \left(1 - x\right)}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   22. lower-*.f32 99.0

       \[\leadsto 0.5 \cdot \left(\mathsf{log1p}\left(-\frac{\left(x \cdot 2\right) \cdot \left(x \cdot 2\right)}{\color{blue}{\left(1 - x\right) \cdot \left(1 - x\right)}}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

5. Applied rewrites 99.0%

   \[\leadsto 0.5 \cdot \left(\mathsf{log1p}\left(-\color{blue}{\frac{\left(x \cdot 2\right) \cdot \left(x \cdot 2\right)}{\left(1 - x\right) \cdot \left(1 - x\right)}}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

6. Taylor expanded in x around 0

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

   1. *-commutative N/A

      \[\leadsto \left(1 + \frac{1}{3} \cdot {x}^{2}\right) \cdot \color{blue}{x} \]

   2. lower-*.f32 N/A

      \[\leadsto \left(1 + \frac{1}{3} \cdot {x}^{2}\right) \cdot \color{blue}{x} \]

   3. +-commutative N/A

      \[\leadsto \left(\frac{1}{3} \cdot {x}^{2} + 1\right) \cdot x \]

   4. *-commutative N/A

      \[\leadsto \left({x}^{2} \cdot \frac{1}{3} + 1\right) \cdot x \]

   5. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left({x}^{2}, \frac{1}{3}, 1\right) \cdot x \]

   6. pow2 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{3}, 1\right) \cdot x \]

   7. lower-*.f32 98.4

      \[\leadsto \mathsf{fma}\left(x \cdot x, 0.3333333333333333, 1\right) \cdot x \]

8. Applied rewrites 98.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, 0.3333333333333333, 1\right) \cdot x} \]
9. Add Preprocessing

Alternative 6: 96.9% accurate, 125.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x) :precision binary32 x)

C

float code(float x) {
	return x;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(x)
use fmin_fmax_functions
    real(4), intent (in) :: x
    code = x
end function

Julia

function code(x)
	return x
end

MATLAB

function tmp = code(x)
	tmp = x;
end

Derivation

1. Initial program 99.8%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
2. Step-by-step derivation

   1. lift-log1p.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\log \left(1 + \frac{2 \cdot x}{1 - x}\right)} \]

   2. flip-+ N/A

      \[\leadsto \frac{1}{2} \cdot \log \color{blue}{\left(\frac{1 \cdot 1 - \frac{2 \cdot x}{1 - x} \cdot \frac{2 \cdot x}{1 - x}}{1 - \frac{2 \cdot x}{1 - x}}\right)} \]

   3. log-div N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\log \left(1 \cdot 1 - \frac{2 \cdot x}{1 - x} \cdot \frac{2 \cdot x}{1 - x}\right) - \log \left(1 - \frac{2 \cdot x}{1 - x}\right)\right)} \]

   4. lower--.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\log \left(1 \cdot 1 - \frac{2 \cdot x}{1 - x} \cdot \frac{2 \cdot x}{1 - x}\right) - \log \left(1 - \frac{2 \cdot x}{1 - x}\right)\right)} \]

3. Applied rewrites 99.1%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\mathsf{log1p}\left(-{\left(\frac{x}{1 - x} \cdot 2\right)}^{2}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right)} \]
4. Step-by-step derivation

   1. lift-pow.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\color{blue}{{\left(\frac{x}{1 - x} \cdot 2\right)}^{2}}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - x} \cdot 2\right) \cdot \left(\frac{x}{1 - x} \cdot 2\right)}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - x} \cdot 2\right)} \cdot \left(\frac{x}{1 - x} \cdot 2\right)\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   4. lift-/.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\left(\color{blue}{\frac{x}{1 - x}} \cdot 2\right) \cdot \left(\frac{x}{1 - x} \cdot 2\right)\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   5. associate-*l/ N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\color{blue}{\frac{x \cdot 2}{1 - x}} \cdot \left(\frac{x}{1 - x} \cdot 2\right)\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{\color{blue}{2 \cdot x}}{1 - x} \cdot \left(\frac{x}{1 - x} \cdot 2\right)\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   7. lift-*.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{\color{blue}{2 \cdot x}}{1 - x} \cdot \left(\frac{x}{1 - x} \cdot 2\right)\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   8. lift-*.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{2 \cdot x}{1 - x} \cdot \color{blue}{\left(\frac{x}{1 - x} \cdot 2\right)}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   9. lift-/.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{2 \cdot x}{1 - x} \cdot \left(\color{blue}{\frac{x}{1 - x}} \cdot 2\right)\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   10. associate-*l/ N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{2 \cdot x}{1 - x} \cdot \color{blue}{\frac{x \cdot 2}{1 - x}}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{2 \cdot x}{1 - x} \cdot \frac{\color{blue}{2 \cdot x}}{1 - x}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   12. lift-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{2 \cdot x}{1 - x} \cdot \frac{\color{blue}{2 \cdot x}}{1 - x}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   13. frac-times N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\color{blue}{\frac{\left(2 \cdot x\right) \cdot \left(2 \cdot x\right)}{\left(1 - x\right) \cdot \left(1 - x\right)}}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   14. lower-/.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\color{blue}{\frac{\left(2 \cdot x\right) \cdot \left(2 \cdot x\right)}{\left(1 - x\right) \cdot \left(1 - x\right)}}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   15. lower-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{\color{blue}{\left(2 \cdot x\right) \cdot \left(2 \cdot x\right)}}{\left(1 - x\right) \cdot \left(1 - x\right)}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   16. lift-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{\color{blue}{\left(2 \cdot x\right)} \cdot \left(2 \cdot x\right)}{\left(1 - x\right) \cdot \left(1 - x\right)}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{\color{blue}{\left(x \cdot 2\right)} \cdot \left(2 \cdot x\right)}{\left(1 - x\right) \cdot \left(1 - x\right)}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   18. lower-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{\color{blue}{\left(x \cdot 2\right)} \cdot \left(2 \cdot x\right)}{\left(1 - x\right) \cdot \left(1 - x\right)}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   19. lift-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{\left(x \cdot 2\right) \cdot \color{blue}{\left(2 \cdot x\right)}}{\left(1 - x\right) \cdot \left(1 - x\right)}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   20. *-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{\left(x \cdot 2\right) \cdot \color{blue}{\left(x \cdot 2\right)}}{\left(1 - x\right) \cdot \left(1 - x\right)}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   21. lower-*.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(-\frac{\left(x \cdot 2\right) \cdot \color{blue}{\left(x \cdot 2\right)}}{\left(1 - x\right) \cdot \left(1 - x\right)}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

   22. lower-*.f32 99.0

       \[\leadsto 0.5 \cdot \left(\mathsf{log1p}\left(-\frac{\left(x \cdot 2\right) \cdot \left(x \cdot 2\right)}{\color{blue}{\left(1 - x\right) \cdot \left(1 - x\right)}}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

5. Applied rewrites 99.0%

   \[\leadsto 0.5 \cdot \left(\mathsf{log1p}\left(-\color{blue}{\frac{\left(x \cdot 2\right) \cdot \left(x \cdot 2\right)}{\left(1 - x\right) \cdot \left(1 - x\right)}}\right) - \mathsf{log1p}\left(-2 \cdot \frac{x}{1 - x}\right)\right) \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x} \]
7. Step-by-step derivation

8. Applied rewrites 96.9%

   \[\leadsto \color{blue}{x} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025107 
(FPCore (x)
  :name "Rust f32::atanh"
  :precision binary32
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))