Rust f32::atanh

Percentage Accurate: 99.8% → 99.9%

Time: 7.5s

Alternatives: 13

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \tanh^{-1} x \end{array} \]

FPCore

(FPCore (x) :precision binary32 (atanh x))

C

float code(float x) {
	return atanhf(x);
}

Julia

function code(x)
	return atanh(x)
end

MATLAB

function tmp = code(x)
	tmp = atanh(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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
2. Add Preprocessing
3. 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. associate-*l/ N/A

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

   6. lower-/.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

   9. +-commutative N/A

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

   10. distribute-rgt-out N/A

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

   11. lower-*.f32 N/A

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

   12. *-lft-identity N/A

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

   13. lower-fma.f32 N/A

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

   14. metadata-eval N/A

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

   15. lower--.f32 N/A

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

   16. lower-*.f32 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
2. Add Preprocessing

3. Final simplification 99.8%

   \[\leadsto \mathsf{log1p}\left(\frac{x \cdot 2}{1 - x}\right) \cdot 0.5 \]
4. Add Preprocessing

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
2. Add Preprocessing
3. 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

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. frac-2neg N/A

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

   8. lower-/.f32 N/A

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

   9. metadata-eval N/A

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

   10. neg-sub0 N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{-2}{\color{blue}{0 - \left(1 - x\right)}} \cdot x\right) \]

   11. lift--.f32 N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{-2}{0 - \color{blue}{\left(1 - x\right)}} \cdot x\right) \]

   12. sub-neg N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{-2}{0 - \color{blue}{\left(1 + \left(\mathsf{neg}\left(x\right)\right)\right)}} \cdot x\right) \]

   13. +-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{-2}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + 1\right)}} \cdot x\right) \]

   14. associate--r+ N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{-2}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(x\right)\right)\right) - 1}} \cdot x\right) \]

   15. neg-sub0 N/A

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

   16. remove-double-neg N/A

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

   17. lower--.f32 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

   \[\leadsto \mathsf{log1p}\left(\frac{-2}{x - 1} \cdot x\right) \cdot 0.5 \]
6. Add Preprocessing

Alternative 4: 99.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\left(\left(\mathsf{fma}\left(0.13333333333333333, x \cdot x, 0.1111111111111111\right) \cdot x\right) \cdot x\right) \cdot x, x, -1\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.14285714285714285, 0.2\right), x \cdot x, 0.3333333333333333\right), x \cdot x, -1\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (/
  (*
   (fma
    (* (* (* (fma 0.13333333333333333 (* x x) 0.1111111111111111) x) x) x)
    x
    -1.0)
   x)
  (fma
   (fma (fma (* x x) 0.14285714285714285 0.2) (* x x) 0.3333333333333333)
   (* x x)
   -1.0)))

C

float code(float x) {
	return (fmaf((((fmaf(0.13333333333333333f, (x * x), 0.1111111111111111f) * x) * x) * x), x, -1.0f) * x) / fmaf(fmaf(fmaf((x * x), 0.14285714285714285f, 0.2f), (x * x), 0.3333333333333333f), (x * x), -1.0f);
}

Julia

function code(x)
	return Float32(Float32(fma(Float32(Float32(Float32(fma(Float32(0.13333333333333333), Float32(x * x), Float32(0.1111111111111111)) * x) * x) * x), x, Float32(-1.0)) * x) / fma(fma(fma(Float32(x * x), Float32(0.14285714285714285), Float32(0.2)), Float32(x * x), Float32(0.3333333333333333)), Float32(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. Add Preprocessing
3. 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. associate-*l/ N/A

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

   6. lower-/.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

   9. +-commutative N/A

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

   10. distribute-rgt-out N/A

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

   11. lower-*.f32 N/A

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

   12. *-lft-identity N/A

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

   13. lower-fma.f32 N/A

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

   14. metadata-eval N/A

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

   15. lower--.f32 N/A

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

   16. lower-*.f32 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 N/A

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

   15. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{7}, x \cdot x, \frac{1}{5}\right), x \cdot x, \frac{1}{3}\right), \color{blue}{x \cdot x}, 1\right) \cdot x \]

   16. lower-*.f32 99.2

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

7. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.14285714285714285, x \cdot x, 0.2\right), x \cdot x, 0.3333333333333333\right), x \cdot x, 1\right) \cdot x} \]
8. Step-by-step derivation

9. Applied rewrites 99.3%

   \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.14285714285714285, 0.2\right), x \cdot x, 0.3333333333333333\right) \cdot x\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.14285714285714285, 0.2\right), x \cdot x, 0.3333333333333333\right) \cdot x\right), x, -1\right) \cdot x}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.14285714285714285, 0.2\right), x \cdot x, 0.3333333333333333\right), x \cdot x, -1\right)}} \]

10. Taylor expanded in x around 0

    \[\leadsto \frac{\mathsf{fma}\left({x}^{3} \cdot \left(\frac{1}{9} + \frac{2}{15} \cdot {x}^{2}\right), x, -1\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{7}, \frac{1}{5}\right), x \cdot x, \frac{1}{3}\right), x \cdot x, -1\right)} \]
11. Step-by-step derivation

12. Applied rewrites 99.4%

    \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\mathsf{fma}\left(0.13333333333333333, x \cdot x, 0.1111111111111111\right) \cdot x\right) \cdot x\right) \cdot x, x, -1\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{x \cdot x}, 0.14285714285714285, 0.2\right), x \cdot x, 0.3333333333333333\right), x \cdot x, -1\right)} \]
13. Add Preprocessing

Alternative 5: 99.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot 0.1111111111111111, x, -1\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(0.2, x \cdot x, 0.3333333333333333\right), x \cdot x, -1\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (/
  (* (fma (* (* (* x x) x) 0.1111111111111111) x -1.0) x)
  (fma (fma 0.2 (* x x) 0.3333333333333333) (* x x) -1.0)))

C

float code(float x) {
	return (fmaf((((x * x) * x) * 0.1111111111111111f), x, -1.0f) * x) / fmaf(fmaf(0.2f, (x * x), 0.3333333333333333f), (x * x), -1.0f);
}

Julia

function code(x)
	return Float32(Float32(fma(Float32(Float32(Float32(x * x) * x) * Float32(0.1111111111111111)), x, Float32(-1.0)) * x) / fma(fma(Float32(0.2), Float32(x * x), Float32(0.3333333333333333)), Float32(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. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 99.0

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

5. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.2, x \cdot x, 0.3333333333333333\right), x \cdot x, 1\right) \cdot x} \]
6. Step-by-step derivation

7. Applied rewrites 99.0%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{9} \cdot {x}^{3}, x, -1\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5}, x \cdot x, \frac{1}{3}\right), x \cdot x, -1\right)} \]
9. Step-by-step derivation

10. Applied rewrites 99.3%

    \[\leadsto \frac{\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot 0.1111111111111111, x, -1\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(0.2, x \cdot x, 0.3333333333333333\right), x \cdot x, -1\right)} \]
11. Add Preprocessing

Alternative 6: 99.4% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
2. Add Preprocessing
3. 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. associate-*l/ N/A

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

   6. lower-/.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

   9. +-commutative N/A

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

   10. distribute-rgt-out N/A

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

   11. lower-*.f32 N/A

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

   12. *-lft-identity N/A

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

   13. lower-fma.f32 N/A

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

   14. metadata-eval N/A

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

   15. lower--.f32 N/A

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

   16. lower-*.f32 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 N/A

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

   15. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{7}, x \cdot x, \frac{1}{5}\right), x \cdot x, \frac{1}{3}\right), \color{blue}{x \cdot x}, 1\right) \cdot x \]

   16. lower-*.f32 99.2

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

7. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.14285714285714285, x \cdot x, 0.2\right), x \cdot x, 0.3333333333333333\right), x \cdot x, 1\right) \cdot x} \]
8. Step-by-step derivation

9. Applied rewrites 99.3%

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

Alternative 7: 99.3% 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. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 N/A

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

   15. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{7}, x \cdot x, \frac{1}{5}\right), x \cdot x, \frac{1}{3}\right), \color{blue}{x \cdot x}, 1\right) \cdot x \]

   16. lower-*.f32 99.2

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

5. Applied rewrites 99.2%

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

Alternative 8: 99.1% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 99.0

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

5. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.2, x \cdot x, 0.3333333333333333\right), x \cdot x, 1\right) \cdot x} \]
6. Step-by-step derivation

7. Applied rewrites 99.0%

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

Alternative 9: 99.1% 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. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 99.0

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

5. Applied rewrites 99.0%

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

Alternative 10: 98.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float32(Float32(-x) / fma(Float32(0.3333333333333333), Float32(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. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 99.0

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

5. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.2, x \cdot x, 0.3333333333333333\right), x \cdot x, 1\right) \cdot x} \]
6. Step-by-step derivation

7. Applied rewrites 99.0%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{-1 \cdot x}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{5}, x \cdot x, \frac{1}{3}\right)}, x \cdot x, -1\right)} \]
9. Step-by-step derivation

10. Applied rewrites 98.7%

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

11. Taylor expanded in x around 0

    \[\leadsto \frac{\mathsf{neg}\left(x\right)}{\mathsf{fma}\left(\frac{1}{3}, \color{blue}{x} \cdot x, -1\right)} \]
12. Step-by-step derivation

13. Applied rewrites 98.7%

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

Alternative 11: 98.6% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
2. Add Preprocessing
3. 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. associate-*l/ N/A

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

   6. lower-/.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

   9. +-commutative N/A

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

   10. distribute-rgt-out N/A

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

   11. lower-*.f32 N/A

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

   12. *-lft-identity N/A

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

   13. lower-fma.f32 N/A

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

   14. metadata-eval N/A

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

   15. lower--.f32 N/A

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

   16. lower-*.f32 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 98.6

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

7. Applied rewrites 98.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, 0.3333333333333333, 1\right) \cdot x} \]
8. Step-by-step derivation

9. Applied rewrites 98.6%

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

Alternative 12: 98.6% 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. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 98.6

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

5. Applied rewrites 98.6%

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

Alternative 13: 97.0% accurate, 20.8× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x \end{array} \]

FPCore

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

C

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

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 1.0e0 * x
end function

Julia

function code(x)
	return Float32(Float32(1.0) * x)
end

MATLAB

function tmp = code(x)
	tmp = single(1.0) * x;
end

Derivation

1. Initial program 99.8%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 99.0

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

5. Applied rewrites 99.0%

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

6. Taylor expanded in x around 0

   \[\leadsto 1 \cdot x \]
7. Step-by-step derivation

8. Applied rewrites 97.1%

   \[\leadsto 1 \cdot x \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024240 
(FPCore (x)
  :name "Rust f32::atanh"
  :precision binary32
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))