Rust f64::atanh

Percentage Accurate: 100.0% → 100.0%

Time: 5.4s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (x) :precision binary64 (atanh x))

C

double code(double x) {
	return atanh(x);
}

Python

def code(x):
	return math.atanh(x)

Julia

function code(x)
	return atanh(x)
end

MATLAB

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

Wolfram

code[x_] := N[ArcTanh[x], $MachinePrecision]

Initial Program: 100.0% 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 binary64 (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))

C

double code(double x) {
	return 0.5 * log1p(((2.0 * x) / (1.0 - x)));
}

Java

public static double code(double x) {
	return 0.5 * Math.log1p(((2.0 * x) / (1.0 - x)));
}

Python

def code(x):
	return 0.5 * math.log1p(((2.0 * x) / (1.0 - x)))

Julia

function code(x)
	return Float64(0.5 * log1p(Float64(Float64(2.0 * x) / Float64(1.0 - x))))
end

Wolfram

code[x_] := N[(0.5 * N[Log[1 + N[(N[(2.0 * x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (* (log1p (/ (fma (* x x) 2.0 (* 2.0 x)) (fma (- x) x 1.0))) 0.5))

C

double code(double x) {
	return log1p((fma((x * x), 2.0, (2.0 * x)) / fma(-x, x, 1.0))) * 0.5;
}

Julia

function code(x)
	return Float64(log1p(Float64(fma(Float64(x * x), 2.0, Float64(2.0 * x)) / fma(Float64(-x), x, 1.0))) * 0.5)
end

Wolfram

code[x_] := N[(N[Log[1 + N[(N[(N[(x * x), $MachinePrecision] * 2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision] / N[((-x) * x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 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-/.f64 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-*.f64 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-*.f64 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.f64 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. sub-neg N/A

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

   16. +-commutative N/A

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

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

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

   18. lower-fma.f64 N/A

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

   19. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-rgt-in N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. lower-fma.f64 100.0

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 100.0

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

6. Applied rewrites 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (* (log1p (/ (* (fma x x x) 2.0) (fma (- x) x 1.0))) 0.5))

C

double code(double x) {
	return log1p(((fma(x, x, x) * 2.0) / fma(-x, x, 1.0))) * 0.5;
}

Julia

function code(x)
	return Float64(log1p(Float64(Float64(fma(x, x, x) * 2.0) / fma(Float64(-x), x, 1.0))) * 0.5)
end

Wolfram

code[x_] := N[(N[Log[1 + N[(N[(N[(x * x + x), $MachinePrecision] * 2.0), $MachinePrecision] / N[((-x) * x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 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-/.f64 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-*.f64 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-*.f64 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.f64 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. sub-neg N/A

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

   16. +-commutative N/A

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

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

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

   18. lower-fma.f64 N/A

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

   19. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* (log1p (/ (* 2.0 x) (- 1.0 x))) 0.5))

C

double code(double x) {
	return log1p(((2.0 * x) / (1.0 - x))) * 0.5;
}

Java

public static double code(double x) {
	return Math.log1p(((2.0 * x) / (1.0 - x))) * 0.5;
}

Python

def code(x):
	return math.log1p(((2.0 * x) / (1.0 - x))) * 0.5

Julia

function code(x)
	return Float64(log1p(Float64(Float64(2.0 * x) / Float64(1.0 - x))) * 0.5)
end

Wolfram

code[x_] := N[(N[Log[1 + N[(N[(2.0 * x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 4: 99.8% accurate, 3.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (*
   (* (fma (fma 0.14285714285714285 (* x x) 0.2) (* x x) 0.3333333333333333) x)
   x)
  x
  x))

C

double code(double x) {
	return fma(((fma(fma(0.14285714285714285, (x * x), 0.2), (x * x), 0.3333333333333333) * x) * x), x, x);
}

Julia

function code(x)
	return fma(Float64(Float64(fma(fma(0.14285714285714285, Float64(x * x), 0.2), Float64(x * x), 0.3333333333333333) * x) * x), x, x)
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(0.14285714285714285 * N[(x * x), $MachinePrecision] + 0.2), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x + x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
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-*.f64 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.f64 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.f64 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.f64 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-*.f64 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-*.f64 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-*.f64 99.4

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

5. Applied rewrites 99.4%

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

7. Applied rewrites 99.4%

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

Alternative 5: 99.8% 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 binary64
 (*
  (fma
   (fma (fma 0.14285714285714285 (* x x) 0.2) (* x x) 0.3333333333333333)
   (* x x)
   1.0)
  x))

C

double code(double x) {
	return fma(fma(fma(0.14285714285714285, (x * x), 0.2), (x * x), 0.3333333333333333), (x * x), 1.0) * x;
}

Julia

function code(x)
	return Float64(fma(fma(fma(0.14285714285714285, Float64(x * x), 0.2), Float64(x * x), 0.3333333333333333), Float64(x * x), 1.0) * x)
end

Wolfram

code[x_] := N[(N[(N[(N[(0.14285714285714285 * N[(x * x), $MachinePrecision] + 0.2), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
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-*.f64 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.f64 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.f64 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.f64 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-*.f64 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-*.f64 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-*.f64 99.4

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

5. Applied rewrites 99.4%

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

Alternative 6: 99.7% 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 binary64
 (* (fma (fma 0.2 (* x x) 0.3333333333333333) (* x x) 1.0) x))

C

double code(double x) {
	return fma(fma(0.2, (x * x), 0.3333333333333333), (x * x), 1.0) * x;
}

Julia

function code(x)
	return Float64(fma(fma(0.2, Float64(x * x), 0.3333333333333333), Float64(x * x), 1.0) * x)
end

Wolfram

code[x_] := N[(N[(N[(0.2 * N[(x * x), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
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-*.f64 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.f64 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.f64 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-*.f64 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-*.f64 99.3

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

5. Applied rewrites 99.3%

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

Alternative 7: 99.5% 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 binary64 (fma (* 0.3333333333333333 (* x x)) x x))

C

double code(double x) {
	return fma((0.3333333333333333 * (x * x)), x, x);
}

Julia

function code(x)
	return fma(Float64(0.3333333333333333 * Float64(x * x)), x, x)
end

Wolfram

code[x_] := N[(N[(0.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
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-*.f64 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.f64 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-*.f64 99.1

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

5. Applied rewrites 99.1%

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

7. Applied rewrites 99.1%

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

Alternative 8: 99.5% 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 binary64 (* (fma (* x x) 0.3333333333333333 1.0) x))

C

double code(double x) {
	return fma((x * x), 0.3333333333333333, 1.0) * x;
}

Julia

function code(x)
	return Float64(fma(Float64(x * x), 0.3333333333333333, 1.0) * x)
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
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-*.f64 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.f64 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-*.f64 99.1

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

5. Applied rewrites 99.1%

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

Alternative 9: 99.1% accurate, 20.8× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* 1.0 x))

C

double code(double x) {
	return 1.0 * x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 * x
end function

Java

public static double code(double x) {
	return 1.0 * x;
}

Python

def code(x):
	return 1.0 * x

Julia

function code(x)
	return Float64(1.0 * x)
end

MATLAB

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

Wolfram

code[x_] := N[(1.0 * x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
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-*.f64 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.f64 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.f64 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.f64 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-*.f64 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-*.f64 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-*.f64 99.4

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.2%

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

Reproduce

herbie shell --seed 2024235 
(FPCore (x)
  :name "Rust f64::atanh"
  :precision binary64
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))