Rust f64::atanh

Percentage Accurate: 100.0% → 100.0%

Time: 8.2s

Alternatives: 7

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(0.5 * N[Log[1 + N[(N[(x * N[(x * N[(x * -2.0 + -2.0), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] / N[(x * N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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. flip3-- N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{\color{blue}{\frac{{1}^{3} - {x}^{3}}{1 \cdot 1 + \left(x \cdot x + 1 \cdot x\right)}}}\right) \]

   4. frac-2neg N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{\color{blue}{\frac{\mathsf{neg}\left(\left({1}^{3} - {x}^{3}\right)\right)}{\mathsf{neg}\left(\left(1 \cdot 1 + \left(x \cdot x + 1 \cdot x\right)\right)\right)}}}\right) \]

   5. associate-/r/ N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\color{blue}{\frac{2 \cdot x}{\mathsf{neg}\left(\left({1}^{3} - {x}^{3}\right)\right)} \cdot \left(\mathsf{neg}\left(\left(1 \cdot 1 + \left(x \cdot x + 1 \cdot x\right)\right)\right)\right)}\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\color{blue}{\frac{2 \cdot x}{\mathsf{neg}\left(\left({1}^{3} - {x}^{3}\right)\right)} \cdot \left(\mathsf{neg}\left(\left(1 \cdot 1 + \left(x \cdot x + 1 \cdot x\right)\right)\right)\right)}\right) \]

4. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 100.0

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

   6. lift-neg.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. distribute-neg-in N/A

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

   9. unsub-neg N/A

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

   10. lower--.f64 N/A

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

   11. lower-neg.f64 100.0

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lower-fma.f64 100.0

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. sub-neg N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 100.0

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

9. Applied rewrites 100.0%

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

Alternative 2: 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]

Derivation

1. Initial program 100.0%

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

Alternative 3: 99.8% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.14285714285714285), $MachinePrecision] + 0.2), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $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. +-commutative N/A

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

   3. distribute-lft1-in N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. unpow2 N/A

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

   7. unpow3 N/A

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

   8. lower-fma.f64 N/A

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

5. Applied rewrites 99.6%

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

Alternative 4: 99.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.4, 0.6666666666666666\right), 2\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* 0.5 (* x (fma (* x x) (fma x (* x 0.4) 0.6666666666666666) 2.0))))

C

double code(double x) {
	return 0.5 * (x * fma((x * x), fma(x, (x * 0.4), 0.6666666666666666), 2.0));
}

Julia

function code(x)
	return Float64(0.5 * Float64(x * fma(Float64(x * x), fma(x, Float64(x * 0.4), 0.6666666666666666), 2.0)))
end

Wolfram

code[x_] := N[(0.5 * N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.4), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $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

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. unpow2 N/A

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

   9. associate-*l* N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 99.5

       \[\leadsto 0.5 \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.4}, 0.6666666666666666\right), 2\right)\right) \]

5. Applied rewrites 99.5%

   \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.4, 0.6666666666666666\right), 2\right)\right)} \]
6. Add Preprocessing

Alternative 5: 99.7% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] * 0.2 + 0.3333333333333333), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $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. distribute-rgt-in N/A

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

   2. *-lft-identity N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. unpow2 N/A

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

   7. unpow3 N/A

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

   8. lower-fma.f64 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. cube-mult N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 99.5

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

5. Applied rewrites 99.5%

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

Alternative 6: 99.5% accurate, 7.4× speedup

Math

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

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(0.3333333333333333, Float64(x * Float64(x * x)), x)
end

Wolfram

code[x_] := N[(0.3333333333333333 * N[(x * N[(x * x), $MachinePrecision]), $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. +-commutative N/A

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

   3. distribute-lft1-in N/A

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

   4. associate-*l* N/A

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

   5. unpow2 N/A

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

   6. unpow3 N/A

      \[\leadsto \frac{1}{3} \cdot \color{blue}{{x}^{3}} + x \]

   7. lower-fma.f64 N/A

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

   8. cube-mult N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 99.3

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

5. Applied rewrites 99.3%

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

Alternative 7: 99.0% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(2 \cdot x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.5 (* 2.0 x)))

C

double code(double x) {
	return 0.5 * (2.0 * x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 * (2.0d0 * x)
end function

Java

public static double code(double x) {
	return 0.5 * (2.0 * x);
}

Python

def code(x):
	return 0.5 * (2.0 * x)

Julia

function code(x)
	return Float64(0.5 * Float64(2.0 * x))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 * (2.0 * x);
end

Wolfram

code[x_] := N[(0.5 * N[(2.0 * x), $MachinePrecision]), $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

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

   1. *-commutative N/A

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

   2. lower-*.f64 98.9

      \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot 2\right)} \]

5. Applied rewrites 98.9%

   \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot 2\right)} \]

6. Final simplification 98.9%

   \[\leadsto 0.5 \cdot \left(2 \cdot x\right) \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024221 
(FPCore (x)
  :name "Rust f64::atanh"
  :precision binary64
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))