Rust f64::atanh

Percentage Accurate: 100.0% → 100.0%

Time: 2.1s

Alternatives: 3

Speedup: N/A×

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, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\\ t_1 := \mathsf{fma}\left(4 \cdot t\_0, t\_0, -2 \cdot t\_0\right)\\ 0.5 \cdot \left(\mathsf{log1p}\left(8 \cdot {t\_0}^{3}\right) - \left(\mathsf{log1p}\left({t\_1}^{3}\right) - \mathsf{log1p}\left(t\_1 \cdot t\_1 - t\_1\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ x (fma -1.0 x 1.0))) (t_1 (fma (* 4.0 t_0) t_0 (* -2.0 t_0))))
   (*
    0.5
    (-
     (log1p (* 8.0 (pow t_0 3.0)))
     (- (log1p (pow t_1 3.0)) (log1p (- (* t_1 t_1) t_1)))))))

C

double code(double x) {
	double t_0 = x / fma(-1.0, x, 1.0);
	double t_1 = fma((4.0 * t_0), t_0, (-2.0 * t_0));
	return 0.5 * (log1p((8.0 * pow(t_0, 3.0))) - (log1p(pow(t_1, 3.0)) - log1p(((t_1 * t_1) - t_1))));
}

Julia

function code(x)
	t_0 = Float64(x / fma(-1.0, x, 1.0))
	t_1 = fma(Float64(4.0 * t_0), t_0, Float64(-2.0 * t_0))
	return Float64(0.5 * Float64(log1p(Float64(8.0 * (t_0 ^ 3.0))) - Float64(log1p((t_1 ^ 3.0)) - log1p(Float64(Float64(t_1 * t_1) - t_1)))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x / N[(-1.0 * x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * t$95$0), $MachinePrecision] * t$95$0 + N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(0.5 * N[(N[Log[1 + N[(8.0 * N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(N[Log[1 + N[Power[t$95$1, 3.0], $MachinePrecision]], $MachinePrecision] - N[Log[1 + N[(N[(t$95$1 * t$95$1), $MachinePrecision] - t$95$1), $MachinePrecision]], $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-log1p.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. flip3-+ N/A

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

   6. log-div N/A

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

   7. lower--.f64 N/A

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

4. Applied rewrites 99.9%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\mathsf{log1p}\left(8 \cdot {\left(\frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\right)}^{3}\right) - \mathsf{log1p}\left(4 \cdot \left(\frac{x}{\mathsf{fma}\left(-1, x, 1\right)} \cdot \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\right) - 2 \cdot \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\right)\right)} \]
5. Step-by-step derivation

   1. lift-log1p.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(8 \cdot {\left(\frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\right)}^{3}\right) - \color{blue}{\log \left(1 + \left(4 \cdot \left(\frac{x}{\mathsf{fma}\left(-1, x, 1\right)} \cdot \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\right) - 2 \cdot \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\right)\right)}\right) \]

   2. lift--.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(8 \cdot {\left(\frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\right)}^{3}\right) - \log \left(1 + \color{blue}{\left(4 \cdot \left(\frac{x}{\mathsf{fma}\left(-1, x, 1\right)} \cdot \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\right) - 2 \cdot \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\right)}\right)\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(8 \cdot {\left(\frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\right)}^{3}\right) - \log \left(1 + \left(\color{blue}{4 \cdot \left(\frac{x}{\mathsf{fma}\left(-1, x, 1\right)} \cdot \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\right)} - 2 \cdot \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\right)\right)\right) \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(8 \cdot {\left(\frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\right)}^{3}\right) - \log \left(1 + \left(4 \cdot \color{blue}{\left(\frac{x}{\mathsf{fma}\left(-1, x, 1\right)} \cdot \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\right)} - 2 \cdot \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\right)\right)\right) \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(8 \cdot {\left(\frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\right)}^{3}\right) - \log \left(1 + \left(4 \cdot \left(\color{blue}{\frac{x}{\mathsf{fma}\left(-1, x, 1\right)}} \cdot \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\right) - 2 \cdot \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\right)\right)\right) \]

   6. lift-fma.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(8 \cdot {\left(\frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\right)}^{3}\right) - \log \left(1 + \left(4 \cdot \left(\frac{x}{\color{blue}{-1 \cdot x + 1}} \cdot \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\right) - 2 \cdot \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\right)\right)\right) \]

   7. lift-/.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(8 \cdot {\left(\frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\right)}^{3}\right) - \log \left(1 + \left(4 \cdot \left(\frac{x}{-1 \cdot x + 1} \cdot \color{blue}{\frac{x}{\mathsf{fma}\left(-1, x, 1\right)}}\right) - 2 \cdot \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\right)\right)\right) \]

   8. lift-fma.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(8 \cdot {\left(\frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\right)}^{3}\right) - \log \left(1 + \left(4 \cdot \left(\frac{x}{-1 \cdot x + 1} \cdot \frac{x}{\color{blue}{-1 \cdot x + 1}}\right) - 2 \cdot \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\right)\right)\right) \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(8 \cdot {\left(\frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\right)}^{3}\right) - \log \left(1 + \left(4 \cdot \left(\frac{x}{-1 \cdot x + 1} \cdot \frac{x}{-1 \cdot x + 1}\right) - \color{blue}{2 \cdot \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}}\right)\right)\right) \]

   10. lift-/.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(8 \cdot {\left(\frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\right)}^{3}\right) - \log \left(1 + \left(4 \cdot \left(\frac{x}{-1 \cdot x + 1} \cdot \frac{x}{-1 \cdot x + 1}\right) - 2 \cdot \color{blue}{\frac{x}{\mathsf{fma}\left(-1, x, 1\right)}}\right)\right)\right) \]

   11. lift-fma.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(8 \cdot {\left(\frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\right)}^{3}\right) - \log \left(1 + \left(4 \cdot \left(\frac{x}{-1 \cdot x + 1} \cdot \frac{x}{-1 \cdot x + 1}\right) - 2 \cdot \frac{x}{\color{blue}{-1 \cdot x + 1}}\right)\right)\right) \]

6. Applied rewrites 100.0%

   \[\leadsto 0.5 \cdot \left(\mathsf{log1p}\left(8 \cdot {\left(\frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\right)}^{3}\right) - \color{blue}{\left(\mathsf{log1p}\left({\left(\mathsf{fma}\left(4 \cdot \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}, \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}, -2 \cdot \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\right)\right)}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(4 \cdot \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}, \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}, -2 \cdot \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\right) \cdot \mathsf{fma}\left(4 \cdot \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}, \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}, -2 \cdot \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\right) - 1 \cdot \mathsf{fma}\left(4 \cdot \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}, \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}, -2 \cdot \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\right)\right)\right)}\right) \]

7. Final simplification 100.0%

   \[\leadsto 0.5 \cdot \left(\mathsf{log1p}\left(8 \cdot {\left(\frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\right)}^{3}\right) - \left(\mathsf{log1p}\left({\left(\mathsf{fma}\left(4 \cdot \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}, \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}, -2 \cdot \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\right)\right)}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(4 \cdot \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}, \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}, -2 \cdot \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\right) \cdot \mathsf{fma}\left(4 \cdot \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}, \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}, -2 \cdot \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\right) - \mathsf{fma}\left(4 \cdot \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}, \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}, -2 \cdot \frac{x}{\mathsf{fma}\left(-1, x, 1\right)}\right)\right)\right)\right) \]
8. Add Preprocessing

Alternative 2: 100.0% accurate, N/A× 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: 99.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.2857142857142857, x \cdot x, 0.4\right), x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  0.5
  (*
   (fma
    (fma (fma 0.2857142857142857 (* x x) 0.4) (* x x) 0.6666666666666666)
    (* x x)
    2.0)
   x)))

C

double code(double x) {
	return 0.5 * (fma(fma(fma(0.2857142857142857, (x * x), 0.4), (x * x), 0.6666666666666666), (x * x), 2.0) * x);
}

Julia

function code(x)
	return Float64(0.5 * Float64(fma(fma(fma(0.2857142857142857, Float64(x * x), 0.4), Float64(x * x), 0.6666666666666666), Float64(x * x), 2.0) * x))
end

Wolfram

code[x_] := N[(0.5 * N[(N[(N[(N[(0.2857142857142857 * N[(x * x), $MachinePrecision] + 0.4), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] * 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

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.9%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.2857142857142857, x \cdot x, 0.4\right), x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right)} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025066 
(FPCore (x)
  :name "Rust f64::atanh"
  :precision binary64
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))