Rust f64::atanh

Percentage Accurate: 100.0% → 100.0%

Time: 3.5s

Alternatives: 5

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, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

code[x_] := N[(0.5 * N[Log[1 + N[(N[(N[(2.0 * x), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[Power[x, 2.0], $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. flip-- 99.9%

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

   2. associate-/r/ 100.0%

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

   3. metadata-eval 100.0%

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

   4. pow2 100.0%

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

   5. +-commutative 100.0%

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

4. Applied egg-rr 100.0%

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

   1. associate-*l/ 100.0%

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

6. Simplified 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

code[x_] := N[(0.5 * N[Log[1 + N[(1.0 / N[(N[(0.5 / x), $MachinePrecision] + -0.5), $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. add-log-exp 9.1%

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

   2. *-un-lft-identity 9.1%

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

   3. log-prod 9.1%

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

   4. metadata-eval 9.1%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{0} + \log \left(e^{\frac{2 \cdot x}{1 - x}}\right)\right) \]

   5. add-log-exp 100.0%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(0 + \color{blue}{\frac{2 \cdot x}{1 - x}}\right) \]

   6. *-commutative 100.0%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(0 + \frac{\color{blue}{x \cdot 2}}{1 - x}\right) \]

   7. associate-/l* 99.9%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(0 + \color{blue}{x \cdot \frac{2}{1 - x}}\right) \]

4. Applied egg-rr 99.9%

   \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{0 + x \cdot \frac{2}{1 - x}}\right) \]
5. Step-by-step derivation

   1. +-lft-identity 99.9%

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

   2. associate-*r/ 100.0%

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

   3. *-commutative 100.0%

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

   4. *-lft-identity 100.0%

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

   5. associate-*l/ 99.9%

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

   6. associate-/r/ 99.7%

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

   7. /-rgt-identity 99.7%

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

   8. /-rgt-identity 99.7%

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

   9. div-sub 99.7%

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

   10. sub-neg 99.7%

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

   11. associate-/r* 99.7%

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

   12. metadata-eval 99.7%

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

   13. *-commutative 99.7%

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

   14. associate-/r* 99.7%

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

   15. *-inverses 99.7%

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

   16. metadata-eval 99.7%

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

   17. metadata-eval 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 3: 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. Final simplification 100.0%

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

Alternative 4: 97.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* 0.5 (log1p (* 2.0 x))))

C

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

Java

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

Python

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

Julia

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

Wolfram

code[x_] := N[(0.5 * N[Log[1 + N[(2.0 * x), $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 97.3%

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

4. Final simplification 97.3%

   \[\leadsto 0.5 \cdot \mathsf{log1p}\left(2 \cdot x\right) \]
5. Add Preprocessing

Alternative 5: 0.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

code[x_] := N[(0.5 * N[Log[1 + -2.0], $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 inf 0.0%

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

4. Final simplification 0.0%

   \[\leadsto 0.5 \cdot \mathsf{log1p}\left(-2\right) \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024044 
(FPCore (x)
  :name "Rust f64::atanh"
  :precision binary64
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))