Rust f64::atanh

Percentage Accurate: 100.0% → 100.0%

Time: 4.6s

Alternatives: 4

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

   1. associate-*l/ 100.0%

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

   2. *-commutative 100.0%

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

   3. sub-neg 100.0%

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

   4. +-commutative 100.0%

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

   5. neg-sub0 100.0%

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

   6. associate-+l- 100.0%

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

   7. sub0-neg 100.0%

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

   8. distribute-frac-neg2 100.0%

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

   9. distribute-neg-frac 100.0%

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

   10. metadata-eval 100.0%

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

   11. sub-neg 100.0%

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

   12. metadata-eval 100.0%

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

3. Simplified 100.0%

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

   1. *-commutative 100.0%

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

   2. frac-2neg 100.0%

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

   3. metadata-eval 100.0%

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

   4. +-commutative 100.0%

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

   5. distribute-neg-in 100.0%

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

   6. metadata-eval 100.0%

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

   7. sub-neg 100.0%

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

   8. associate-*l/ 100.0%

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

   9. flip-- 99.9%

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

   10. associate-/r/ 100.0%

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

   11. *-commutative 100.0%

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

   12. metadata-eval 100.0%

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

   13. +-commutative 100.0%

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

6. Applied egg-rr 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

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

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

code[x_] := N[(0.5 * N[Log[1 + N[(x * N[(-2.0 / N[(x + -1.0), $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. Step-by-step derivation

   1. associate-*l/ 100.0%

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

   2. *-commutative 100.0%

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

   3. sub-neg 100.0%

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

   4. +-commutative 100.0%

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

   5. neg-sub0 100.0%

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

   6. associate-+l- 100.0%

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

   7. sub0-neg 100.0%

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

   8. distribute-frac-neg2 100.0%

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

   9. distribute-neg-frac 100.0%

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

   10. metadata-eval 100.0%

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

   11. sub-neg 100.0%

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

   12. metadata-eval 100.0%

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

3. Simplified 100.0%

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

Alternative 4: 99.0% accurate, 21.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. associate-*l/ 100.0%

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

   2. *-commutative 100.0%

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

   3. sub-neg 100.0%

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

   4. +-commutative 100.0%

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

   5. neg-sub0 100.0%

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

   6. associate-+l- 100.0%

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

   7. sub0-neg 100.0%

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

   8. distribute-frac-neg2 100.0%

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

   9. distribute-neg-frac 100.0%

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

   10. metadata-eval 100.0%

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

   11. sub-neg 100.0%

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

   12. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 98.3%

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

6. Final simplification 98.3%

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

Reproduce

herbie shell --seed 2024097 
(FPCore (x)
  :name "Rust f64::atanh"
  :precision binary64
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))