Hyperbolic arc-(co)tangent

Percentage Accurate: 8.4% → 99.7%

Time: 20.5s

Alternatives: 3

Speedup: 111.0×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / 2.0d0) * log(((1.0d0 + x) / (1.0d0 - x)))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = (1.0 / 2.0) * log(((1.0 + x) / (1.0 - x)));
end

Wolfram

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

Initial Program: 8.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / 2.0d0) * log(((1.0d0 + x) / (1.0d0 - x)))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = (1.0 / 2.0) * log(((1.0 + x) / (1.0 - x)));
end

Wolfram

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

Alternative 1: 99.7% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.3333333333333333 + \left(x \cdot x\right) \cdot 0.2\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (+ 1.0 (* (* x x) (+ 0.3333333333333333 (* (* x x) 0.2))))))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + ((x * x) * (0.3333333333333333d0 + ((x * x) * 0.2d0))))
end function

Java

public static double code(double x) {
	return x * (1.0 + ((x * x) * (0.3333333333333333 + ((x * x) * 0.2))));
}

Python

def code(x):
	return x * (1.0 + ((x * x) * (0.3333333333333333 + ((x * x) * 0.2))))

Julia

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

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + ((x * x) * (0.3333333333333333 + ((x * x) * 0.2))));
end

Wolfram

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

Derivation

1. Initial program 7.4%

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

   1. metadata-eval 7.4%

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

3. Simplified 7.4%

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

5. Taylor expanded in x around 0 100.0%

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.3333333333333333 + 0.2 \cdot {x}^{2}\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative 100.0%

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.3333333333333333 + \color{blue}{{x}^{2} \cdot 0.2}\right)\right) \]

7. Simplified 100.0%

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.3333333333333333 + {x}^{2} \cdot 0.2\right)\right)} \]
8. Step-by-step derivation

   1. unpow2 100.0%

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.3333333333333333 + \color{blue}{\left(x \cdot x\right)} \cdot 0.2\right)\right) \]

9. Applied egg-rr 100.0%

   \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.3333333333333333 + \color{blue}{\left(x \cdot x\right)} \cdot 0.2\right)\right) \]
10. Step-by-step derivation

    1. unpow2 100.0%

       \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.3333333333333333 + \color{blue}{\left(x \cdot x\right)} \cdot 0.2\right)\right) \]

11. Applied egg-rr 100.0%

    \[\leadsto x \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.3333333333333333 + \left(x \cdot x\right) \cdot 0.2\right)\right) \]
12. Add Preprocessing

Alternative 2: 99.6% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + \left(x \cdot x\right) \cdot 0.3333333333333333\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (+ 1.0 (* (* x x) 0.3333333333333333))))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + ((x * x) * 0.3333333333333333d0))
end function

Java

public static double code(double x) {
	return x * (1.0 + ((x * x) * 0.3333333333333333));
}

Python

def code(x):
	return x * (1.0 + ((x * x) * 0.3333333333333333))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64(Float64(x * x) * 0.3333333333333333)))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + ((x * x) * 0.3333333333333333));
end

Wolfram

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

Derivation

1. Initial program 7.4%

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

   1. metadata-eval 7.4%

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

3. Simplified 7.4%

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

5. Taylor expanded in x around 0 100.0%

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.3333333333333333 + 0.2 \cdot {x}^{2}\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative 100.0%

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.3333333333333333 + \color{blue}{{x}^{2} \cdot 0.2}\right)\right) \]

7. Simplified 100.0%

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.3333333333333333 + {x}^{2} \cdot 0.2\right)\right)} \]
8. Step-by-step derivation

   1. unpow2 100.0%

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.3333333333333333 + \color{blue}{\left(x \cdot x\right)} \cdot 0.2\right)\right) \]

9. Applied egg-rr 100.0%

   \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.3333333333333333 + \color{blue}{\left(x \cdot x\right)} \cdot 0.2\right)\right) \]
10. Step-by-step derivation

    1. unpow2 100.0%

       \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.3333333333333333 + \color{blue}{\left(x \cdot x\right)} \cdot 0.2\right)\right) \]

11. Applied egg-rr 100.0%

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

12. Taylor expanded in x around 0 100.0%

    \[\leadsto x \cdot \left(1 + \left(x \cdot x\right) \cdot \color{blue}{0.3333333333333333}\right) \]
13. Add Preprocessing

Alternative 3: 99.1% accurate, 111.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

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

Java

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

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 7.4%

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

   1. metadata-eval 7.4%

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

3. Simplified 7.4%

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

5. Taylor expanded in x around 0 99.6%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024143 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))