Rust f64::atanh

Percentage Accurate: 100.0% → 100.0%

Time: 5.8s

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, 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 2: 99.8% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.3333333333333333 + x \cdot \left(x \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.14285714285714285\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (+
   1.0
   (*
    (* x x)
    (+
     0.3333333333333333
     (* x (* x (+ 0.2 (* (* x x) 0.14285714285714285)))))))))

C

double code(double x) {
	return x * (1.0 + ((x * x) * (0.3333333333333333 + (x * (x * (0.2 + ((x * x) * 0.14285714285714285)))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + ((x * x) * (0.3333333333333333d0 + (x * (x * (0.2d0 + ((x * x) * 0.14285714285714285d0)))))))
end function

Java

public static double code(double x) {
	return x * (1.0 + ((x * x) * (0.3333333333333333 + (x * (x * (0.2 + ((x * x) * 0.14285714285714285)))))));
}

Python

def code(x):
	return x * (1.0 + ((x * x) * (0.3333333333333333 + (x * (x * (0.2 + ((x * x) * 0.14285714285714285)))))))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(0.3333333333333333 + Float64(x * Float64(x * Float64(0.2 + Float64(Float64(x * x) * 0.14285714285714285))))))))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + ((x * x) * (0.3333333333333333 + (x * (x * (0.2 + ((x * x) * 0.14285714285714285)))))));
end

Wolfram

code[x_] := N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.3333333333333333 + N[(x * N[(x * N[(0.2 + N[(N[(x * x), $MachinePrecision] * 0.14285714285714285), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. 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. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{3}} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{3}} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{3}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{5}} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{5}, \color{blue}{\left(\frac{1}{7} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{5}, \left({x}^{2} \cdot \color{blue}{\frac{1}{7}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{5}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{7}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{5}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f64 99.8%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{5}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{7}\right)\right)\right)\right)\right)\right)\right)\right) \]

5. Simplified 99.8%

   \[\leadsto \color{blue}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.3333333333333333 + x \cdot \left(x \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.14285714285714285\right)\right)\right)\right)} \]
6. Add Preprocessing

Alternative 3: 99.7% accurate, 7.3× speedup

Math

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

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(x * Float64(x * Float64(0.3333333333333333 + Float64(x * Float64(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[(x * N[(x * N[(0.3333333333333333 + N[(x * N[(x * 0.2), $MachinePrecision]), $MachinePrecision]), $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. 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. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)}\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{3}} + \frac{1}{5} \cdot {x}^{2}\right)\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{3}, \color{blue}{\left(\frac{1}{5} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{3}, \left({x}^{2} \cdot \color{blue}{\frac{1}{5}}\right)\right)\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{3}, \left(\left(x \cdot x\right) \cdot \frac{1}{5}\right)\right)\right)\right)\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{3}, \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{5}\right)}\right)\right)\right)\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \frac{1}{5}\right)}\right)\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 99.8%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{5}}\right)\right)\right)\right)\right)\right)\right) \]

5. Simplified 99.8%

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

Alternative 4: 99.5% accurate, 12.1× 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 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. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + \frac{1}{3} \cdot {x}^{2}\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{3} \cdot {x}^{2}\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

   5. *-lowering-*.f64 99.7%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

5. Simplified 99.7%

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

6. Final simplification 99.7%

   \[\leadsto x \cdot \left(1 + \left(x \cdot x\right) \cdot 0.3333333333333333\right) \]
7. Add Preprocessing

Alternative 5: 99.0% accurate, 109.0× 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 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

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Simplified 99.4%

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

Reproduce

herbie shell --seed 2024139 
(FPCore (x)
  :name "Rust f64::atanh"
  :precision binary64
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))