Hyperbolic arc-(co)tangent

Percentage Accurate: 8.3% → 100.0%

Time: 14.0s

Alternatives: 10

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: 8.3% 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: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(2 \cdot \mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(0 - x \cdot x\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* 0.5 (- (* 2.0 (log1p x)) (log1p (- 0.0 (* x x))))))

C

double code(double x) {
	return 0.5 * ((2.0 * log1p(x)) - log1p((0.0 - (x * x))));
}

Java

public static double code(double x) {
	return 0.5 * ((2.0 * Math.log1p(x)) - Math.log1p((0.0 - (x * x))));
}

Python

def code(x):
	return 0.5 * ((2.0 * math.log1p(x)) - math.log1p((0.0 - (x * x))))

Julia

function code(x)
	return Float64(0.5 * Float64(Float64(2.0 * log1p(x)) - log1p(Float64(0.0 - Float64(x * x)))))
end

Wolfram

code[x_] := N[(0.5 * N[(N[(2.0 * N[Log[1 + x], $MachinePrecision]), $MachinePrecision] - N[Log[1 + N[(0.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.7%

   \[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right) \]
2. Add Preprocessing

4. Applied egg-rr 100.0%

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

   1. metadata-eval 100.0%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(2, \mathsf{log1p.f64}\left(x\right)\right)}, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(0, x\right)\right)\right)\right)\right) \]

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

   \[\leadsto 0.5 \cdot \left(2 \cdot \mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(0 - x \cdot x\right)\right) \]
8. Add Preprocessing

Alternative 2: 99.8% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{1 + x \cdot \left(x \cdot \left(-0.3333333333333333 + x \cdot \left(x \cdot \left(-0.08888888888888889 + \left(x \cdot x\right) \cdot -0.04656084656084656\right)\right)\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  x
  (+
   1.0
   (*
    x
    (*
     x
     (+
      -0.3333333333333333
      (*
       x
       (* x (+ -0.08888888888888889 (* (* x x) -0.04656084656084656))))))))))

C

double code(double x) {
	return x / (1.0 + (x * (x * (-0.3333333333333333 + (x * (x * (-0.08888888888888889 + ((x * x) * -0.04656084656084656))))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x / (1.0d0 + (x * (x * ((-0.3333333333333333d0) + (x * (x * ((-0.08888888888888889d0) + ((x * x) * (-0.04656084656084656d0)))))))))
end function

Java

public static double code(double x) {
	return x / (1.0 + (x * (x * (-0.3333333333333333 + (x * (x * (-0.08888888888888889 + ((x * x) * -0.04656084656084656))))))));
}

Python

def code(x):
	return x / (1.0 + (x * (x * (-0.3333333333333333 + (x * (x * (-0.08888888888888889 + ((x * x) * -0.04656084656084656))))))))

Julia

function code(x)
	return Float64(x / Float64(1.0 + Float64(x * Float64(x * Float64(-0.3333333333333333 + Float64(x * Float64(x * Float64(-0.08888888888888889 + Float64(Float64(x * x) * -0.04656084656084656)))))))))
end

MATLAB

function tmp = code(x)
	tmp = x / (1.0 + (x * (x * (-0.3333333333333333 + (x * (x * (-0.08888888888888889 + ((x * x) * -0.04656084656084656))))))));
end

Wolfram

code[x_] := N[(x / N[(1.0 + N[(x * N[(x * N[(-0.3333333333333333 + N[(x * N[(x * N[(-0.08888888888888889 + N[(N[(x * x), $MachinePrecision] * -0.04656084656084656), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.7%

   \[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right) \]
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. *-lowering-*.f64 N/A

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

   8. unpow2 N/A

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

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

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

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

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f64 99.7%

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

5. Simplified 99.7%

   \[\leadsto \color{blue}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.2 + x \cdot \left(x \cdot 0.14285714285714285\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

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

   2. flip-+ N/A

      \[\leadsto \frac{1 \cdot 1 - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + x \cdot \left(x \cdot \frac{1}{7}\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + x \cdot \left(x \cdot \frac{1}{7}\right)\right)\right)\right)}{1 - \left(x \cdot x\right) \cdot \left(\frac{1}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + x \cdot \left(x \cdot \frac{1}{7}\right)\right)\right)} \cdot x \]

   3. associate-*l/ N/A

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

   4. clear-num N/A

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

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

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

7. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\frac{1}{\frac{1 - \left(x \cdot x\right) \cdot \left(0.3333333333333333 + x \cdot \left(x \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.14285714285714285\right)\right)\right)}{\left(1 - \left(\left(x \cdot x\right) \cdot \left(0.3333333333333333 + x \cdot \left(x \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.14285714285714285\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.3333333333333333 + x \cdot \left(x \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.14285714285714285\right)\right)\right)\right)\right) \cdot x}}} \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-44}{945} \cdot {x}^{2} - \frac{4}{45}\right) - \frac{1}{3}\right)}{x}\right)}\right) \]
9. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-44}{945} \cdot {x}^{2} - \frac{4}{45}\right) - \frac{1}{3}\right)\right), \color{blue}{x}\right)\right) \]

10. Simplified 99.5%

    \[\leadsto \frac{1}{\color{blue}{\frac{1 + \left(x \cdot x\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot \left(-0.08888888888888889 + \left(x \cdot x\right) \cdot -0.04656084656084656\right)\right)}{x}}} \]
11. Step-by-step derivation

    1. clear-num N/A

       \[\leadsto \frac{x}{\color{blue}{1 + \left(x \cdot x\right) \cdot \left(\frac{-1}{3} + \left(x \cdot x\right) \cdot \left(\frac{-4}{45} + \left(x \cdot x\right) \cdot \frac{-44}{945}\right)\right)}} \]

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

       \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(\frac{-1}{3} + \left(x \cdot x\right) \cdot \left(\frac{-4}{45} + \left(x \cdot x\right) \cdot \frac{-44}{945}\right)\right)\right)}\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\left(x \cdot x\right) \cdot \left(\frac{-1}{3} + \left(x \cdot x\right) \cdot \left(\frac{-4}{45} + \left(x \cdot x\right) \cdot \frac{-44}{945}\right)\right)\right)}\right)\right) \]

    4. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{-1}{3} + \left(x \cdot x\right) \cdot \left(\frac{-4}{45} + \left(x \cdot x\right) \cdot \frac{-44}{945}\right)\right)\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{-1}{3} + \left(x \cdot x\right) \cdot \left(\frac{-4}{45} + \left(x \cdot x\right) \cdot \frac{-44}{945}\right)\right)\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{3} + \left(x \cdot x\right) \cdot \left(\frac{-4}{45} + \left(x \cdot x\right) \cdot \frac{-44}{945}\right)\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{\left(\left(x \cdot x\right) \cdot \left(\frac{-4}{45} + \left(x \cdot x\right) \cdot \frac{-44}{945}\right)\right)}\right)\right)\right)\right)\right) \]

    8. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{3}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{-4}{45} + \left(x \cdot x\right) \cdot \frac{-44}{945}\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{-4}{45} + \left(x \cdot x\right) \cdot \frac{-44}{945}\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-4}{45} + \left(x \cdot x\right) \cdot \frac{-44}{945}\right)}\right)\right)\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-4}{45}, \color{blue}{\left(\left(x \cdot x\right) \cdot \frac{-44}{945}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-4}{45}, \mathsf{*.f64}\left(\left(x \cdot x\right), \color{blue}{\frac{-44}{945}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

    13. *-lowering-*.f64 99.7%

        \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-4}{45}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-44}{945}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

12. Applied egg-rr 99.7%

    \[\leadsto \color{blue}{\frac{x}{1 + x \cdot \left(x \cdot \left(-0.3333333333333333 + x \cdot \left(x \cdot \left(-0.08888888888888889 + \left(x \cdot x\right) \cdot -0.04656084656084656\right)\right)\right)\right)}} \]
13. Add Preprocessing

Alternative 3: 99.8% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(x + N[(N[(0.3333333333333333 + N[(x * N[(x * N[(0.2 + N[(N[(x * x), $MachinePrecision] * 0.14285714285714285), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.7%

   \[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right) \]
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. *-lowering-*.f64 N/A

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

   8. unpow2 N/A

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

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

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

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

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f64 99.7%

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

5. Simplified 99.7%

   \[\leadsto \color{blue}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.2 + x \cdot \left(x \cdot 0.14285714285714285\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

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

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

7. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\left(0.3333333333333333 + x \cdot \left(x \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.14285714285714285\right)\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) + x} \]

8. Final simplification 99.7%

   \[\leadsto x + \left(0.3333333333333333 + x \cdot \left(x \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.14285714285714285\right)\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) \]
9. Add Preprocessing

Alternative 4: 99.8% accurate, 5.3× speedup

Math

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

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(Float64(x * x) * Float64(0.2 + Float64(x * Float64(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[(N[(x * x), $MachinePrecision] * N[(0.2 + N[(x * N[(x * 0.14285714285714285), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.7%

   \[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right) \]
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. *-lowering-*.f64 N/A

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

   8. unpow2 N/A

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

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

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

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

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f64 99.7%

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

5. Simplified 99.7%

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

Alternative 5: 99.7% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \frac{x}{1 + x \cdot \left(x \cdot \left(-0.3333333333333333 + x \cdot \left(x \cdot -0.08888888888888889\right)\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  x
  (+
   1.0
   (* x (* x (+ -0.3333333333333333 (* x (* x -0.08888888888888889))))))))

C

double code(double x) {
	return x / (1.0 + (x * (x * (-0.3333333333333333 + (x * (x * -0.08888888888888889))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x / (1.0d0 + (x * (x * ((-0.3333333333333333d0) + (x * (x * (-0.08888888888888889d0)))))))
end function

Java

public static double code(double x) {
	return x / (1.0 + (x * (x * (-0.3333333333333333 + (x * (x * -0.08888888888888889))))));
}

Python

def code(x):
	return x / (1.0 + (x * (x * (-0.3333333333333333 + (x * (x * -0.08888888888888889))))))

Julia

function code(x)
	return Float64(x / Float64(1.0 + Float64(x * Float64(x * Float64(-0.3333333333333333 + Float64(x * Float64(x * -0.08888888888888889)))))))
end

MATLAB

function tmp = code(x)
	tmp = x / (1.0 + (x * (x * (-0.3333333333333333 + (x * (x * -0.08888888888888889))))));
end

Wolfram

code[x_] := N[(x / N[(1.0 + N[(x * N[(x * N[(-0.3333333333333333 + N[(x * N[(x * -0.08888888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.7%

   \[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right) \]
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. *-lowering-*.f64 N/A

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

   8. unpow2 N/A

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

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

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

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

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f64 99.7%

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

5. Simplified 99.7%

   \[\leadsto \color{blue}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.2 + x \cdot \left(x \cdot 0.14285714285714285\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

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

   2. flip-+ N/A

      \[\leadsto \frac{1 \cdot 1 - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + x \cdot \left(x \cdot \frac{1}{7}\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + x \cdot \left(x \cdot \frac{1}{7}\right)\right)\right)\right)}{1 - \left(x \cdot x\right) \cdot \left(\frac{1}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + x \cdot \left(x \cdot \frac{1}{7}\right)\right)\right)} \cdot x \]

   3. associate-*l/ N/A

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

   4. clear-num N/A

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

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

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

7. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\frac{1}{\frac{1 - \left(x \cdot x\right) \cdot \left(0.3333333333333333 + x \cdot \left(x \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.14285714285714285\right)\right)\right)}{\left(1 - \left(\left(x \cdot x\right) \cdot \left(0.3333333333333333 + x \cdot \left(x \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.14285714285714285\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.3333333333333333 + x \cdot \left(x \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.14285714285714285\right)\right)\right)\right)\right) \cdot x}}} \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 + {x}^{2} \cdot \left(\frac{-4}{45} \cdot {x}^{2} - \frac{1}{3}\right)}{x}\right)}\right) \]
9. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(1 + {x}^{2} \cdot \left(\frac{-4}{45} \cdot {x}^{2} - \frac{1}{3}\right)\right), \color{blue}{x}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{-4}{45} \cdot {x}^{2} - \frac{1}{3}\right)\right)\right), x\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{-4}{45} \cdot {x}^{2} - \frac{1}{3}\right)\right)\right), x\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{-4}{45} \cdot {x}^{2} - \frac{1}{3}\right)\right)\right), x\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-4}{45} \cdot {x}^{2} - \frac{1}{3}\right)\right)\right), x\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-4}{45} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right)\right), x\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-4}{45} \cdot {x}^{2} + \frac{-1}{3}\right)\right)\right), x\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{3} + \frac{-4}{45} \cdot {x}^{2}\right)\right)\right), x\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \left(\frac{-4}{45} \cdot {x}^{2}\right)\right)\right)\right), x\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \left({x}^{2} \cdot \frac{-4}{45}\right)\right)\right)\right), x\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{-4}{45}\right)\right)\right)\right), x\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-4}{45}\right)\right)\right)\right), x\right)\right) \]

   13. *-lowering-*.f64 99.3%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-4}{45}\right)\right)\right)\right), x\right)\right) \]

10. Simplified 99.3%

    \[\leadsto \frac{1}{\color{blue}{\frac{1 + \left(x \cdot x\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot -0.08888888888888889\right)}{x}}} \]
11. Step-by-step derivation

    1. clear-num N/A

       \[\leadsto \frac{x}{\color{blue}{1 + \left(x \cdot x\right) \cdot \left(\frac{-1}{3} + \left(x \cdot x\right) \cdot \frac{-4}{45}\right)}} \]

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

       \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(\frac{-1}{3} + \left(x \cdot x\right) \cdot \frac{-4}{45}\right)\right)}\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\left(x \cdot x\right) \cdot \left(\frac{-1}{3} + \left(x \cdot x\right) \cdot \frac{-4}{45}\right)\right)}\right)\right) \]

    4. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{-1}{3} + \left(x \cdot x\right) \cdot \frac{-4}{45}\right)\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{-1}{3} + \left(x \cdot x\right) \cdot \frac{-4}{45}\right)\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{3} + \left(x \cdot x\right) \cdot \frac{-4}{45}\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{\left(\left(x \cdot x\right) \cdot \frac{-4}{45}\right)}\right)\right)\right)\right)\right) \]

    8. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{3}, \left(x \cdot \color{blue}{\left(x \cdot \frac{-4}{45}\right)}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \frac{-4}{45}\right)}\right)\right)\right)\right)\right)\right) \]

    10. *-lowering-*.f64 99.5%

        \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{-4}{45}}\right)\right)\right)\right)\right)\right)\right) \]

12. Applied egg-rr 99.5%

    \[\leadsto \color{blue}{\frac{x}{1 + x \cdot \left(x \cdot \left(-0.3333333333333333 + x \cdot \left(x \cdot -0.08888888888888889\right)\right)\right)}} \]
13. Add Preprocessing

Alternative 6: 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 8.7%

   \[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right) \]
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. *-lowering-*.f64 N/A

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

   4. unpow2 N/A

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

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

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

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

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

   7. *-commutative N/A

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

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

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 99.4%

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

5. Simplified 99.4%

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

Alternative 7: 99.6% accurate, 12.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.7%

   \[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right) \]
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. *-lowering-*.f64 N/A

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

   8. unpow2 N/A

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

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

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

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

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f64 99.7%

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

5. Simplified 99.7%

   \[\leadsto \color{blue}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.2 + x \cdot \left(x \cdot 0.14285714285714285\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

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

   2. flip-+ N/A

      \[\leadsto \frac{1 \cdot 1 - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + x \cdot \left(x \cdot \frac{1}{7}\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + x \cdot \left(x \cdot \frac{1}{7}\right)\right)\right)\right)}{1 - \left(x \cdot x\right) \cdot \left(\frac{1}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + x \cdot \left(x \cdot \frac{1}{7}\right)\right)\right)} \cdot x \]

   3. associate-*l/ N/A

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

   4. clear-num N/A

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

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

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

7. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\frac{1}{\frac{1 - \left(x \cdot x\right) \cdot \left(0.3333333333333333 + x \cdot \left(x \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.14285714285714285\right)\right)\right)}{\left(1 - \left(\left(x \cdot x\right) \cdot \left(0.3333333333333333 + x \cdot \left(x \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.14285714285714285\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.3333333333333333 + x \cdot \left(x \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.14285714285714285\right)\right)\right)\right)\right) \cdot x}}} \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 + \frac{-1}{3} \cdot {x}^{2}}{x}\right)}\right) \]
9. Step-by-step derivation

   1. /-lowering-/.f64 N/A

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

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

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

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

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

   7. *-lowering-*.f64 98.8%

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

10. Simplified 98.8%

    \[\leadsto \frac{1}{\color{blue}{\frac{1 + x \cdot \left(x \cdot -0.3333333333333333\right)}{x}}} \]
11. Step-by-step derivation

    1. clear-num N/A

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

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

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

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

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

    4. associate-*r* N/A

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

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

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

    6. *-lowering-*.f64 99.0%

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

12. Applied egg-rr 99.0%

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

13. Final simplification 99.0%

    \[\leadsto \frac{x}{1 + -0.3333333333333333 \cdot \left(x \cdot x\right)} \]
14. Add Preprocessing

Alternative 8: 99.6% accurate, 12.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.7%

   \[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right) \]
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. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

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

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

   7. *-lowering-*.f64 99.0%

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

5. Simplified 99.0%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

      \[\leadsto x \cdot \left(x \cdot \left(x \cdot \frac{1}{3}\right)\right) + x \]

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

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

   5. associate-*r* N/A

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

   6. associate-*r* N/A

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

   7. cube-mult N/A

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

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

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

   9. cube-mult N/A

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

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

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

   11. *-lowering-*.f64 99.0%

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

7. Applied egg-rr 99.0%

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

8. Final simplification 99.0%

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

Alternative 9: 99.6% accurate, 12.3× speedup

Math

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

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(x * Float64(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[(x * N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.7%

   \[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right) \]
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. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

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

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

   7. *-lowering-*.f64 99.0%

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

5. Simplified 99.0%

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

Alternative 10: 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 8.7%

   \[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Simplified 98.5%

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

Reproduce

herbie shell --seed 2024191 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))