Hyperbolic arc-(co)tangent

Percentage Accurate: 8.6% → 99.7%

Time: 12.2s

Alternatives: 7

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: 8.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(x * N[(N[(1.0 + N[(N[(0.2 + N[(N[(x * x), $MachinePrecision] * 0.14285714285714285), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.2%

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

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

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

   2. metadata-eval N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\left(\frac{1 + x}{1 - x}\right)\right)\right) \]

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

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

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

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

   6. --lowering--.f64 9.2%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right)\right) \]

3. Simplified 9.2%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)} \]
6. 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 99.9%

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

7. Simplified 99.9%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-+l+ N/A

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

   4. *-commutative N/A

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

   5. +-commutative N/A

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

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

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

9. Applied egg-rr 99.9%

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

Alternative 2: 99.7% accurate, 5.3× 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 9.2%

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

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

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

   2. metadata-eval N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\left(\frac{1 + x}{1 - x}\right)\right)\right) \]

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

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

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

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

   6. --lowering--.f64 9.2%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right)\right) \]

3. Simplified 9.2%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)} \]
6. 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 99.9%

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

7. Simplified 99.9%

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

Alternative 3: 99.6% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.2%

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

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

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

   2. metadata-eval N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\left(\frac{1 + x}{1 - x}\right)\right)\right) \]

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

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

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

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

   6. --lowering--.f64 9.2%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right)\right) \]

3. Simplified 9.2%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)} \]
6. 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 99.7%

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

7. Simplified 99.7%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

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

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

   5. associate-*l* N/A

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

   6. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

   12. *-lowering-*.f64 99.7%

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

9. Applied egg-rr 99.7%

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

10. Final simplification 99.7%

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

Alternative 4: 99.6% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.2%

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

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

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

   2. metadata-eval N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\left(\frac{1 + x}{1 - x}\right)\right)\right) \]

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

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

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

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

   6. --lowering--.f64 9.2%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right)\right) \]

3. Simplified 9.2%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)} \]
6. 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 99.7%

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

   \[\leadsto x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.3333333333333333 + 0.2 \cdot \left(x \cdot x\right)\right)\right) \]
9. Add Preprocessing

Alternative 5: 99.4% accurate, 12.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.2%

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

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

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

   2. metadata-eval N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\left(\frac{1 + x}{1 - x}\right)\right)\right) \]

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

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

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

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

   6. --lowering--.f64 9.2%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right)\right) \]

3. Simplified 9.2%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)} \]
6. 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 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in x around 0

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

10. Simplified 99.4%

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

    1. +-commutative N/A

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

    2. distribute-rgt-in N/A

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

    3. *-lft-identity N/A

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

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

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

    5. *-commutative N/A

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

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

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

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

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

    8. *-lowering-*.f64 99.4%

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

12. Applied egg-rr 99.4%

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

13. Final simplification 99.4%

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

Alternative 6: 99.4% 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 9.2%

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

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

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

   2. metadata-eval N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\left(\frac{1 + x}{1 - x}\right)\right)\right) \]

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

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

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

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

   6. --lowering--.f64 9.2%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right)\right) \]

3. Simplified 9.2%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)} \]
6. 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 99.4%

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

7. Simplified 99.4%

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

Alternative 7: 98.9% 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 9.2%

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

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

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

   2. metadata-eval N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\left(\frac{1 + x}{1 - x}\right)\right)\right) \]

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

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

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

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

   6. --lowering--.f64 9.2%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, x\right), \mathsf{\_.f64}\left(1, x\right)\right)\right)\right) \]

3. Simplified 9.2%

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

5. Taylor expanded in x around 0

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

7. Simplified 98.9%

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

Reproduce

herbie shell --seed 2024164 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))