Hyperbolic tangent

Percentage Accurate: 9.2% → 100.0%

Time: 14.3s

Alternatives: 8

Speedup: 409.0×

• could not determine a ground truth

  1. x = 4.1167894729186936e+24

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \frac{e^{x} - t\_0}{e^{x} + t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x)))) (/ (- (exp x) t_0) (+ (exp x) t_0))))

C

double code(double x) {
	double t_0 = exp(-x);
	return (exp(x) - t_0) / (exp(x) + t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = exp(-x)
    code = (exp(x) - t_0) / (exp(x) + t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.exp(-x);
	return (Math.exp(x) - t_0) / (Math.exp(x) + t_0);
}

Python

def code(x):
	t_0 = math.exp(-x)
	return (math.exp(x) - t_0) / (math.exp(x) + t_0)

Julia

function code(x)
	t_0 = exp(Float64(-x))
	return Float64(Float64(exp(x) - t_0) / Float64(exp(x) + t_0))
end

MATLAB

function tmp = code(x)
	t_0 = exp(-x);
	tmp = (exp(x) - t_0) / (exp(x) + t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, N[(N[(N[Exp[x], $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[Exp[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Initial Program: 9.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \frac{e^{x} - t\_0}{e^{x} + t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x)))) (/ (- (exp x) t_0) (+ (exp x) t_0))))

C

double code(double x) {
	double t_0 = exp(-x);
	return (exp(x) - t_0) / (exp(x) + t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = exp(-x)
    code = (exp(x) - t_0) / (exp(x) + t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.exp(-x);
	return (Math.exp(x) - t_0) / (Math.exp(x) + t_0);
}

Python

def code(x):
	t_0 = math.exp(-x)
	return (math.exp(x) - t_0) / (math.exp(x) + t_0)

Julia

function code(x)
	t_0 = exp(Float64(-x))
	return Float64(Float64(exp(x) - t_0) / Float64(exp(x) + t_0))
end

MATLAB

function tmp = code(x)
	t_0 = exp(-x);
	tmp = (exp(x) - t_0) / (exp(x) + t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, N[(N[(N[Exp[x], $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[Exp[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 100.0% accurate, 4.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (tanh x))

C

double code(double x) {
	return tanh(x);
}

Fortran

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

Java

public static double code(double x) {
	return Math.tanh(x);
}

Python

def code(x):
	return math.tanh(x)

Julia

function code(x)
	return tanh(x)
end

MATLAB

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

Wolfram

code[x_] := N[Tanh[x], $MachinePrecision]

Derivation

1. Initial program 9.0%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. tanh-undef N/A

      \[\leadsto \tanh x \]

   2. tanh-lowering-tanh.f64 100.0%

      \[\leadsto \mathsf{tanh.f64}\left(x\right) \]

4. Applied egg-rr 100.0%

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

Alternative 2: 97.2% accurate, 19.5× speedup

Math

\[\begin{array}{l} \\ \frac{x}{1 + x \cdot \left(x \cdot \left(0.3333333333333333 + \left(x \cdot x\right) \cdot \left(-0.022222222222222223 + x \cdot \left(x \cdot 0.0021164021164021165\right)\right)\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  x
  (+
   1.0
   (*
    x
    (*
     x
     (+
      0.3333333333333333
      (*
       (* x x)
       (+ -0.022222222222222223 (* x (* x 0.0021164021164021165))))))))))

C

double code(double x) {
	return x / (1.0 + (x * (x * (0.3333333333333333 + ((x * x) * (-0.022222222222222223 + (x * (x * 0.0021164021164021165))))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x / (1.0d0 + (x * (x * (0.3333333333333333d0 + ((x * x) * ((-0.022222222222222223d0) + (x * (x * 0.0021164021164021165d0))))))))
end function

Java

public static double code(double x) {
	return x / (1.0 + (x * (x * (0.3333333333333333 + ((x * x) * (-0.022222222222222223 + (x * (x * 0.0021164021164021165))))))));
}

Python

def code(x):
	return x / (1.0 + (x * (x * (0.3333333333333333 + ((x * x) * (-0.022222222222222223 + (x * (x * 0.0021164021164021165))))))))

Julia

function code(x)
	return Float64(x / Float64(1.0 + Float64(x * Float64(x * Float64(0.3333333333333333 + Float64(Float64(x * x) * Float64(-0.022222222222222223 + Float64(x * Float64(x * 0.0021164021164021165)))))))))
end

MATLAB

function tmp = code(x)
	tmp = x / (1.0 + (x * (x * (0.3333333333333333 + ((x * x) * (-0.022222222222222223 + (x * (x * 0.0021164021164021165))))))));
end

Wolfram

code[x_] := N[(x / N[(1.0 + N[(x * N[(x * N[(0.3333333333333333 + N[(N[(x * x), $MachinePrecision] * N[(-0.022222222222222223 + N[(x * N[(x * 0.0021164021164021165), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.0%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)} \]
4. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)}\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right)} - \frac{1}{3}\right)\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)}\right)\right)\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right)\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) + \frac{-1}{3}\right)\right)\right)\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{3} + \color{blue}{{x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{3}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{2}{15}} + \frac{-17}{315} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right) \]

   12. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{3}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{3}, \left(x \cdot \left(\left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{15}, \color{blue}{\left(\frac{-17}{315} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

   18. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{15}, \left({x}^{2} \cdot \color{blue}{\frac{-17}{315}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

5. Simplified 96.8%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.3333333333333333 + x \cdot \left(x \cdot \left(0.13333333333333333 + x \cdot \left(x \cdot -0.05396825396825397\right)\right)\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(1 + x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)\right) \cdot \color{blue}{x} \]

   2. flip-+ N/A

      \[\leadsto \frac{1 \cdot 1 - \left(x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)\right)}{1 - x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)} \cdot x \]

   3. associate-*l/ N/A

      \[\leadsto \frac{\left(1 \cdot 1 - \left(x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)\right)\right) \cdot x}{\color{blue}{1 - x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)}} \]

   4. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1 - x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)}{\left(1 \cdot 1 - \left(x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)\right)\right) \cdot x}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 - x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)}{\left(1 \cdot 1 - \left(x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)\right)\right) \cdot x}\right)}\right) \]

7. Applied egg-rr 96.5%

   \[\leadsto \color{blue}{\frac{1}{\frac{1 - x \cdot \left(x \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.13333333333333333 + x \cdot \left(x \cdot -0.05396825396825397\right)\right)\right)\right)}{\left(1 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(-0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.13333333333333333 + x \cdot \left(x \cdot -0.05396825396825397\right)\right)\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.13333333333333333 + x \cdot \left(x \cdot -0.05396825396825397\right)\right)\right)\right)\right) \cdot x}}} \]

8. Taylor expanded in x around 0

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

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

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

10. Simplified 96.8%

    \[\leadsto \frac{1}{\color{blue}{\frac{1 + \left(x \cdot x\right) \cdot \left(0.3333333333333333 + \left(x \cdot x\right) \cdot \left(-0.022222222222222223 + x \cdot \left(x \cdot 0.0021164021164021165\right)\right)\right)}{x}}} \]
11. Step-by-step derivation

    1. clear-num N/A

       \[\leadsto \frac{x}{\color{blue}{1 + \left(x \cdot x\right) \cdot \left(\frac{1}{3} + \left(x \cdot x\right) \cdot \left(\frac{-1}{45} + x \cdot \left(x \cdot \frac{2}{945}\right)\right)\right)}} \]

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

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

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

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

    4. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    12. *-lowering-*.f64 97.1%

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

12. Applied egg-rr 97.1%

    \[\leadsto \color{blue}{\frac{x}{1 + x \cdot \left(x \cdot \left(0.3333333333333333 + \left(x \cdot x\right) \cdot \left(-0.022222222222222223 + x \cdot \left(x \cdot 0.0021164021164021165\right)\right)\right)\right)}} \]
13. Add Preprocessing

Alternative 3: 97.1% accurate, 27.3× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.3333333333333333 + x \cdot \left(x \cdot 0.13333333333333333\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (+ 1.0 (* (* x x) (+ -0.3333333333333333 (* x (* x 0.13333333333333333)))))))

C

double code(double x) {
	return x * (1.0 + ((x * x) * (-0.3333333333333333 + (x * (x * 0.13333333333333333)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + ((x * x) * ((-0.3333333333333333d0) + (x * (x * 0.13333333333333333d0)))))
end function

Java

public static double code(double x) {
	return x * (1.0 + ((x * x) * (-0.3333333333333333 + (x * (x * 0.13333333333333333)))));
}

Python

def code(x):
	return x * (1.0 + ((x * x) * (-0.3333333333333333 + (x * (x * 0.13333333333333333)))))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(-0.3333333333333333 + Float64(x * Float64(x * 0.13333333333333333))))))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + ((x * x) * (-0.3333333333333333 + (x * (x * 0.13333333333333333)))));
end

Wolfram

code[x_] := N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(-0.3333333333333333 + N[(x * N[(x * 0.13333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.0%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
4. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{2}{15} \cdot {x}^{2}} - \frac{1}{3}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{2}{15} \cdot {x}^{2}} - \frac{1}{3}\right)\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{2}{15} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{2}{15} \cdot {x}^{2} + \frac{-1}{3}\right)\right)\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{3} + \color{blue}{\frac{2}{15} \cdot {x}^{2}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{\left(\frac{2}{15} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \left(\frac{2}{15} \cdot \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

   11. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \left(\left(\frac{2}{15} \cdot x\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \left(x \cdot \color{blue}{\left(\frac{2}{15} \cdot x\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{2}{15} \cdot x\right)}\right)\right)\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{2}{15}}\right)\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f64 96.8%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{2}{15}}\right)\right)\right)\right)\right)\right) \]

5. Simplified 96.8%

   \[\leadsto \color{blue}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.3333333333333333 + x \cdot \left(x \cdot 0.13333333333333333\right)\right)\right)} \]
6. Add Preprocessing

Alternative 4: 97.1% accurate, 45.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return Float64(x / Float64(1.0 + Float64(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 9.0%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)} \]
4. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)}\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right)} - \frac{1}{3}\right)\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)}\right)\right)\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right)\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) + \frac{-1}{3}\right)\right)\right)\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{3} + \color{blue}{{x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{3}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{2}{15}} + \frac{-17}{315} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right) \]

   12. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{3}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{3}, \left(x \cdot \left(\left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{15}, \color{blue}{\left(\frac{-17}{315} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

   18. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{15}, \left({x}^{2} \cdot \color{blue}{\frac{-17}{315}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

5. Simplified 96.8%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.3333333333333333 + x \cdot \left(x \cdot \left(0.13333333333333333 + x \cdot \left(x \cdot -0.05396825396825397\right)\right)\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(1 + x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)\right) \cdot \color{blue}{x} \]

   2. flip-+ N/A

      \[\leadsto \frac{1 \cdot 1 - \left(x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)\right)}{1 - x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)} \cdot x \]

   3. associate-*l/ N/A

      \[\leadsto \frac{\left(1 \cdot 1 - \left(x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)\right)\right) \cdot x}{\color{blue}{1 - x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)}} \]

   4. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1 - x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)}{\left(1 \cdot 1 - \left(x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)\right)\right) \cdot x}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 - x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)}{\left(1 \cdot 1 - \left(x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)\right)\right) \cdot x}\right)}\right) \]

7. Applied egg-rr 96.5%

   \[\leadsto \color{blue}{\frac{1}{\frac{1 - x \cdot \left(x \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.13333333333333333 + x \cdot \left(x \cdot -0.05396825396825397\right)\right)\right)\right)}{\left(1 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(-0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.13333333333333333 + x \cdot \left(x \cdot -0.05396825396825397\right)\right)\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.13333333333333333 + x \cdot \left(x \cdot -0.05396825396825397\right)\right)\right)\right)\right) \cdot x}}} \]

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. *-lowering-*.f64 N/A

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

   5. unpow2 N/A

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

   6. *-lowering-*.f64 96.3%

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

10. Simplified 96.3%

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

    1. clear-num N/A

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

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

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

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

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

    4. associate-*l* N/A

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

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

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

    6. *-lowering-*.f64 96.6%

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

12. Applied egg-rr 96.6%

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

Alternative 5: 96.9% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x \cdot 0.3333333333333333 + \frac{1}{x}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.0%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)} \]
4. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)}\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right)} - \frac{1}{3}\right)\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)}\right)\right)\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right)\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) + \frac{-1}{3}\right)\right)\right)\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{3} + \color{blue}{{x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{3}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{2}{15}} + \frac{-17}{315} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right) \]

   12. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{3}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{3}, \left(x \cdot \left(\left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{15}, \color{blue}{\left(\frac{-17}{315} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

   18. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{15}, \left({x}^{2} \cdot \color{blue}{\frac{-17}{315}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

5. Simplified 96.8%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.3333333333333333 + x \cdot \left(x \cdot \left(0.13333333333333333 + x \cdot \left(x \cdot -0.05396825396825397\right)\right)\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(1 + x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)\right) \cdot \color{blue}{x} \]

   2. flip-+ N/A

      \[\leadsto \frac{1 \cdot 1 - \left(x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)\right)}{1 - x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)} \cdot x \]

   3. associate-*l/ N/A

      \[\leadsto \frac{\left(1 \cdot 1 - \left(x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)\right)\right) \cdot x}{\color{blue}{1 - x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)}} \]

   4. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1 - x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)}{\left(1 \cdot 1 - \left(x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)\right)\right) \cdot x}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 - x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)}{\left(1 \cdot 1 - \left(x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-1}{3} + x \cdot \left(x \cdot \left(\frac{2}{15} + x \cdot \left(x \cdot \frac{-17}{315}\right)\right)\right)\right)\right)\right)\right) \cdot x}\right)}\right) \]

7. Applied egg-rr 96.5%

   \[\leadsto \color{blue}{\frac{1}{\frac{1 - x \cdot \left(x \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.13333333333333333 + x \cdot \left(x \cdot -0.05396825396825397\right)\right)\right)\right)}{\left(1 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(-0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.13333333333333333 + x \cdot \left(x \cdot -0.05396825396825397\right)\right)\right) \cdot \left(-0.3333333333333333 + \left(x \cdot x\right) \cdot \left(0.13333333333333333 + x \cdot \left(x \cdot -0.05396825396825397\right)\right)\right)\right)\right) \cdot x}}} \]

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. *-lowering-*.f64 N/A

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

   5. unpow2 N/A

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

   6. *-lowering-*.f64 96.3%

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

10. Simplified 96.3%

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

11. Taylor expanded in x around 0

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

    1. lft-mult-inverse N/A

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

    2. distribute-rgt-in N/A

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

    3. unpow2 N/A

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

    4. +-commutative N/A

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

    5. associate-*l* N/A

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

    6. *-rgt-identity N/A

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

    7. times-frac N/A

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

    8. *-inverses N/A

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

    9. /-rgt-identity N/A

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

    10. *-lft-identity N/A

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

    11. +-commutative N/A

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

    12. distribute-rgt-in N/A

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

    13. unpow2 N/A

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

    14. associate-/r* N/A

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

    15. associate-*l/ N/A

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

    16. lft-mult-inverse N/A

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

    17. metadata-eval N/A

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

    18. lft-mult-inverse N/A

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

    19. associate-*l* N/A

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

    20. associate-*r* N/A

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

    21. unpow2 N/A

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

    22. unpow3 N/A

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

13. Simplified 96.3%

    \[\leadsto \frac{1}{\color{blue}{\frac{1}{x} + x \cdot 0.3333333333333333}} \]

14. Final simplification 96.3%

    \[\leadsto \frac{1}{x \cdot 0.3333333333333333 + \frac{1}{x}} \]
15. Add Preprocessing

Alternative 6: 96.7% accurate, 45.4× speedup

Math

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

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

Derivation

1. Initial program 9.0%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

   5. *-lowering-*.f64 96.2%

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

5. Simplified 96.2%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

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

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

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

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

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

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

   9. *-commutative N/A

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

   10. *-lowering-*.f64 96.2%

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

7. Applied egg-rr 96.2%

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

8. Final simplification 96.2%

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

Alternative 7: 96.7% accurate, 45.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.0%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

   5. *-lowering-*.f64 96.2%

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

5. Simplified 96.2%

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

6. Final simplification 96.2%

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

Alternative 8: 96.6% accurate, 409.0× 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 9.0%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Simplified 96.0%

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

Reproduce

herbie shell --seed 2024192 
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))