Hyperbolic tangent

Percentage Accurate: 9.2% → 100.0%

Time: 12.9s

Alternatives: 8

Speedup: 409.0×

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 10.2%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
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: 97.3% accurate, 17.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 10.2%

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

   1. div-sub N/A

      \[\leadsto \frac{e^{x}}{e^{x} + e^{\mathsf{neg}\left(x\right)}} - \color{blue}{\frac{e^{\mathsf{neg}\left(x\right)}}{e^{x} + e^{\mathsf{neg}\left(x\right)}}} \]

   2. sub-neg N/A

      \[\leadsto \frac{e^{x}}{e^{x} + e^{\mathsf{neg}\left(x\right)}} + \color{blue}{\left(\mathsf{neg}\left(\frac{e^{\mathsf{neg}\left(x\right)}}{e^{x} + e^{\mathsf{neg}\left(x\right)}}\right)\right)} \]

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

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{e^{x}}{e^{x} + e^{\mathsf{neg}\left(x\right)}}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{e^{\mathsf{neg}\left(x\right)}}{e^{x} + e^{\mathsf{neg}\left(x\right)}}\right)\right)}\right) \]

3. Simplified 11.3%

   \[\leadsto \color{blue}{\frac{1}{1 + e^{x \cdot -2}} + \frac{1}{-1 - e^{x \cdot 2}}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)} \]
6. 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. *-lowering-*.f64 N/A

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

   4. unpow2 N/A

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

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

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. *-commutative N/A

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

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

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

   14. *-commutative N/A

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

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

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

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

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

   17. *-commutative N/A

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

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

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

   19. unpow2 N/A

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

   20. *-lowering-*.f64 97.6%

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

7. Simplified 97.6%

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-+l+ N/A

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

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

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

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

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

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

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

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

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

9. Applied egg-rr 97.6%

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

    1. *-commutative N/A

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

    2. associate-+r+ N/A

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

    3. *-commutative N/A

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

    4. associate-*l* N/A

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

    5. distribute-lft-in N/A

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

    6. distribute-rgt1-in N/A

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

    7. *-commutative N/A

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

    8. flip-+ N/A

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

    9. clear-num N/A

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

11. Applied egg-rr 58.0%

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

12. Taylor expanded in x around 0

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

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

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

14. Simplified 97.7%

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

Alternative 3: 97.2% accurate, 19.5× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.3333333333333333 + x \cdot \left(x \cdot \left(0.13333333333333333 + \left(x \cdot x\right) \cdot -0.05396825396825397\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (+
   1.0
   (*
    (* x x)
    (+
     -0.3333333333333333
     (* x (* x (+ 0.13333333333333333 (* (* x x) -0.05396825396825397)))))))))

C

double code(double x) {
	return x * (1.0 + ((x * x) * (-0.3333333333333333 + (x * (x * (0.13333333333333333 + ((x * x) * -0.05396825396825397)))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + ((x * x) * ((-0.3333333333333333d0) + (x * (x * (0.13333333333333333d0 + ((x * x) * (-0.05396825396825397d0))))))))
end function

Java

public static double code(double x) {
	return x * (1.0 + ((x * x) * (-0.3333333333333333 + (x * (x * (0.13333333333333333 + ((x * x) * -0.05396825396825397)))))));
}

Python

def code(x):
	return x * (1.0 + ((x * x) * (-0.3333333333333333 + (x * (x * (0.13333333333333333 + ((x * x) * -0.05396825396825397)))))))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(-0.3333333333333333 + Float64(x * Float64(x * Float64(0.13333333333333333 + Float64(Float64(x * x) * -0.05396825396825397))))))))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + ((x * x) * (-0.3333333333333333 + (x * (x * (0.13333333333333333 + ((x * x) * -0.05396825396825397)))))));
end

Wolfram

code[x_] := N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(-0.3333333333333333 + N[(x * N[(x * N[(0.13333333333333333 + N[(N[(x * x), $MachinePrecision] * -0.05396825396825397), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 10.2%

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

   1. div-sub N/A

      \[\leadsto \frac{e^{x}}{e^{x} + e^{\mathsf{neg}\left(x\right)}} - \color{blue}{\frac{e^{\mathsf{neg}\left(x\right)}}{e^{x} + e^{\mathsf{neg}\left(x\right)}}} \]

   2. sub-neg N/A

      \[\leadsto \frac{e^{x}}{e^{x} + e^{\mathsf{neg}\left(x\right)}} + \color{blue}{\left(\mathsf{neg}\left(\frac{e^{\mathsf{neg}\left(x\right)}}{e^{x} + e^{\mathsf{neg}\left(x\right)}}\right)\right)} \]

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

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{e^{x}}{e^{x} + e^{\mathsf{neg}\left(x\right)}}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{e^{\mathsf{neg}\left(x\right)}}{e^{x} + e^{\mathsf{neg}\left(x\right)}}\right)\right)}\right) \]

3. Simplified 11.3%

   \[\leadsto \color{blue}{\frac{1}{1 + e^{x \cdot -2}} + \frac{1}{-1 - e^{x \cdot 2}}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)} \]
6. 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. *-lowering-*.f64 N/A

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

   4. unpow2 N/A

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

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

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. *-commutative N/A

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

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

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

   14. *-commutative N/A

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

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

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

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

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

   17. *-commutative N/A

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

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

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

   19. unpow2 N/A

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

   20. *-lowering-*.f64 97.6%

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

7. Simplified 97.6%

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

Alternative 4: 97.4% accurate, 27.3× speedup

Math

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

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(Float64(x * 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[(N[(x * x), $MachinePrecision] * 0.13333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 10.2%

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

   1. div-sub N/A

      \[\leadsto \frac{e^{x}}{e^{x} + e^{\mathsf{neg}\left(x\right)}} - \color{blue}{\frac{e^{\mathsf{neg}\left(x\right)}}{e^{x} + e^{\mathsf{neg}\left(x\right)}}} \]

   2. sub-neg N/A

      \[\leadsto \frac{e^{x}}{e^{x} + e^{\mathsf{neg}\left(x\right)}} + \color{blue}{\left(\mathsf{neg}\left(\frac{e^{\mathsf{neg}\left(x\right)}}{e^{x} + e^{\mathsf{neg}\left(x\right)}}\right)\right)} \]

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

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{e^{x}}{e^{x} + e^{\mathsf{neg}\left(x\right)}}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{e^{\mathsf{neg}\left(x\right)}}{e^{x} + e^{\mathsf{neg}\left(x\right)}}\right)\right)}\right) \]

3. Simplified 11.3%

   \[\leadsto \color{blue}{\frac{1}{1 + e^{x \cdot -2}} + \frac{1}{-1 - e^{x \cdot 2}}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
6. 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. *-commutative N/A

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

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

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

   12. unpow2 N/A

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

   13. *-lowering-*.f64 97.5%

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

7. Simplified 97.5%

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

Alternative 5: 97.1% accurate, 37.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 10.2%

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

   1. div-sub N/A

      \[\leadsto \frac{e^{x}}{e^{x} + e^{\mathsf{neg}\left(x\right)}} - \color{blue}{\frac{e^{\mathsf{neg}\left(x\right)}}{e^{x} + e^{\mathsf{neg}\left(x\right)}}} \]

   2. sub-neg N/A

      \[\leadsto \frac{e^{x}}{e^{x} + e^{\mathsf{neg}\left(x\right)}} + \color{blue}{\left(\mathsf{neg}\left(\frac{e^{\mathsf{neg}\left(x\right)}}{e^{x} + e^{\mathsf{neg}\left(x\right)}}\right)\right)} \]

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

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{e^{x}}{e^{x} + e^{\mathsf{neg}\left(x\right)}}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{e^{\mathsf{neg}\left(x\right)}}{e^{x} + e^{\mathsf{neg}\left(x\right)}}\right)\right)}\right) \]

3. Simplified 11.3%

   \[\leadsto \color{blue}{\frac{1}{1 + e^{x \cdot -2}} + \frac{1}{-1 - e^{x \cdot 2}}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)} \]
6. 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. *-lowering-*.f64 N/A

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

   4. unpow2 N/A

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

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

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. *-commutative N/A

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

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

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

   14. *-commutative N/A

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

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

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

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

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

   17. *-commutative N/A

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

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

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

   19. unpow2 N/A

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

   20. *-lowering-*.f64 97.6%

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

7. Simplified 97.6%

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-+l+ N/A

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

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

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

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

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

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

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

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

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

9. Applied egg-rr 97.6%

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

    1. *-commutative N/A

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

    2. associate-+r+ N/A

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

    3. *-commutative N/A

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

    4. associate-*l* N/A

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

    5. distribute-lft-in N/A

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

    6. distribute-rgt1-in N/A

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

    7. *-commutative N/A

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

    8. flip-+ N/A

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

    9. clear-num N/A

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

11. Applied egg-rr 58.0%

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

12. Taylor expanded in x around 0

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

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

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

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

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

    3. *-commutative N/A

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

    4. unpow2 N/A

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

    5. associate-*l* N/A

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

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

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

    7. *-lowering-*.f64 97.2%

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

14. Simplified 97.2%

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

Alternative 6: 97.0% accurate, 45.4× 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 10.2%

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

   1. div-sub N/A

      \[\leadsto \frac{e^{x}}{e^{x} + e^{\mathsf{neg}\left(x\right)}} - \color{blue}{\frac{e^{\mathsf{neg}\left(x\right)}}{e^{x} + e^{\mathsf{neg}\left(x\right)}}} \]

   2. sub-neg N/A

      \[\leadsto \frac{e^{x}}{e^{x} + e^{\mathsf{neg}\left(x\right)}} + \color{blue}{\left(\mathsf{neg}\left(\frac{e^{\mathsf{neg}\left(x\right)}}{e^{x} + e^{\mathsf{neg}\left(x\right)}}\right)\right)} \]

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

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{e^{x}}{e^{x} + e^{\mathsf{neg}\left(x\right)}}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{e^{\mathsf{neg}\left(x\right)}}{e^{x} + e^{\mathsf{neg}\left(x\right)}}\right)\right)}\right) \]

3. Simplified 11.3%

   \[\leadsto \color{blue}{\frac{1}{1 + e^{x \cdot -2}} + \frac{1}{-1 - e^{x \cdot 2}}} \]
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. *-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 97.1%

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

7. Simplified 97.1%

   \[\leadsto \color{blue}{x \cdot \left(1 + -0.3333333333333333 \cdot \left(x \cdot x\right)\right)} \]
8. 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. *-lowering-*.f64 N/A

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

   7. *-lowering-*.f64 97.2%

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

9. Applied egg-rr 97.2%

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

10. Final simplification 97.2%

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

Alternative 7: 97.0% 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 10.2%

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

   1. div-sub N/A

      \[\leadsto \frac{e^{x}}{e^{x} + e^{\mathsf{neg}\left(x\right)}} - \color{blue}{\frac{e^{\mathsf{neg}\left(x\right)}}{e^{x} + e^{\mathsf{neg}\left(x\right)}}} \]

   2. sub-neg N/A

      \[\leadsto \frac{e^{x}}{e^{x} + e^{\mathsf{neg}\left(x\right)}} + \color{blue}{\left(\mathsf{neg}\left(\frac{e^{\mathsf{neg}\left(x\right)}}{e^{x} + e^{\mathsf{neg}\left(x\right)}}\right)\right)} \]

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

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{e^{x}}{e^{x} + e^{\mathsf{neg}\left(x\right)}}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{e^{\mathsf{neg}\left(x\right)}}{e^{x} + e^{\mathsf{neg}\left(x\right)}}\right)\right)}\right) \]

3. Simplified 11.3%

   \[\leadsto \color{blue}{\frac{1}{1 + e^{x \cdot -2}} + \frac{1}{-1 - e^{x \cdot 2}}} \]
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. *-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 97.1%

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

7. Simplified 97.1%

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

8. Final simplification 97.1%

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

Alternative 8: 96.7% 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 10.2%

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

   1. div-sub N/A

      \[\leadsto \frac{e^{x}}{e^{x} + e^{\mathsf{neg}\left(x\right)}} - \color{blue}{\frac{e^{\mathsf{neg}\left(x\right)}}{e^{x} + e^{\mathsf{neg}\left(x\right)}}} \]

   2. sub-neg N/A

      \[\leadsto \frac{e^{x}}{e^{x} + e^{\mathsf{neg}\left(x\right)}} + \color{blue}{\left(\mathsf{neg}\left(\frac{e^{\mathsf{neg}\left(x\right)}}{e^{x} + e^{\mathsf{neg}\left(x\right)}}\right)\right)} \]

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

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{e^{x}}{e^{x} + e^{\mathsf{neg}\left(x\right)}}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{e^{\mathsf{neg}\left(x\right)}}{e^{x} + e^{\mathsf{neg}\left(x\right)}}\right)\right)}\right) \]

3. Simplified 11.3%

   \[\leadsto \color{blue}{\frac{1}{1 + e^{x \cdot -2}} + \frac{1}{-1 - e^{x \cdot 2}}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

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

7. Simplified 96.4%

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

Reproduce

herbie shell --seed 2024150 
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))