Hyperbolic sine

Percentage Accurate: 54.9% → 100.0%

Time: 9.7s

Alternatives: 11

Speedup: 22.9×

• 50.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{e^{x} - e^{-x}}{2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- (exp x) (exp (- x))) 2.0))

C

double code(double x) {
	return (exp(x) - exp(-x)) / 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (exp(x) - exp(-x)) / 2.0d0
end function

Java

public static double code(double x) {
	return (Math.exp(x) - Math.exp(-x)) / 2.0;
}

Python

def code(x):
	return (math.exp(x) - math.exp(-x)) / 2.0

Julia

function code(x)
	return Float64(Float64(exp(x) - exp(Float64(-x))) / 2.0)
end

MATLAB

function tmp = code(x)
	tmp = (exp(x) - exp(-x)) / 2.0;
end

Wolfram

code[x_] := N[(N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Initial Program: 54.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x} - e^{-x}}{2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- (exp x) (exp (- x))) 2.0))

C

double code(double x) {
	return (exp(x) - exp(-x)) / 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (exp(x) - exp(-x)) / 2.0d0
end function

Java

public static double code(double x) {
	return (Math.exp(x) - Math.exp(-x)) / 2.0;
}

Python

def code(x):
	return (math.exp(x) - math.exp(-x)) / 2.0

Julia

function code(x)
	return Float64(Float64(exp(x) - exp(Float64(-x))) / 2.0)
end

MATLAB

function tmp = code(x)
	tmp = (exp(x) - exp(-x)) / 2.0;
end

Wolfram

code[x_] := N[(N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Alternative 1: 100.0% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (sinh x))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return sinh(x)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.2%

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

   1. sinh-def N/A

      \[\leadsto \sinh x \]

   2. sinh-lowering-sinh.f64 100.0%

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

4. Applied egg-rr 100.0%

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

Alternative 2: 75.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot 0.0001984126984126984\right)\right)\right)\\ \mathbf{if}\;x \leq 5 \cdot 10^{+44}:\\ \;\;\;\;\frac{x \cdot \left(1 - t\_0 \cdot t\_0\right)}{1 - t\_0}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.16666666666666666 + x \cdot \left(0.0001984126984126984 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (*
          x
          (*
           x
           (+
            0.16666666666666666
            (*
             (* x x)
             (+ 0.008333333333333333 (* (* x x) 0.0001984126984126984))))))))
   (if (<= x 5e+44)
     (/ (* x (- 1.0 (* t_0 t_0))) (- 1.0 t_0))
     (*
      x
      (+
       1.0
       (*
        (* x x)
        (+
         0.16666666666666666
         (* x (* 0.0001984126984126984 (* x (* x x)))))))))))

C

double code(double x) {
	double t_0 = x * (x * (0.16666666666666666 + ((x * x) * (0.008333333333333333 + ((x * x) * 0.0001984126984126984)))));
	double tmp;
	if (x <= 5e+44) {
		tmp = (x * (1.0 - (t_0 * t_0))) / (1.0 - t_0);
	} else {
		tmp = x * (1.0 + ((x * x) * (0.16666666666666666 + (x * (0.0001984126984126984 * (x * (x * x)))))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (x * (0.16666666666666666d0 + ((x * x) * (0.008333333333333333d0 + ((x * x) * 0.0001984126984126984d0)))))
    if (x <= 5d+44) then
        tmp = (x * (1.0d0 - (t_0 * t_0))) / (1.0d0 - t_0)
    else
        tmp = x * (1.0d0 + ((x * x) * (0.16666666666666666d0 + (x * (0.0001984126984126984d0 * (x * (x * x)))))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * (x * (0.16666666666666666 + ((x * x) * (0.008333333333333333 + ((x * x) * 0.0001984126984126984)))));
	double tmp;
	if (x <= 5e+44) {
		tmp = (x * (1.0 - (t_0 * t_0))) / (1.0 - t_0);
	} else {
		tmp = x * (1.0 + ((x * x) * (0.16666666666666666 + (x * (0.0001984126984126984 * (x * (x * x)))))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x * (0.16666666666666666 + ((x * x) * (0.008333333333333333 + ((x * x) * 0.0001984126984126984)))))
	tmp = 0
	if x <= 5e+44:
		tmp = (x * (1.0 - (t_0 * t_0))) / (1.0 - t_0)
	else:
		tmp = x * (1.0 + ((x * x) * (0.16666666666666666 + (x * (0.0001984126984126984 * (x * (x * x)))))))
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x * Float64(0.16666666666666666 + Float64(Float64(x * x) * Float64(0.008333333333333333 + Float64(Float64(x * x) * 0.0001984126984126984))))))
	tmp = 0.0
	if (x <= 5e+44)
		tmp = Float64(Float64(x * Float64(1.0 - Float64(t_0 * t_0))) / Float64(1.0 - t_0));
	else
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(0.16666666666666666 + Float64(x * Float64(0.0001984126984126984 * Float64(x * Float64(x * x))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * (0.16666666666666666 + ((x * x) * (0.008333333333333333 + ((x * x) * 0.0001984126984126984)))));
	tmp = 0.0;
	if (x <= 5e+44)
		tmp = (x * (1.0 - (t_0 * t_0))) / (1.0 - t_0);
	else
		tmp = x * (1.0 + ((x * x) * (0.16666666666666666 + (x * (0.0001984126984126984 * (x * (x * x)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * N[(0.008333333333333333 + N[(N[(x * x), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5e+44], N[(N[(x * N[(1.0 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.16666666666666666 + N[(x * N[(0.0001984126984126984 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.9999999999999996e44

   1. Initial program 41.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{120}} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{120}} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{1}{5040} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \left({x}^{2} \cdot \color{blue}{\frac{1}{5040}}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{5040}}\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 89.5%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 89.5%

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot 0.0001984126984126984\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + \left(x \cdot x\right) \cdot \left(\frac{1}{6} + \left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{1}{5040}\right)\right)\right) \cdot \color{blue}{x} \]

      2. flip-+ N/A

         \[\leadsto \frac{1 \cdot 1 - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{6} + \left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{1}{5040}\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{6} + \left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{1}{5040}\right)\right)\right)}{1 - \left(x \cdot x\right) \cdot \left(\frac{1}{6} + \left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{1}{5040}\right)\right)} \cdot x \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\left(1 \cdot 1 - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{6} + \left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{1}{5040}\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{6} + \left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{1}{5040}\right)\right)\right)\right) \cdot x}{\color{blue}{1 - \left(x \cdot x\right) \cdot \left(\frac{1}{6} + \left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{1}{5040}\right)\right)}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 \cdot 1 - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{6} + \left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{1}{5040}\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{6} + \left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{1}{5040}\right)\right)\right)\right) \cdot x\right), \color{blue}{\left(1 - \left(x \cdot x\right) \cdot \left(\frac{1}{6} + \left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{1}{5040}\right)\right)\right)}\right) \]

   7. Applied egg-rr 70.5%

      \[\leadsto \color{blue}{\frac{\left(1 - \left(x \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot 0.0001984126984126984\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right) \cdot x}{1 - x \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot 0.0001984126984126984\right)\right)\right)}} \]

   if 4.9999999999999996e44 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{120}} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{120}} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{1}{5040} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \left({x}^{2} \cdot \color{blue}{\frac{1}{5040}}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{5040}}\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot 0.0001984126984126984\right)\right)\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{1}{5040} \cdot {x}^{4}\right)}\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{5040} \cdot {x}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right)\right)\right) \]

      2. pow-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{5040} \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right)\right)\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(\frac{1}{5040} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(\frac{1}{5040} \cdot {x}^{2}\right) \cdot \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(\left(\frac{1}{5040} \cdot {x}^{2}\right) \cdot x\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\left(\left(\frac{1}{5040} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{5040} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \left(\frac{1}{5040} \cdot \color{blue}{\left({x}^{2} \cdot x\right)}\right)\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \left(\frac{1}{5040} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)\right)\right)\right)\right) \]

      10. unpow3 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \left(\frac{1}{5040} \cdot {x}^{\color{blue}{3}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{5040}, \color{blue}{\left({x}^{3}\right)}\right)\right)\right)\right)\right)\right) \]

      12. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{5040}, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{5040}, \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{5040}, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{5040}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.16666666666666666 + \color{blue}{x \cdot \left(0.0001984126984126984 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+44}:\\ \;\;\;\;\frac{x \cdot \left(1 - \left(x \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot 0.0001984126984126984\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)}{1 - x \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot 0.0001984126984126984\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.16666666666666666 + x \cdot \left(0.0001984126984126984 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.3% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ x \cdot \left(1 + \frac{\left(x \cdot x\right) \cdot \left(0.004629629629629629 + t\_0 \cdot \left(t\_0 \cdot 5.787037037037037 \cdot 10^{-7}\right)\right)}{0.027777777777777776}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (*
    x
    (+
     1.0
     (/
      (* (* x x) (+ 0.004629629629629629 (* t_0 (* t_0 5.787037037037037e-7))))
      0.027777777777777776)))))

C

double code(double x) {
	double t_0 = x * (x * x);
	return x * (1.0 + (((x * x) * (0.004629629629629629 + (t_0 * (t_0 * 5.787037037037037e-7)))) / 0.027777777777777776));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = x * (x * x)
    code = x * (1.0d0 + (((x * x) * (0.004629629629629629d0 + (t_0 * (t_0 * 5.787037037037037d-7)))) / 0.027777777777777776d0))
end function

Java

public static double code(double x) {
	double t_0 = x * (x * x);
	return x * (1.0 + (((x * x) * (0.004629629629629629 + (t_0 * (t_0 * 5.787037037037037e-7)))) / 0.027777777777777776));
}

Python

def code(x):
	t_0 = x * (x * x)
	return x * (1.0 + (((x * x) * (0.004629629629629629 + (t_0 * (t_0 * 5.787037037037037e-7)))) / 0.027777777777777776))

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	return Float64(x * Float64(1.0 + Float64(Float64(Float64(x * x) * Float64(0.004629629629629629 + Float64(t_0 * Float64(t_0 * 5.787037037037037e-7)))) / 0.027777777777777776)))
end

MATLAB

function tmp = code(x)
	t_0 = x * (x * x);
	tmp = x * (1.0 + (((x * x) * (0.004629629629629629 + (t_0 * (t_0 * 5.787037037037037e-7)))) / 0.027777777777777776));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, N[(x * N[(1.0 + N[(N[(N[(x * x), $MachinePrecision] * N[(0.004629629629629629 + N[(t$95$0 * N[(t$95$0 * 5.787037037037037e-7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 53.2%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)} \]
4. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)}\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{6}} + \frac{1}{120} \cdot {x}^{2}\right)\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left({x}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

   11. *-lowering-*.f64 89.3%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

5. Simplified 89.3%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot 0.008333333333333333\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \color{blue}{\left(\frac{1}{6} + \left(x \cdot x\right) \cdot \frac{1}{120}\right)}\right)\right)\right) \]

   2. flip3-+ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{{\frac{1}{6}}^{3} + {\left(\left(x \cdot x\right) \cdot \frac{1}{120}\right)}^{3}}{\color{blue}{\frac{1}{6} \cdot \frac{1}{6} + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{120}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{120}\right) - \frac{1}{6} \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{120}\right)\right)}}\right)\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\left(x \cdot x\right) \cdot \left({\frac{1}{6}}^{3} + {\left(\left(x \cdot x\right) \cdot \frac{1}{120}\right)}^{3}\right)}{\color{blue}{\frac{1}{6} \cdot \frac{1}{6} + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{120}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{120}\right) - \frac{1}{6} \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{120}\right)\right)}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(x \cdot x\right) \cdot \left({\frac{1}{6}}^{3} + {\left(\left(x \cdot x\right) \cdot \frac{1}{120}\right)}^{3}\right)\right), \color{blue}{\left(\frac{1}{6} \cdot \frac{1}{6} + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{120}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{120}\right) - \frac{1}{6} \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{120}\right)\right)\right)}\right)\right)\right) \]

7. Applied egg-rr 57.8%

   \[\leadsto x \cdot \left(1 + \color{blue}{\frac{\left(x \cdot x\right) \cdot \left(0.004629629629629629 + \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 5.787037037037037 \cdot 10^{-7}\right)\right)}{0.027777777777777776 + \left(\left(x \cdot x\right) \cdot 0.008333333333333333\right) \cdot \left(\left(x \cdot x\right) \cdot 0.008333333333333333 + -0.16666666666666666\right)}}\right) \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \frac{1}{1728000}\right)\right)\right)\right), \color{blue}{\frac{1}{36}}\right)\right)\right) \]
9. Step-by-step derivation

10. Simplified 93.9%

    \[\leadsto x \cdot \left(1 + \frac{\left(x \cdot x\right) \cdot \left(0.004629629629629629 + \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 5.787037037037037 \cdot 10^{-7}\right)\right)}{\color{blue}{0.027777777777777776}}\right) \]
11. Add Preprocessing

Alternative 4: 92.8% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + x \cdot \left(x \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (+
   1.0
   (*
    x
    (*
     x
     (+
      0.16666666666666666
      (*
       x
       (* x (+ 0.008333333333333333 (* (* x x) 0.0001984126984126984))))))))))

C

double code(double x) {
	return x * (1.0 + (x * (x * (0.16666666666666666 + (x * (x * (0.008333333333333333 + ((x * x) * 0.0001984126984126984))))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + (x * (x * (0.16666666666666666d0 + (x * (x * (0.008333333333333333d0 + ((x * x) * 0.0001984126984126984d0))))))))
end function

Java

public static double code(double x) {
	return x * (1.0 + (x * (x * (0.16666666666666666 + (x * (x * (0.008333333333333333 + ((x * x) * 0.0001984126984126984))))))));
}

Python

def code(x):
	return x * (1.0 + (x * (x * (0.16666666666666666 + (x * (x * (0.008333333333333333 + ((x * x) * 0.0001984126984126984))))))))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64(x * Float64(x * Float64(0.16666666666666666 + Float64(x * Float64(x * Float64(0.008333333333333333 + Float64(Float64(x * x) * 0.0001984126984126984)))))))))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + (x * (x * (0.16666666666666666 + (x * (x * (0.008333333333333333 + ((x * x) * 0.0001984126984126984))))))));
end

Wolfram

code[x_] := N[(x * N[(1.0 + N[(x * N[(x * N[(0.16666666666666666 + N[(x * N[(x * N[(0.008333333333333333 + N[(N[(x * x), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.2%

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

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{2}{e^{x} - e^{\mathsf{neg}\left(x\right)}}}} \]

   2. inv-pow N/A

      \[\leadsto {\left(\frac{2}{e^{x} - e^{\mathsf{neg}\left(x\right)}}\right)}^{\color{blue}{-1}} \]

   3. sqr-pow N/A

      \[\leadsto {\left(\frac{2}{e^{x} - e^{\mathsf{neg}\left(x\right)}}\right)}^{\left(\frac{-1}{2}\right)} \cdot \color{blue}{{\left(\frac{2}{e^{x} - e^{\mathsf{neg}\left(x\right)}}\right)}^{\left(\frac{-1}{2}\right)}} \]

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

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

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

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

   9. clear-num N/A

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

   10. sinh-def N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\frac{1}{\sinh x}\right), \left(\frac{1}{\mathsf{neg}\left(2\right)}\right)\right), \left({\left(\frac{2}{e^{x} - e^{\mathsf{neg}\left(x\right)}}\right)}^{\left(\frac{-1}{2}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \sinh x\right), \left(\frac{1}{\mathsf{neg}\left(2\right)}\right)\right), \left({\left(\frac{\color{blue}{2}}{e^{x} - e^{\mathsf{neg}\left(x\right)}}\right)}^{\left(\frac{-1}{2}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sinh.f64}\left(x\right)\right), \left(\frac{1}{\mathsf{neg}\left(2\right)}\right)\right), \left({\left(\frac{2}{e^{x} - e^{\mathsf{neg}\left(x\right)}}\right)}^{\left(\frac{-1}{2}\right)}\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sinh.f64}\left(x\right)\right), \left(\frac{1}{-2}\right)\right), \left({\left(\frac{2}{e^{x} - \color{blue}{e^{\mathsf{neg}\left(x\right)}}}\right)}^{\left(\frac{-1}{2}\right)}\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sinh.f64}\left(x\right)\right), \frac{-1}{2}\right), \left({\left(\frac{2}{\color{blue}{e^{x} - e^{\mathsf{neg}\left(x\right)}}}\right)}^{\left(\frac{-1}{2}\right)}\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sinh.f64}\left(x\right)\right), \frac{-1}{2}\right), \left({\left(\frac{2}{e^{x} - e^{\mathsf{neg}\left(x\right)}}\right)}^{\frac{-1}{2}}\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sinh.f64}\left(x\right)\right), \frac{-1}{2}\right), \left({\left(\frac{2}{e^{x} - e^{\mathsf{neg}\left(x\right)}}\right)}^{\left(\frac{1}{\color{blue}{-2}}\right)}\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sinh.f64}\left(x\right)\right), \frac{-1}{2}\right), \left({\left(\frac{2}{e^{x} - e^{\mathsf{neg}\left(x\right)}}\right)}^{\left(\frac{1}{\mathsf{neg}\left(2\right)}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sinh.f64}\left(x\right)\right), \frac{-1}{2}\right), \mathsf{pow.f64}\left(\left(\frac{2}{e^{x} - e^{\mathsf{neg}\left(x\right)}}\right), \color{blue}{\left(\frac{1}{\mathsf{neg}\left(2\right)}\right)}\right)\right) \]

4. Applied egg-rr 43.9%

   \[\leadsto \color{blue}{{\left(\frac{1}{\sinh x}\right)}^{-0.5} \cdot {\left(\frac{1}{\sinh x}\right)}^{-0.5}} \]

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{120}} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{1}{5040} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \left({x}^{2} \cdot \color{blue}{\frac{1}{5040}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{5040}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   16. *-lowering-*.f64 91.6%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

7. Simplified 91.6%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)} \]
8. Add Preprocessing

Alternative 5: 92.6% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.16666666666666666 + x \cdot \left(0.0001984126984126984 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (+
   1.0
   (*
    (* x x)
    (+ 0.16666666666666666 (* x (* 0.0001984126984126984 (* x (* x x)))))))))

C

double code(double x) {
	return x * (1.0 + ((x * x) * (0.16666666666666666 + (x * (0.0001984126984126984 * (x * (x * x)))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + ((x * x) * (0.16666666666666666d0 + (x * (0.0001984126984126984d0 * (x * (x * x)))))))
end function

Java

public static double code(double x) {
	return x * (1.0 + ((x * x) * (0.16666666666666666 + (x * (0.0001984126984126984 * (x * (x * x)))))));
}

Python

def code(x):
	return x * (1.0 + ((x * x) * (0.16666666666666666 + (x * (0.0001984126984126984 * (x * (x * x)))))))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(0.16666666666666666 + Float64(x * Float64(0.0001984126984126984 * Float64(x * Float64(x * x))))))))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + ((x * x) * (0.16666666666666666 + (x * (0.0001984126984126984 * (x * (x * x)))))));
end

Wolfram

code[x_] := N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.16666666666666666 + N[(x * N[(0.0001984126984126984 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.2%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{120}} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{120}} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{1}{5040} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \left({x}^{2} \cdot \color{blue}{\frac{1}{5040}}\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{5040}}\right)\right)\right)\right)\right)\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

   14. *-lowering-*.f64 91.6%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

5. Simplified 91.6%

   \[\leadsto \color{blue}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot 0.0001984126984126984\right)\right)\right)} \]

6. Taylor expanded in x around inf

   \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{1}{5040} \cdot {x}^{4}\right)}\right)\right)\right)\right) \]
7. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{5040} \cdot {x}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right)\right)\right) \]

   2. pow-sqr N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{5040} \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right)\right)\right)\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(\frac{1}{5040} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}}\right)\right)\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(\frac{1}{5040} \cdot {x}^{2}\right) \cdot \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(\left(\frac{1}{5040} \cdot {x}^{2}\right) \cdot x\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\left(\left(\frac{1}{5040} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{5040} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right)\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \left(\frac{1}{5040} \cdot \color{blue}{\left({x}^{2} \cdot x\right)}\right)\right)\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \left(\frac{1}{5040} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)\right)\right)\right)\right) \]

   10. unpow3 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \left(\frac{1}{5040} \cdot {x}^{\color{blue}{3}}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{5040}, \color{blue}{\left({x}^{3}\right)}\right)\right)\right)\right)\right)\right) \]

   12. cube-mult N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{5040}, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right)\right)\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{5040}, \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{5040}, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{5040}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right)\right)\right) \]

   16. *-lowering-*.f64 91.6%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right)\right)\right)\right) \]

8. Simplified 91.6%

   \[\leadsto x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.16666666666666666 + \color{blue}{x \cdot \left(0.0001984126984126984 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \]
9. Add Preprocessing

Alternative 6: 70.3% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot 0.008333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.2)
   x
   (* x (* (* x x) (+ 0.16666666666666666 (* (* x x) 0.008333333333333333))))))

C

double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = x;
	} else {
		tmp = x * ((x * x) * (0.16666666666666666 + ((x * x) * 0.008333333333333333)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.2d0) then
        tmp = x
    else
        tmp = x * ((x * x) * (0.16666666666666666d0 + ((x * x) * 0.008333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = x;
	} else {
		tmp = x * ((x * x) * (0.16666666666666666 + ((x * x) * 0.008333333333333333)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.2:
		tmp = x
	else:
		tmp = x * ((x * x) * (0.16666666666666666 + ((x * x) * 0.008333333333333333)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.2)
		tmp = x;
	else
		tmp = Float64(x * Float64(Float64(x * x) * Float64(0.16666666666666666 + Float64(Float64(x * x) * 0.008333333333333333))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.2)
		tmp = x;
	else
		tmp = x * ((x * x) * (0.16666666666666666 + ((x * x) * 0.008333333333333333)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.2], x, N[(x * N[(N[(x * x), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 39.5%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 66.5%

      \[\leadsto \color{blue}{x} \]

   if 2.2000000000000002 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{6}} + \frac{1}{120} \cdot {x}^{2}\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left({x}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 82.2%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 82.2%

      \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot 0.008333333333333333\right)\right)\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{4} \cdot \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{x}^{2}}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left({x}^{\left(2 \cdot 2\right)} \cdot \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

      2. pow-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left({x}^{2} \cdot {x}^{2}\right) \cdot \left(\color{blue}{\frac{1}{120}} + \frac{1}{6} \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\left(x \cdot x\right) \cdot {x}^{2}\right) \cdot \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(x \cdot \left(x \cdot {x}^{2}\right)\right) \cdot \left(\color{blue}{\frac{1}{120}} + \frac{1}{6} \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(x \cdot {x}^{3}\right) \cdot \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({x}^{3} \cdot \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{x}^{2}}\right)\right)}\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\frac{1}{120}} + \frac{1}{6} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \left(\left(x \cdot {x}^{2}\right) \cdot \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{x}^{2}}\right)\right)}\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{x}^{2}}\right)\right)}\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\color{blue}{{x}^{2}} \cdot \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{x}^{2}}\right) \cdot \color{blue}{{x}^{2}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{2}\right)}\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{x}^{2}}\right)} \cdot {x}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{x}^{2}}\right)} \cdot {x}^{2}\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \color{blue}{\left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{x}^{2}}\right)}\right)\right)\right) \]

      18. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{{x}^{2}} + \color{blue}{\frac{1}{120}}\right)\right)\right)\right) \]

      19. distribute-rgt-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(\frac{1}{6} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{2} + \color{blue}{\frac{1}{120} \cdot {x}^{2}}\right)\right)\right) \]

   8. Simplified 82.2%

      \[\leadsto x \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot 0.008333333333333333\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 70.3% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.3:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.008333333333333333 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.3) x (* x (* x (* 0.008333333333333333 (* x (* x x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 3.3) {
		tmp = x;
	} else {
		tmp = x * (x * (0.008333333333333333 * (x * (x * x))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.3d0) then
        tmp = x
    else
        tmp = x * (x * (0.008333333333333333d0 * (x * (x * x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.3) {
		tmp = x;
	} else {
		tmp = x * (x * (0.008333333333333333 * (x * (x * x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.3:
		tmp = x
	else:
		tmp = x * (x * (0.008333333333333333 * (x * (x * x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.3)
		tmp = x;
	else
		tmp = Float64(x * Float64(x * Float64(0.008333333333333333 * Float64(x * Float64(x * x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.3)
		tmp = x;
	else
		tmp = x * (x * (0.008333333333333333 * (x * (x * x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.3], x, N[(x * N[(x * N[(0.008333333333333333 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.2999999999999998

   1. Initial program 39.5%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 66.5%

      \[\leadsto \color{blue}{x} \]

   if 3.2999999999999998 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{6}} + \frac{1}{120} \cdot {x}^{2}\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left({x}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 82.2%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 82.2%

      \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot 0.008333333333333333\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{120}\right)}\right)\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(x \cdot \frac{1}{120}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot \frac{1}{120}\right), \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 82.2%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{120}\right), x\right)\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 82.2%

      \[\leadsto x \cdot \left(1 + x \cdot \left(x \cdot \left(0.16666666666666666 + \color{blue}{\left(x \cdot 0.008333333333333333\right) \cdot x}\right)\right)\right) \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{120} \cdot {x}^{5}} \]
   9. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{1}{120} \cdot {x}^{\left(4 + \color{blue}{1}\right)} \]

      2. pow-plus N/A

         \[\leadsto \frac{1}{120} \cdot \left({x}^{4} \cdot \color{blue}{x}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\frac{1}{120} \cdot {x}^{4}\right) \cdot \color{blue}{x} \]

      4. metadata-eval N/A

         \[\leadsto \left(\frac{1}{120} \cdot {x}^{\left(2 \cdot 2\right)}\right) \cdot x \]

      5. pow-sqr N/A

         \[\leadsto \left(\frac{1}{120} \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot x \]

      6. associate-*l* N/A

         \[\leadsto \left(\left(\frac{1}{120} \cdot {x}^{2}\right) \cdot {x}^{2}\right) \cdot x \]

      7. *-commutative N/A

         \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2}\right)\right) \cdot x \]

      8. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2}\right)\right)}\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{120}} \cdot {x}^{2}\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{120} \cdot {x}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{120} \cdot {x}^{2}\right)\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{120} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot \color{blue}{\left({x}^{2} \cdot x\right)}\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)\right) \]

      16. unpow3 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot {x}^{\color{blue}{3}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{120}, \color{blue}{\left({x}^{3}\right)}\right)\right)\right) \]

      18. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{120}, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{120}, \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right) \]

      21. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      22. *-lowering-*.f64 82.2%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right) \]

   10. Simplified 82.2%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(0.008333333333333333 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 89.8% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + x \cdot \left(x \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot 0.008333333333333333\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (+
   1.0
   (* x (* x (+ 0.16666666666666666 (* x (* x 0.008333333333333333))))))))

C

double code(double x) {
	return x * (1.0 + (x * (x * (0.16666666666666666 + (x * (x * 0.008333333333333333))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + (x * (x * (0.16666666666666666d0 + (x * (x * 0.008333333333333333d0))))))
end function

Java

public static double code(double x) {
	return x * (1.0 + (x * (x * (0.16666666666666666 + (x * (x * 0.008333333333333333))))));
}

Python

def code(x):
	return x * (1.0 + (x * (x * (0.16666666666666666 + (x * (x * 0.008333333333333333))))))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64(x * Float64(x * Float64(0.16666666666666666 + Float64(x * Float64(x * 0.008333333333333333)))))))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + (x * (x * (0.16666666666666666 + (x * (x * 0.008333333333333333))))));
end

Wolfram

code[x_] := N[(x * N[(1.0 + N[(x * N[(x * N[(0.16666666666666666 + N[(x * N[(x * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.2%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)} \]
4. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)}\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{6}} + \frac{1}{120} \cdot {x}^{2}\right)\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left({x}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

   11. *-lowering-*.f64 89.3%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

5. Simplified 89.3%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot 0.008333333333333333\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{120}\right)}\right)\right)\right)\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(x \cdot \frac{1}{120}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot \frac{1}{120}\right), \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

   4. *-lowering-*.f64 89.3%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{120}\right), x\right)\right)\right)\right)\right)\right) \]

7. Applied egg-rr 89.3%

   \[\leadsto x \cdot \left(1 + x \cdot \left(x \cdot \left(0.16666666666666666 + \color{blue}{\left(x \cdot 0.008333333333333333\right) \cdot x}\right)\right)\right) \]

8. Final simplification 89.3%

   \[\leadsto x \cdot \left(1 + x \cdot \left(x \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot 0.008333333333333333\right)\right)\right)\right) \]
9. Add Preprocessing

Alternative 9: 66.9% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.45:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.45) x (* 0.16666666666666666 (* x (* x x)))))

C

double code(double x) {
	double tmp;
	if (x <= 2.45) {
		tmp = x;
	} else {
		tmp = 0.16666666666666666 * (x * (x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.45d0) then
        tmp = x
    else
        tmp = 0.16666666666666666d0 * (x * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.45) {
		tmp = x;
	} else {
		tmp = 0.16666666666666666 * (x * (x * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.45:
		tmp = x
	else:
		tmp = 0.16666666666666666 * (x * (x * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.45)
		tmp = x;
	else
		tmp = Float64(0.16666666666666666 * Float64(x * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.45)
		tmp = x;
	else
		tmp = 0.16666666666666666 * (x * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.45], x, N[(0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.4500000000000002

   1. Initial program 39.5%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 66.5%

      \[\leadsto \color{blue}{x} \]

   if 2.4500000000000002 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{6} \cdot {x}^{2}\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + \frac{1}{6} \cdot {x}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {x}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(\frac{1}{6} \cdot x\right) \cdot \color{blue}{x}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(\frac{1}{6} \cdot x\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} \cdot x\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 67.8%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

   5. Simplified 67.8%

      \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot 0.16666666666666666\right)\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{3}} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{3}\right)}\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \left(x \cdot {x}^{\color{blue}{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      6. *-lowering-*.f64 67.8%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   8. Simplified 67.8%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 83.1% accurate, 22.9× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + x \cdot \left(x \cdot 0.16666666666666666\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (+ 1.0 (* x (* x 0.16666666666666666)))))

C

double code(double x) {
	return x * (1.0 + (x * (x * 0.16666666666666666)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + (x * (x * 0.16666666666666666d0)))
end function

Java

public static double code(double x) {
	return x * (1.0 + (x * (x * 0.16666666666666666)));
}

Python

def code(x):
	return x * (1.0 + (x * (x * 0.16666666666666666)))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64(x * Float64(x * 0.16666666666666666))))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + (x * (x * 0.16666666666666666)));
end

Wolfram

code[x_] := N[(x * N[(1.0 + N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.2%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{6} \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + \frac{1}{6} \cdot {x}^{2}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {x}^{2}\right)}\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(\frac{1}{6} \cdot x\right) \cdot \color{blue}{x}\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(\frac{1}{6} \cdot x\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} \cdot x\right)}\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

   8. *-lowering-*.f64 82.6%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

5. Simplified 82.6%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot 0.16666666666666666\right)\right)} \]
6. Add Preprocessing

Alternative 11: 51.7% accurate, 206.0× 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 53.2%

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

3. Taylor expanded in x around 0

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

5. Simplified 52.6%

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

Reproduce

herbie shell --seed 2024155 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))