Hyperbolic sine

Percentage Accurate: 54.6% → 100.0%

Time: 10.1s

Alternatives: 15

Speedup: 22.9×

• 50.2% 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: 54.6% 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 62.9%

   \[\frac{e^{x} - e^{-x}}{2} \]
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: 75.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.008333333333333333 + x \cdot \left(x \cdot 0.0001984126984126984\right)\\ t_1 := \left(x \cdot x\right) \cdot 0.0001984126984126984\\ t_2 := \left(x \cdot x\right) \cdot t\_0\\ \mathbf{if}\;x \leq 10^{+44}:\\ \;\;\;\;x \cdot \left(\left(\frac{\left(x \cdot \left(\left(0.16666666666666666 + t\_2\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(0.027777777777777776 - x \cdot \left(t\_2 \cdot \left(x \cdot t\_0\right)\right)\right)}{0.16666666666666666 - t\_2} + -1\right) \cdot \frac{1}{\left(x \cdot x\right) \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot \left(0.008333333333333333 + t\_1\right)\right)\right) + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot t\_1\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 0.008333333333333333 (* x (* x 0.0001984126984126984))))
        (t_1 (* (* x x) 0.0001984126984126984))
        (t_2 (* (* x x) t_0)))
   (if (<= x 1e+44)
     (*
      x
      (*
       (+
        (/
         (*
          (* x (* (+ 0.16666666666666666 t_2) (* x (* x x))))
          (- 0.027777777777777776 (* x (* t_2 (* x t_0)))))
         (- 0.16666666666666666 t_2))
        -1.0)
       (/
        1.0
        (+
         (*
          (* x x)
          (+ 0.16666666666666666 (* x (* x (+ 0.008333333333333333 t_1)))))
         -1.0))))
     (* x (* x (* x (* x (* x t_1))))))))

C

double code(double x) {
	double t_0 = 0.008333333333333333 + (x * (x * 0.0001984126984126984));
	double t_1 = (x * x) * 0.0001984126984126984;
	double t_2 = (x * x) * t_0;
	double tmp;
	if (x <= 1e+44) {
		tmp = x * (((((x * ((0.16666666666666666 + t_2) * (x * (x * x)))) * (0.027777777777777776 - (x * (t_2 * (x * t_0))))) / (0.16666666666666666 - t_2)) + -1.0) * (1.0 / (((x * x) * (0.16666666666666666 + (x * (x * (0.008333333333333333 + t_1))))) + -1.0)));
	} else {
		tmp = x * (x * (x * (x * (x * t_1))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 0.008333333333333333d0 + (x * (x * 0.0001984126984126984d0))
    t_1 = (x * x) * 0.0001984126984126984d0
    t_2 = (x * x) * t_0
    if (x <= 1d+44) then
        tmp = x * (((((x * ((0.16666666666666666d0 + t_2) * (x * (x * x)))) * (0.027777777777777776d0 - (x * (t_2 * (x * t_0))))) / (0.16666666666666666d0 - t_2)) + (-1.0d0)) * (1.0d0 / (((x * x) * (0.16666666666666666d0 + (x * (x * (0.008333333333333333d0 + t_1))))) + (-1.0d0))))
    else
        tmp = x * (x * (x * (x * (x * t_1))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 0.008333333333333333 + (x * (x * 0.0001984126984126984));
	double t_1 = (x * x) * 0.0001984126984126984;
	double t_2 = (x * x) * t_0;
	double tmp;
	if (x <= 1e+44) {
		tmp = x * (((((x * ((0.16666666666666666 + t_2) * (x * (x * x)))) * (0.027777777777777776 - (x * (t_2 * (x * t_0))))) / (0.16666666666666666 - t_2)) + -1.0) * (1.0 / (((x * x) * (0.16666666666666666 + (x * (x * (0.008333333333333333 + t_1))))) + -1.0)));
	} else {
		tmp = x * (x * (x * (x * (x * t_1))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.008333333333333333 + (x * (x * 0.0001984126984126984))
	t_1 = (x * x) * 0.0001984126984126984
	t_2 = (x * x) * t_0
	tmp = 0
	if x <= 1e+44:
		tmp = x * (((((x * ((0.16666666666666666 + t_2) * (x * (x * x)))) * (0.027777777777777776 - (x * (t_2 * (x * t_0))))) / (0.16666666666666666 - t_2)) + -1.0) * (1.0 / (((x * x) * (0.16666666666666666 + (x * (x * (0.008333333333333333 + t_1))))) + -1.0)))
	else:
		tmp = x * (x * (x * (x * (x * t_1))))
	return tmp

Julia

function code(x)
	t_0 = Float64(0.008333333333333333 + Float64(x * Float64(x * 0.0001984126984126984)))
	t_1 = Float64(Float64(x * x) * 0.0001984126984126984)
	t_2 = Float64(Float64(x * x) * t_0)
	tmp = 0.0
	if (x <= 1e+44)
		tmp = Float64(x * Float64(Float64(Float64(Float64(Float64(x * Float64(Float64(0.16666666666666666 + t_2) * Float64(x * Float64(x * x)))) * Float64(0.027777777777777776 - Float64(x * Float64(t_2 * Float64(x * t_0))))) / Float64(0.16666666666666666 - t_2)) + -1.0) * Float64(1.0 / Float64(Float64(Float64(x * x) * Float64(0.16666666666666666 + Float64(x * Float64(x * Float64(0.008333333333333333 + t_1))))) + -1.0))));
	else
		tmp = Float64(x * Float64(x * Float64(x * Float64(x * Float64(x * t_1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.008333333333333333 + (x * (x * 0.0001984126984126984));
	t_1 = (x * x) * 0.0001984126984126984;
	t_2 = (x * x) * t_0;
	tmp = 0.0;
	if (x <= 1e+44)
		tmp = x * (((((x * ((0.16666666666666666 + t_2) * (x * (x * x)))) * (0.027777777777777776 - (x * (t_2 * (x * t_0))))) / (0.16666666666666666 - t_2)) + -1.0) * (1.0 / (((x * x) * (0.16666666666666666 + (x * (x * (0.008333333333333333 + t_1))))) + -1.0)));
	else
		tmp = x * (x * (x * (x * (x * t_1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.008333333333333333 + N[(x * N[(x * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[x, 1e+44], N[(x * N[(N[(N[(N[(N[(x * N[(N[(0.16666666666666666 + t$95$2), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.027777777777777776 - N[(x * N[(t$95$2 * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.16666666666666666 - t$95$2), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] * N[(1.0 / N[(N[(N[(x * x), $MachinePrecision] * N[(0.16666666666666666 + N[(x * N[(x * N[(0.008333333333333333 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(x * N[(x * N[(x * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.0000000000000001e44

   1. Initial program 52.3%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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. 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. *-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. unpow2 N/A

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

      13. associate-*l* N/A

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

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

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

      15. *-lowering-*.f64 89.8%

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

   7. Simplified 89.8%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

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

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

   9. Applied egg-rr 58.4%

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

       1. associate-*r* N/A

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

       2. flip-+ N/A

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

       3. associate-*r/ N/A

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

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

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

   11. Applied egg-rr 59.3%

       \[\leadsto x \cdot \left(\left(\color{blue}{\frac{\left(x \cdot \left(\left(0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot 0.0001984126984126984\right)\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(0.027777777777777776 - x \cdot \left(\left(x \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot 0.0001984126984126984\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot 0.0001984126984126984\right)\right)\right)\right)\right)}{0.16666666666666666 - \left(x \cdot x\right) \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot 0.0001984126984126984\right)\right)}} - 1\right) \cdot \frac{1}{\left(x \cdot x\right) \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot 0.0001984126984126984\right)\right)\right) - 1}\right) \]

   if 1.0000000000000001e44 < x

   1. Initial program 100.0%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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. 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. *-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. unpow2 N/A

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

      13. associate-*l* N/A

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

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

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

      15. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left({x}^{\left(2 \cdot 3\right)} \cdot \frac{1}{5040}\right)\right) \]

      3. pow-sqr N/A

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

      4. cube-prod N/A

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

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left({\left({x}^{2}\right)}^{3} \cdot \frac{1}{5040}\right)\right) \]

      6. cube-unmult N/A

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

      7. pow-sqr N/A

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

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left({x}^{2} \cdot {x}^{4}\right) \cdot \frac{1}{5040}\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \color{blue}{\left({x}^{4} \cdot \frac{1}{5040}\right)}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{5040} \cdot \color{blue}{{x}^{4}}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{5040}} \cdot {x}^{4}\right)\right)\right) \]

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

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

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

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

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

      19. unpow2 N/A

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

      20. associate-*l* N/A

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

      21. *-commutative N/A

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

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

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

      23. *-commutative N/A

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

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

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

      25. *-commutative N/A

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

   10. Simplified 100.0%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+44}:\\ \;\;\;\;x \cdot \left(\left(\frac{\left(x \cdot \left(\left(0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot 0.0001984126984126984\right)\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(0.027777777777777776 - x \cdot \left(\left(\left(x \cdot x\right) \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot 0.0001984126984126984\right)\right)\right) \cdot \left(x \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot 0.0001984126984126984\right)\right)\right)\right)\right)}{0.16666666666666666 - \left(x \cdot x\right) \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot 0.0001984126984126984\right)\right)} + -1\right) \cdot \frac{1}{\left(x \cdot x\right) \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot 0.0001984126984126984\right)\right)\right) + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := 0.008333333333333333 + x \cdot \left(x \cdot 0.0001984126984126984\right)\\ t_2 := \left(x \cdot x\right) \cdot t\_1\\ \mathbf{if}\;x \leq 10^{+62}:\\ \;\;\;\;x + \frac{t\_0 \cdot \left(0.027777777777777776 - x \cdot \left(t\_2 \cdot \left(x \cdot t\_1\right)\right)\right)}{0.16666666666666666 - t\_2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.008333333333333333 \cdot t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x)))
        (t_1 (+ 0.008333333333333333 (* x (* x 0.0001984126984126984))))
        (t_2 (* (* x x) t_1)))
   (if (<= x 1e+62)
     (+
      x
      (/
       (* t_0 (- 0.027777777777777776 (* x (* t_2 (* x t_1)))))
       (- 0.16666666666666666 t_2)))
     (* x (* x (* 0.008333333333333333 t_0))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = 0.008333333333333333 + (x * (x * 0.0001984126984126984));
	double t_2 = (x * x) * t_1;
	double tmp;
	if (x <= 1e+62) {
		tmp = x + ((t_0 * (0.027777777777777776 - (x * (t_2 * (x * t_1))))) / (0.16666666666666666 - t_2));
	} else {
		tmp = x * (x * (0.008333333333333333 * t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x * (x * x)
    t_1 = 0.008333333333333333d0 + (x * (x * 0.0001984126984126984d0))
    t_2 = (x * x) * t_1
    if (x <= 1d+62) then
        tmp = x + ((t_0 * (0.027777777777777776d0 - (x * (t_2 * (x * t_1))))) / (0.16666666666666666d0 - t_2))
    else
        tmp = x * (x * (0.008333333333333333d0 * t_0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = 0.008333333333333333 + (x * (x * 0.0001984126984126984));
	double t_2 = (x * x) * t_1;
	double tmp;
	if (x <= 1e+62) {
		tmp = x + ((t_0 * (0.027777777777777776 - (x * (t_2 * (x * t_1))))) / (0.16666666666666666 - t_2));
	} else {
		tmp = x * (x * (0.008333333333333333 * t_0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x * x)
	t_1 = 0.008333333333333333 + (x * (x * 0.0001984126984126984))
	t_2 = (x * x) * t_1
	tmp = 0
	if x <= 1e+62:
		tmp = x + ((t_0 * (0.027777777777777776 - (x * (t_2 * (x * t_1))))) / (0.16666666666666666 - t_2))
	else:
		tmp = x * (x * (0.008333333333333333 * t_0))
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	t_1 = Float64(0.008333333333333333 + Float64(x * Float64(x * 0.0001984126984126984)))
	t_2 = Float64(Float64(x * x) * t_1)
	tmp = 0.0
	if (x <= 1e+62)
		tmp = Float64(x + Float64(Float64(t_0 * Float64(0.027777777777777776 - Float64(x * Float64(t_2 * Float64(x * t_1))))) / Float64(0.16666666666666666 - t_2)));
	else
		tmp = Float64(x * Float64(x * Float64(0.008333333333333333 * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * x);
	t_1 = 0.008333333333333333 + (x * (x * 0.0001984126984126984));
	t_2 = (x * x) * t_1;
	tmp = 0.0;
	if (x <= 1e+62)
		tmp = x + ((t_0 * (0.027777777777777776 - (x * (t_2 * (x * t_1))))) / (0.16666666666666666 - t_2));
	else
		tmp = x * (x * (0.008333333333333333 * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.008333333333333333 + N[(x * N[(x * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * x), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[x, 1e+62], N[(x + N[(N[(t$95$0 * N[(0.027777777777777776 - N[(x * N[(t$95$2 * N[(x * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.16666666666666666 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(0.008333333333333333 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000004e62

   1. Initial program 53.0%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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. 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. *-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. unpow2 N/A

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

      13. associate-*l* N/A

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

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

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

      15. *-lowering-*.f64 90.0%

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

   7. Simplified 90.0%

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

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

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

   9. Applied egg-rr 90.0%

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

       1. flip-+ N/A

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

       2. associate-*l/ N/A

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

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

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

   11. Applied egg-rr 62.5%

       \[\leadsto \color{blue}{\frac{\left(0.027777777777777776 - x \cdot \left(\left(x \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot 0.0001984126984126984\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot 0.0001984126984126984\right)\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)}{0.16666666666666666 - \left(x \cdot x\right) \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot 0.0001984126984126984\right)\right)}} + x \]

   if 1.00000000000000004e62 < 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 100.0%

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

   5. Simplified 100.0%

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

   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 100.0%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{120} \cdot {x}^{5}} \]
   10. 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. *-commutative N/A

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

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

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

       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 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+62}:\\ \;\;\;\;x + \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.027777777777777776 - x \cdot \left(\left(\left(x \cdot x\right) \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot 0.0001984126984126984\right)\right)\right) \cdot \left(x \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot 0.0001984126984126984\right)\right)\right)\right)\right)}{0.16666666666666666 - \left(x \cdot x\right) \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot 0.0001984126984126984\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.008333333333333333 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.6% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq 5.6:\\ \;\;\;\;x + t\_0 \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot 0.008333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(0.008333333333333333 + \left(x \cdot x\right) \cdot 0.0001984126984126984\right) \cdot \left(x \cdot t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= x 5.6)
     (+ x (* t_0 (+ 0.16666666666666666 (* (* x x) 0.008333333333333333))))
     (*
      x
      (*
       (+ 0.008333333333333333 (* (* x x) 0.0001984126984126984))
       (* x t_0))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= 5.6) {
		tmp = x + (t_0 * (0.16666666666666666 + ((x * x) * 0.008333333333333333)));
	} else {
		tmp = x * ((0.008333333333333333 + ((x * x) * 0.0001984126984126984)) * (x * t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (x * x)
    if (x <= 5.6d0) then
        tmp = x + (t_0 * (0.16666666666666666d0 + ((x * x) * 0.008333333333333333d0)))
    else
        tmp = x * ((0.008333333333333333d0 + ((x * x) * 0.0001984126984126984d0)) * (x * t_0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= 5.6) {
		tmp = x + (t_0 * (0.16666666666666666 + ((x * x) * 0.008333333333333333)));
	} else {
		tmp = x * ((0.008333333333333333 + ((x * x) * 0.0001984126984126984)) * (x * t_0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x * x)
	tmp = 0
	if x <= 5.6:
		tmp = x + (t_0 * (0.16666666666666666 + ((x * x) * 0.008333333333333333)))
	else:
		tmp = x * ((0.008333333333333333 + ((x * x) * 0.0001984126984126984)) * (x * t_0))
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (x <= 5.6)
		tmp = Float64(x + Float64(t_0 * Float64(0.16666666666666666 + Float64(Float64(x * x) * 0.008333333333333333))));
	else
		tmp = Float64(x * Float64(Float64(0.008333333333333333 + Float64(Float64(x * x) * 0.0001984126984126984)) * Float64(x * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * x);
	tmp = 0.0;
	if (x <= 5.6)
		tmp = x + (t_0 * (0.16666666666666666 + ((x * x) * 0.008333333333333333)));
	else
		tmp = x * ((0.008333333333333333 + ((x * x) * 0.0001984126984126984)) * (x * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5.6], N[(x + N[(t$95$0 * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(0.008333333333333333 + N[(N[(x * x), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision] * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999996

   1. Initial program 48.9%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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 92.0%

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

   5. Simplified 92.0%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. cube-unmult N/A

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

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

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

      9. cube-unmult N/A

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

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

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

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

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

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

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

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

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

      14. *-lowering-*.f64 92.0%

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

   7. Applied egg-rr 92.0%

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

   if 5.5999999999999996 < x

   1. Initial program 100.0%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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. 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. *-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. unpow2 N/A

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

      13. associate-*l* N/A

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

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

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

      15. *-lowering-*.f64 82.5%

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

   7. Simplified 82.5%

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

   8. Taylor expanded in x around inf

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

   10. Simplified 82.5%

       \[\leadsto x \cdot \color{blue}{\left(\left(0.008333333333333333 + \left(x \cdot x\right) \cdot 0.0001984126984126984\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.6:\\ \;\;\;\;x + \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot 0.008333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(0.008333333333333333 + \left(x \cdot x\right) \cdot 0.0001984126984126984\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.2% accurate, 9.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	return x + (x * (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 + (x * (x * (x * (0.16666666666666666d0 + ((x * x) * (0.008333333333333333d0 + (x * (x * 0.0001984126984126984d0))))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.9%

   \[\frac{e^{x} - e^{-x}}{2} \]
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. 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. *-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. unpow2 N/A

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

   13. associate-*l* N/A

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

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

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

   15. *-lowering-*.f64 92.1%

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

7. Simplified 92.1%

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

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

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

9. Applied egg-rr 92.1%

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

    1. associate-*r* N/A

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

    2. associate-*r* N/A

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

    3. *-commutative N/A

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

    4. distribute-rgt-in N/A

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

    5. associate-*r* N/A

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

    6. distribute-rgt-in N/A

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

    7. remove-double-div N/A

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

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

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

11. Applied egg-rr 92.1%

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

12. Final simplification 92.1%

    \[\leadsto x + x \cdot \left(x \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot 0.0001984126984126984\right)\right)\right)\right)\right) \]
13. Add Preprocessing

Alternative 6: 93.2% accurate, 9.8× speedup

Math

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

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(Float64(x * x) * Float64(0.16666666666666666 + Float64(Float64(x * x) * Float64(0.008333333333333333 + Float64(x * Float64(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[(N[(x * x), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * N[(0.008333333333333333 + N[(x * N[(x * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.9%

   \[\frac{e^{x} - e^{-x}}{2} \]
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. 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. *-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. unpow2 N/A

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

   13. associate-*l* N/A

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

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

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

   15. *-lowering-*.f64 92.1%

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

7. Simplified 92.1%

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

Alternative 7: 91.6% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.6:\\ \;\;\;\;x + \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot 0.008333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 7.6)
   (+
    x
    (* (* x (* x x)) (+ 0.16666666666666666 (* (* x x) 0.008333333333333333))))
   (* x (* x (* x (* x (* x (* (* x x) 0.0001984126984126984))))))))

C

double code(double x) {
	double tmp;
	if (x <= 7.6) {
		tmp = x + ((x * (x * x)) * (0.16666666666666666 + ((x * x) * 0.008333333333333333)));
	} else {
		tmp = x * (x * (x * (x * (x * ((x * x) * 0.0001984126984126984)))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 7.6d0) then
        tmp = x + ((x * (x * x)) * (0.16666666666666666d0 + ((x * x) * 0.008333333333333333d0)))
    else
        tmp = x * (x * (x * (x * (x * ((x * x) * 0.0001984126984126984d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 7.6) {
		tmp = x + ((x * (x * x)) * (0.16666666666666666 + ((x * x) * 0.008333333333333333)));
	} else {
		tmp = x * (x * (x * (x * (x * ((x * x) * 0.0001984126984126984)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 7.6:
		tmp = x + ((x * (x * x)) * (0.16666666666666666 + ((x * x) * 0.008333333333333333)))
	else:
		tmp = x * (x * (x * (x * (x * ((x * x) * 0.0001984126984126984)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 7.6)
		tmp = Float64(x + Float64(Float64(x * Float64(x * x)) * Float64(0.16666666666666666 + Float64(Float64(x * x) * 0.008333333333333333))));
	else
		tmp = Float64(x * Float64(x * Float64(x * Float64(x * Float64(x * Float64(Float64(x * x) * 0.0001984126984126984))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 7.6)
		tmp = x + ((x * (x * x)) * (0.16666666666666666 + ((x * x) * 0.008333333333333333)));
	else
		tmp = x * (x * (x * (x * (x * ((x * x) * 0.0001984126984126984)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 7.6], N[(x + N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(x * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.5999999999999996

   1. Initial program 48.9%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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 92.0%

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

   5. Simplified 92.0%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. cube-unmult N/A

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

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

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

      9. cube-unmult N/A

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

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

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

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

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

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

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

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

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

      14. *-lowering-*.f64 92.0%

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

   7. Applied egg-rr 92.0%

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

   if 7.5999999999999996 < x

   1. Initial program 100.0%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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. 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. *-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. unpow2 N/A

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

      13. associate-*l* N/A

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

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

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

      15. *-lowering-*.f64 82.5%

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

   7. Simplified 82.5%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left({x}^{\left(2 \cdot 3\right)} \cdot \frac{1}{5040}\right)\right) \]

      3. pow-sqr N/A

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

      4. cube-prod N/A

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

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left({\left({x}^{2}\right)}^{3} \cdot \frac{1}{5040}\right)\right) \]

      6. cube-unmult N/A

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

      7. pow-sqr N/A

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

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left({x}^{2} \cdot {x}^{4}\right) \cdot \frac{1}{5040}\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \color{blue}{\left({x}^{4} \cdot \frac{1}{5040}\right)}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{5040} \cdot \color{blue}{{x}^{4}}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{5040}} \cdot {x}^{4}\right)\right)\right) \]

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

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

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

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

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

      19. unpow2 N/A

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

      20. associate-*l* N/A

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

      21. *-commutative N/A

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

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

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

      23. *-commutative N/A

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

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

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

      25. *-commutative N/A

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

   10. Simplified 82.5%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.6:\\ \;\;\;\;x + \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot 0.008333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.6% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.6:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot 0.008333333333333333\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 7.6)
   (*
    x
    (+
     1.0
     (* x (* x (+ 0.16666666666666666 (* x (* x 0.008333333333333333)))))))
   (* x (* x (* x (* x (* x (* (* x x) 0.0001984126984126984))))))))

C

double code(double x) {
	double tmp;
	if (x <= 7.6) {
		tmp = x * (1.0 + (x * (x * (0.16666666666666666 + (x * (x * 0.008333333333333333))))));
	} else {
		tmp = x * (x * (x * (x * (x * ((x * x) * 0.0001984126984126984)))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 7.6d0) then
        tmp = x * (1.0d0 + (x * (x * (0.16666666666666666d0 + (x * (x * 0.008333333333333333d0))))))
    else
        tmp = x * (x * (x * (x * (x * ((x * x) * 0.0001984126984126984d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 7.6) {
		tmp = x * (1.0 + (x * (x * (0.16666666666666666 + (x * (x * 0.008333333333333333))))));
	} else {
		tmp = x * (x * (x * (x * (x * ((x * x) * 0.0001984126984126984)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 7.6:
		tmp = x * (1.0 + (x * (x * (0.16666666666666666 + (x * (x * 0.008333333333333333))))))
	else:
		tmp = x * (x * (x * (x * (x * ((x * x) * 0.0001984126984126984)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 7.6)
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(x * Float64(0.16666666666666666 + Float64(x * Float64(x * 0.008333333333333333)))))));
	else
		tmp = Float64(x * Float64(x * Float64(x * Float64(x * Float64(x * Float64(Float64(x * x) * 0.0001984126984126984))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 7.6)
		tmp = x * (1.0 + (x * (x * (0.16666666666666666 + (x * (x * 0.008333333333333333))))));
	else
		tmp = x * (x * (x * (x * (x * ((x * x) * 0.0001984126984126984)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 7.6], 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], N[(x * N[(x * N[(x * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.5999999999999996

   1. Initial program 48.9%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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 92.0%

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

   5. Simplified 92.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot 0.008333333333333333\right)\right)\right)} \]
   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 92.0%

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

   7. Applied egg-rr 92.0%

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

   if 7.5999999999999996 < x

   1. Initial program 100.0%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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. 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. *-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. unpow2 N/A

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

      13. associate-*l* N/A

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

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

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

      15. *-lowering-*.f64 82.5%

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

   7. Simplified 82.5%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left({x}^{\left(2 \cdot 3\right)} \cdot \frac{1}{5040}\right)\right) \]

      3. pow-sqr N/A

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

      4. cube-prod N/A

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

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left({\left({x}^{2}\right)}^{3} \cdot \frac{1}{5040}\right)\right) \]

      6. cube-unmult N/A

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

      7. pow-sqr N/A

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

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left({x}^{2} \cdot {x}^{4}\right) \cdot \frac{1}{5040}\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \color{blue}{\left({x}^{4} \cdot \frac{1}{5040}\right)}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{5040} \cdot \color{blue}{{x}^{4}}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{5040}} \cdot {x}^{4}\right)\right)\right) \]

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

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

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

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

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

      19. unpow2 N/A

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

      20. associate-*l* N/A

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

      21. *-commutative N/A

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

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

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

      23. *-commutative N/A

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

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

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

      25. *-commutative N/A

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

   10. Simplified 82.5%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.6:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot 0.008333333333333333\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 88.5% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.6:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.6)
   (* x (+ 1.0 (* x (* x 0.16666666666666666))))
   (* x (* x (* x (* x (* x (* (* x x) 0.0001984126984126984))))))))

C

double code(double x) {
	double tmp;
	if (x <= 5.6) {
		tmp = x * (1.0 + (x * (x * 0.16666666666666666)));
	} else {
		tmp = x * (x * (x * (x * (x * ((x * x) * 0.0001984126984126984)))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5.6d0) then
        tmp = x * (1.0d0 + (x * (x * 0.16666666666666666d0)))
    else
        tmp = x * (x * (x * (x * (x * ((x * x) * 0.0001984126984126984d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.6) {
		tmp = x * (1.0 + (x * (x * 0.16666666666666666)));
	} else {
		tmp = x * (x * (x * (x * (x * ((x * x) * 0.0001984126984126984)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.6:
		tmp = x * (1.0 + (x * (x * 0.16666666666666666)))
	else:
		tmp = x * (x * (x * (x * (x * ((x * x) * 0.0001984126984126984)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.6)
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(x * 0.16666666666666666))));
	else
		tmp = Float64(x * Float64(x * Float64(x * Float64(x * Float64(x * Float64(Float64(x * x) * 0.0001984126984126984))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.6)
		tmp = x * (1.0 + (x * (x * 0.16666666666666666)));
	else
		tmp = x * (x * (x * (x * (x * ((x * x) * 0.0001984126984126984)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.6], N[(x * N[(1.0 + N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(x * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999996

   1. Initial program 48.9%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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 84.0%

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

   5. Simplified 84.0%

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

   if 5.5999999999999996 < x

   1. Initial program 100.0%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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. 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. *-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. unpow2 N/A

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

      13. associate-*l* N/A

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

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

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

      15. *-lowering-*.f64 82.5%

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

   7. Simplified 82.5%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left({x}^{\left(2 \cdot 3\right)} \cdot \frac{1}{5040}\right)\right) \]

      3. pow-sqr N/A

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

      4. cube-prod N/A

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

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left({\left({x}^{2}\right)}^{3} \cdot \frac{1}{5040}\right)\right) \]

      6. cube-unmult N/A

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

      7. pow-sqr N/A

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

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left({x}^{2} \cdot {x}^{4}\right) \cdot \frac{1}{5040}\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \color{blue}{\left({x}^{4} \cdot \frac{1}{5040}\right)}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{5040} \cdot \color{blue}{{x}^{4}}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{5040}} \cdot {x}^{4}\right)\right)\right) \]

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

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

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

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

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

      19. unpow2 N/A

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

      20. associate-*l* N/A

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

      21. *-commutative N/A

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

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

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

      23. *-commutative N/A

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

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

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

      25. *-commutative N/A

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

   10. Simplified 82.5%

       \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 87.2% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.3:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot 0.008333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.3)
   (* x (+ 1.0 (* x (* x 0.16666666666666666))))
   (* x (* (* x x) (+ 0.16666666666666666 (* (* x x) 0.008333333333333333))))))

C

double code(double x) {
	double tmp;
	if (x <= 3.3) {
		tmp = x * (1.0 + (x * (x * 0.16666666666666666)));
	} 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 <= 3.3d0) then
        tmp = x * (1.0d0 + (x * (x * 0.16666666666666666d0)))
    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 <= 3.3) {
		tmp = x * (1.0 + (x * (x * 0.16666666666666666)));
	} else {
		tmp = x * ((x * x) * (0.16666666666666666 + ((x * x) * 0.008333333333333333)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.3:
		tmp = x * (1.0 + (x * (x * 0.16666666666666666)))
	else:
		tmp = x * ((x * x) * (0.16666666666666666 + ((x * x) * 0.008333333333333333)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.3)
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(x * 0.16666666666666666))));
	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 <= 3.3)
		tmp = x * (1.0 + (x * (x * 0.16666666666666666)));
	else
		tmp = x * ((x * x) * (0.16666666666666666 + ((x * x) * 0.008333333333333333)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.3], N[(x * N[(1.0 + N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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 < 3.2999999999999998

   1. Initial program 48.9%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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 84.0%

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

   5. Simplified 84.0%

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

   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 78.4%

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

   5. Simplified 78.4%

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

   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 78.4%

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

Alternative 11: 92.8% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ x + t\_0 \cdot \left(x \cdot \left(0.0001984126984126984 \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (+ x (* t_0 (* x (* 0.0001984126984126984 t_0))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	return x + (t_0 * (x * (0.0001984126984126984 * t_0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = x * (x * x)
    code = x + (t_0 * (x * (0.0001984126984126984d0 * t_0)))
end function

Java

public static double code(double x) {
	double t_0 = x * (x * x);
	return x + (t_0 * (x * (0.0001984126984126984 * t_0)));
}

Python

def code(x):
	t_0 = x * (x * x)
	return x + (t_0 * (x * (0.0001984126984126984 * t_0)))

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	return Float64(x + Float64(t_0 * Float64(x * Float64(0.0001984126984126984 * t_0))))
end

MATLAB

function tmp = code(x)
	t_0 = x * (x * x);
	tmp = x + (t_0 * (x * (0.0001984126984126984 * t_0)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, N[(x + N[(t$95$0 * N[(x * N[(0.0001984126984126984 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.9%

   \[\frac{e^{x} - e^{-x}}{2} \]
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. 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. *-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. unpow2 N/A

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

   13. associate-*l* N/A

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

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

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

   15. *-lowering-*.f64 92.1%

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

7. Simplified 92.1%

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

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

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

9. Applied egg-rr 92.1%

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

10. Taylor expanded in x around inf

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

    1. metadata-eval N/A

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

    2. pow-sqr N/A

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

    3. associate-*l* N/A

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

    4. unpow2 N/A

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

    5. associate-*r* N/A

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

    6. *-commutative N/A

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

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

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

    8. associate-*r* N/A

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

    9. unpow2 N/A

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

    10. unpow3 N/A

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

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

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

    12. cube-mult N/A

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

    13. unpow2 N/A

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

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

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

    15. unpow2 N/A

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

    16. *-lowering-*.f64 91.3%

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

12. Simplified 91.3%

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

13. Final simplification 91.3%

    \[\leadsto x + \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(0.0001984126984126984 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \]
14. Add Preprocessing

Alternative 12: 87.2% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.9:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.008333333333333333 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.9)
   (* x (+ 1.0 (* x (* x 0.16666666666666666))))
   (* x (* x (* 0.008333333333333333 (* x (* x x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 4.9) {
		tmp = x * (1.0 + (x * (x * 0.16666666666666666)));
	} 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 <= 4.9d0) then
        tmp = x * (1.0d0 + (x * (x * 0.16666666666666666d0)))
    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 <= 4.9) {
		tmp = x * (1.0 + (x * (x * 0.16666666666666666)));
	} else {
		tmp = x * (x * (0.008333333333333333 * (x * (x * x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 4.9:
		tmp = x * (1.0 + (x * (x * 0.16666666666666666)))
	else:
		tmp = x * (x * (0.008333333333333333 * (x * (x * x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.9)
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(x * 0.16666666666666666))));
	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 <= 4.9)
		tmp = x * (1.0 + (x * (x * 0.16666666666666666)));
	else
		tmp = x * (x * (0.008333333333333333 * (x * (x * x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 4.9], N[(x * N[(1.0 + N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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 < 4.9000000000000004

   1. Initial program 48.9%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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 84.0%

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

   5. Simplified 84.0%

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

   if 4.9000000000000004 < 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 78.4%

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

   5. Simplified 78.4%

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

   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 78.4%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{120} \cdot {x}^{5}} \]
   10. 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. *-commutative N/A

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

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

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

       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 78.4%

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

   11. Simplified 78.4%

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

Alternative 13: 67.5% 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 48.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 57.4%

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

   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 64.9%

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

   5. Simplified 64.9%

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

   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 64.9%

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

   8. Simplified 64.9%

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

Alternative 14: 83.8% 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 62.9%

   \[\frac{e^{x} - e^{-x}}{2} \]
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 78.8%

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

5. Simplified 78.8%

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

Alternative 15: 51.9% 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 62.9%

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

3. Taylor expanded in x around 0

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

5. Simplified 43.2%

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

Reproduce

herbie shell --seed 2024152 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))