Hyperbolic sine

Percentage Accurate: 54.1% → 100.0%

Time: 9.9s

Alternatives: 13

Speedup: 22.9×

• 49.6% 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: 54.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return sinh(x)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.1%

   \[\frac{e^{x} - e^{-x}}{2} \]
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: 74.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.16666666666666666 + x \cdot \left(x \cdot 0.008333333333333333\right)\\ t_1 := \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(t\_0 \cdot t\_0\right)\\ \mathbf{if}\;x \leq 1.12 \cdot 10^{+39}:\\ \;\;\;\;\frac{\left(1 - t\_1 \cdot t\_1\right) \cdot \frac{x}{1 + \left(x \cdot x\right) \cdot -0.16666666666666666}}{1 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot \left(0.0001984126984126984 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 0.16666666666666666 (* x (* x 0.008333333333333333))))
        (t_1 (* (* (* x x) (* x x)) (* t_0 t_0))))
   (if (<= x 1.12e+39)
     (/
      (* (- 1.0 (* t_1 t_1)) (/ x (+ 1.0 (* (* x x) -0.16666666666666666))))
      (+ 1.0 t_1))
     (* x (* x (* x (* 0.0001984126984126984 (* x (* x (* x x))))))))))

C

double code(double x) {
	double t_0 = 0.16666666666666666 + (x * (x * 0.008333333333333333));
	double t_1 = ((x * x) * (x * x)) * (t_0 * t_0);
	double tmp;
	if (x <= 1.12e+39) {
		tmp = ((1.0 - (t_1 * t_1)) * (x / (1.0 + ((x * x) * -0.16666666666666666)))) / (1.0 + t_1);
	} else {
		tmp = x * (x * (x * (0.0001984126984126984 * (x * (x * (x * x))))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.16666666666666666d0 + (x * (x * 0.008333333333333333d0))
    t_1 = ((x * x) * (x * x)) * (t_0 * t_0)
    if (x <= 1.12d+39) then
        tmp = ((1.0d0 - (t_1 * t_1)) * (x / (1.0d0 + ((x * x) * (-0.16666666666666666d0))))) / (1.0d0 + t_1)
    else
        tmp = x * (x * (x * (0.0001984126984126984d0 * (x * (x * (x * x))))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 0.16666666666666666 + (x * (x * 0.008333333333333333));
	double t_1 = ((x * x) * (x * x)) * (t_0 * t_0);
	double tmp;
	if (x <= 1.12e+39) {
		tmp = ((1.0 - (t_1 * t_1)) * (x / (1.0 + ((x * x) * -0.16666666666666666)))) / (1.0 + t_1);
	} else {
		tmp = x * (x * (x * (0.0001984126984126984 * (x * (x * (x * x))))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.16666666666666666 + (x * (x * 0.008333333333333333))
	t_1 = ((x * x) * (x * x)) * (t_0 * t_0)
	tmp = 0
	if x <= 1.12e+39:
		tmp = ((1.0 - (t_1 * t_1)) * (x / (1.0 + ((x * x) * -0.16666666666666666)))) / (1.0 + t_1)
	else:
		tmp = x * (x * (x * (0.0001984126984126984 * (x * (x * (x * x))))))
	return tmp

Julia

function code(x)
	t_0 = Float64(0.16666666666666666 + Float64(x * Float64(x * 0.008333333333333333)))
	t_1 = Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(t_0 * t_0))
	tmp = 0.0
	if (x <= 1.12e+39)
		tmp = Float64(Float64(Float64(1.0 - Float64(t_1 * t_1)) * Float64(x / Float64(1.0 + Float64(Float64(x * x) * -0.16666666666666666)))) / Float64(1.0 + t_1));
	else
		tmp = Float64(x * Float64(x * Float64(x * Float64(0.0001984126984126984 * Float64(x * Float64(x * Float64(x * x)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.16666666666666666 + (x * (x * 0.008333333333333333));
	t_1 = ((x * x) * (x * x)) * (t_0 * t_0);
	tmp = 0.0;
	if (x <= 1.12e+39)
		tmp = ((1.0 - (t_1 * t_1)) * (x / (1.0 + ((x * x) * -0.16666666666666666)))) / (1.0 + t_1);
	else
		tmp = x * (x * (x * (0.0001984126984126984 * (x * (x * (x * x))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.16666666666666666 + N[(x * N[(x * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.12e+39], N[(N[(N[(1.0 - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(x / N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(x * N[(0.0001984126984126984 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.12e39

   1. Initial program 42.3%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 89.2%

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

   5. Simplified 89.2%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

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

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

   7. Applied egg-rr 65.0%

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

   8. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 73.2%

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

   10. Simplified 73.2%

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

       1. associate-/l* N/A

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

       2. flip-- N/A

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

       3. associate-*l/ N/A

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

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

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

   12. Applied egg-rr 67.2%

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

   if 1.12e39 < 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. Step-by-step derivation

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. associate-*r* N/A

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

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

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

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

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

      12. associate-*l* N/A

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

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

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

      14. *-lowering-*.f64 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around inf

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

       1. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{5040} \cdot {x}^{\left(5 + \color{blue}{1}\right)}\right)\right) \]

       2. pow-plus N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{5040} \cdot \left({x}^{5} \cdot \color{blue}{x}\right)\right)\right) \]

       3. metadata-eval N/A

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

       4. pow-plus N/A

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

       5. associate-*r* N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. unpow2 N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

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

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

       12. *-commutative N/A

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

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

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

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

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

       15. metadata-eval N/A

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

       16. pow-sqr N/A

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

       17. unpow2 N/A

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

       18. associate-*l* N/A

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

       19. unpow2 N/A

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

       20. cube-mult N/A

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

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

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

       22. cube-mult N/A

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

       23. unpow2 N/A

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

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

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

       25. unpow2 N/A

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

       26. *-lowering-*.f64 100.0%

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

   12. Simplified 100.0%

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

Alternative 3: 74.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot 0.0001984126984126984\right)\right)\right)\\ \mathbf{if}\;x \leq 4 \cdot 10^{+42}:\\ \;\;\;\;\frac{x \cdot \left(1 - t\_0 \cdot t\_0\right)}{1 - t\_0}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot \left(0.0001984126984126984 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (*
          (* x x)
          (+
           0.16666666666666666
           (*
            x
            (*
             x
             (+ 0.008333333333333333 (* (* x x) 0.0001984126984126984))))))))
   (if (<= x 4e+42)
     (/ (* x (- 1.0 (* t_0 t_0))) (- 1.0 t_0))
     (* x (* x (* x (* 0.0001984126984126984 (* x (* x (* x x))))))))))

C

double code(double x) {
	double t_0 = (x * x) * (0.16666666666666666 + (x * (x * (0.008333333333333333 + ((x * x) * 0.0001984126984126984)))));
	double tmp;
	if (x <= 4e+42) {
		tmp = (x * (1.0 - (t_0 * t_0))) / (1.0 - t_0);
	} else {
		tmp = x * (x * (x * (0.0001984126984126984 * (x * (x * (x * x))))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * x) * (0.16666666666666666d0 + (x * (x * (0.008333333333333333d0 + ((x * x) * 0.0001984126984126984d0)))))
    if (x <= 4d+42) then
        tmp = (x * (1.0d0 - (t_0 * t_0))) / (1.0d0 - t_0)
    else
        tmp = x * (x * (x * (0.0001984126984126984d0 * (x * (x * (x * x))))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x * x) * (0.16666666666666666 + (x * (x * (0.008333333333333333 + ((x * x) * 0.0001984126984126984)))));
	double tmp;
	if (x <= 4e+42) {
		tmp = (x * (1.0 - (t_0 * t_0))) / (1.0 - t_0);
	} else {
		tmp = x * (x * (x * (0.0001984126984126984 * (x * (x * (x * x))))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * x) * (0.16666666666666666 + (x * (x * (0.008333333333333333 + ((x * x) * 0.0001984126984126984)))))
	tmp = 0
	if x <= 4e+42:
		tmp = (x * (1.0 - (t_0 * t_0))) / (1.0 - t_0)
	else:
		tmp = x * (x * (x * (0.0001984126984126984 * (x * (x * (x * x))))))
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * Float64(0.16666666666666666 + Float64(x * Float64(x * Float64(0.008333333333333333 + Float64(Float64(x * x) * 0.0001984126984126984))))))
	tmp = 0.0
	if (x <= 4e+42)
		tmp = Float64(Float64(x * Float64(1.0 - Float64(t_0 * t_0))) / Float64(1.0 - t_0));
	else
		tmp = Float64(x * Float64(x * Float64(x * Float64(0.0001984126984126984 * Float64(x * Float64(x * Float64(x * x)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * x) * (0.16666666666666666 + (x * (x * (0.008333333333333333 + ((x * x) * 0.0001984126984126984)))));
	tmp = 0.0;
	if (x <= 4e+42)
		tmp = (x * (1.0 - (t_0 * t_0))) / (1.0 - t_0);
	else
		tmp = x * (x * (x * (0.0001984126984126984 * (x * (x * (x * x))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(0.16666666666666666 + N[(x * N[(x * N[(0.008333333333333333 + N[(N[(x * x), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 4e+42], N[(N[(x * N[(1.0 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(x * N[(0.0001984126984126984 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.00000000000000018e42

   1. Initial program 42.3%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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 90.2%

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

   7. Simplified 90.2%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

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

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

   9. Applied egg-rr 66.9%

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

   if 4.00000000000000018e42 < 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. Step-by-step derivation

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. associate-*r* N/A

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

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

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

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

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

      12. associate-*l* N/A

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

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

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

      14. *-lowering-*.f64 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around inf

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

       1. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{5040} \cdot {x}^{\left(5 + \color{blue}{1}\right)}\right)\right) \]

       2. pow-plus N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{5040} \cdot \left({x}^{5} \cdot \color{blue}{x}\right)\right)\right) \]

       3. metadata-eval N/A

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

       4. pow-plus N/A

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

       5. associate-*r* N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. unpow2 N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

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

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

       12. *-commutative N/A

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

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

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

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

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

       15. metadata-eval N/A

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

       16. pow-sqr N/A

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

       17. unpow2 N/A

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

       18. associate-*l* N/A

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

       19. unpow2 N/A

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

       20. cube-mult N/A

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

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

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

       22. cube-mult N/A

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

       23. unpow2 N/A

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

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

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

       25. unpow2 N/A

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

       26. *-lowering-*.f64 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+42}:\\ \;\;\;\;\frac{x \cdot \left(1 - \left(\left(x \cdot x\right) \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot 0.0001984126984126984\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)}{1 - \left(x \cdot x\right) \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot 0.0001984126984126984\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot \left(0.0001984126984126984 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.16666666666666666 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.0001984126984126984\right)\\ t_1 := \left(x \cdot x\right) \cdot t\_0\\ \mathbf{if}\;x \leq 4 \cdot 10^{+42}:\\ \;\;\;\;\frac{x \cdot \left(1 - \left(x \cdot x\right) \cdot \left(t\_0 \cdot t\_1\right)\right)}{1 - t\_1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot \left(0.0001984126984126984 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (+ 0.16666666666666666 (* (* x x) (* (* x x) 0.0001984126984126984))))
        (t_1 (* (* x x) t_0)))
   (if (<= x 4e+42)
     (/ (* x (- 1.0 (* (* x x) (* t_0 t_1)))) (- 1.0 t_1))
     (* x (* x (* x (* 0.0001984126984126984 (* x (* x (* x x))))))))))

C

double code(double x) {
	double t_0 = 0.16666666666666666 + ((x * x) * ((x * x) * 0.0001984126984126984));
	double t_1 = (x * x) * t_0;
	double tmp;
	if (x <= 4e+42) {
		tmp = (x * (1.0 - ((x * x) * (t_0 * t_1)))) / (1.0 - t_1);
	} else {
		tmp = x * (x * (x * (0.0001984126984126984 * (x * (x * (x * x))))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.16666666666666666d0 + ((x * x) * ((x * x) * 0.0001984126984126984d0))
    t_1 = (x * x) * t_0
    if (x <= 4d+42) then
        tmp = (x * (1.0d0 - ((x * x) * (t_0 * t_1)))) / (1.0d0 - t_1)
    else
        tmp = x * (x * (x * (0.0001984126984126984d0 * (x * (x * (x * x))))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 0.16666666666666666 + ((x * x) * ((x * x) * 0.0001984126984126984));
	double t_1 = (x * x) * t_0;
	double tmp;
	if (x <= 4e+42) {
		tmp = (x * (1.0 - ((x * x) * (t_0 * t_1)))) / (1.0 - t_1);
	} else {
		tmp = x * (x * (x * (0.0001984126984126984 * (x * (x * (x * x))))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.16666666666666666 + ((x * x) * ((x * x) * 0.0001984126984126984))
	t_1 = (x * x) * t_0
	tmp = 0
	if x <= 4e+42:
		tmp = (x * (1.0 - ((x * x) * (t_0 * t_1)))) / (1.0 - t_1)
	else:
		tmp = x * (x * (x * (0.0001984126984126984 * (x * (x * (x * x))))))
	return tmp

Julia

function code(x)
	t_0 = Float64(0.16666666666666666 + Float64(Float64(x * x) * Float64(Float64(x * x) * 0.0001984126984126984)))
	t_1 = Float64(Float64(x * x) * t_0)
	tmp = 0.0
	if (x <= 4e+42)
		tmp = Float64(Float64(x * Float64(1.0 - Float64(Float64(x * x) * Float64(t_0 * t_1)))) / Float64(1.0 - t_1));
	else
		tmp = Float64(x * Float64(x * Float64(x * Float64(0.0001984126984126984 * Float64(x * Float64(x * Float64(x * x)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.16666666666666666 + ((x * x) * ((x * x) * 0.0001984126984126984));
	t_1 = (x * x) * t_0;
	tmp = 0.0;
	if (x <= 4e+42)
		tmp = (x * (1.0 - ((x * x) * (t_0 * t_1)))) / (1.0 - t_1);
	else
		tmp = x * (x * (x * (0.0001984126984126984 * (x * (x * (x * x))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[x, 4e+42], N[(N[(x * N[(1.0 - N[(N[(x * x), $MachinePrecision] * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - t$95$1), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(x * N[(0.0001984126984126984 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.00000000000000018e42

   1. Initial program 42.3%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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 90.2%

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

   7. Simplified 90.2%

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

   8. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. unpow3 N/A

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

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

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

      12. cube-mult N/A

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

      13. unpow2 N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 90.1%

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

   10. Simplified 90.1%

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

       1. *-commutative N/A

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

       2. flip-+ N/A

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

       3. associate-*l/ N/A

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

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

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

   12. Applied egg-rr 66.8%

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

   if 4.00000000000000018e42 < 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. Step-by-step derivation

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. associate-*r* N/A

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

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

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

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

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

      12. associate-*l* N/A

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

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

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

      14. *-lowering-*.f64 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around inf

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

       1. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{5040} \cdot {x}^{\left(5 + \color{blue}{1}\right)}\right)\right) \]

       2. pow-plus N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{5040} \cdot \left({x}^{5} \cdot \color{blue}{x}\right)\right)\right) \]

       3. metadata-eval N/A

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

       4. pow-plus N/A

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

       5. associate-*r* N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. unpow2 N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

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

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

       12. *-commutative N/A

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

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

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

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

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

       15. metadata-eval N/A

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

       16. pow-sqr N/A

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

       17. unpow2 N/A

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

       18. associate-*l* N/A

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

       19. unpow2 N/A

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

       20. cube-mult N/A

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

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

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

       22. cube-mult N/A

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

       23. unpow2 N/A

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

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

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

       25. unpow2 N/A

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

       26. *-lowering-*.f64 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+42}:\\ \;\;\;\;\frac{x \cdot \left(1 - \left(x \cdot x\right) \cdot \left(\left(0.16666666666666666 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.0001984126984126984\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)}{1 - \left(x \cdot x\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.0001984126984126984\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot \left(0.0001984126984126984 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.5% 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 53.1%

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

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

7. Simplified 92.0%

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

Alternative 6: 88.7% accurate, 10.3× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 5.6) {
		tmp = x * (1.0 + (x * (x * 0.16666666666666666)));
	} else {
		tmp = x * (x * (x * (0.0001984126984126984 * (x * (x * (x * x))))));
	}
	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 * (0.0001984126984126984d0 * (x * (x * (x * x))))))
    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 * (0.0001984126984126984 * (x * (x * (x * x))))));
	}
	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 * (0.0001984126984126984 * (x * (x * (x * x))))))
	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(0.0001984126984126984 * Float64(x * Float64(x * Float64(x * x)))))));
	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 * (0.0001984126984126984 * (x * (x * (x * x))))));
	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[(0.0001984126984126984 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999996

   1. Initial program 39.4%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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 87.7%

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

   5. Simplified 87.7%

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

   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 83.6%

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

   7. Simplified 83.6%

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. associate-*r* N/A

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

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

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

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

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

      12. associate-*l* N/A

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

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

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

      14. *-lowering-*.f64 83.6%

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

   9. Applied egg-rr 83.6%

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

   10. Taylor expanded in x around inf

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

       1. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{5040} \cdot {x}^{\left(5 + \color{blue}{1}\right)}\right)\right) \]

       2. pow-plus N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{5040} \cdot \left({x}^{5} \cdot \color{blue}{x}\right)\right)\right) \]

       3. metadata-eval N/A

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

       4. pow-plus N/A

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

       5. associate-*r* N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. unpow2 N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

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

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

       12. *-commutative N/A

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

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

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

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

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

       15. metadata-eval N/A

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

       16. pow-sqr N/A

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

       17. unpow2 N/A

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

       18. associate-*l* N/A

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

       19. unpow2 N/A

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

       20. cube-mult N/A

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

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

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

       22. cube-mult N/A

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

       23. unpow2 N/A

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

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

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

       25. unpow2 N/A

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

       26. *-lowering-*.f64 83.6%

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

   12. Simplified 83.6%

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

Alternative 7: 93.3% accurate, 10.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.1%

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

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

7. Simplified 92.0%

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

8. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. pow-sqr N/A

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

   3. associate-*l* N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

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

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

   8. associate-*l* N/A

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

   9. unpow2 N/A

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

   10. unpow3 N/A

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

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

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

   12. cube-mult N/A

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

   13. unpow2 N/A

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f64 91.9%

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

10. Simplified 91.9%

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

Alternative 8: 87.6% 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 39.4%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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 87.7%

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

   5. Simplified 87.7%

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

   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 80.4%

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

   5. Simplified 80.4%

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

   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 80.4%

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

Alternative 9: 87.6% accurate, 12.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.0)
   (* x (+ 1.0 (* x (* x 0.16666666666666666))))
   (* x (* 0.008333333333333333 (* x (* x (* x x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 5.0) {
		tmp = x * (1.0 + (x * (x * 0.16666666666666666)));
	} else {
		tmp = x * (0.008333333333333333 * (x * (x * (x * x))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5.0d0) then
        tmp = x * (1.0d0 + (x * (x * 0.16666666666666666d0)))
    else
        tmp = x * (0.008333333333333333d0 * (x * (x * (x * x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.0) {
		tmp = x * (1.0 + (x * (x * 0.16666666666666666)));
	} else {
		tmp = x * (0.008333333333333333 * (x * (x * (x * x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.0:
		tmp = x * (1.0 + (x * (x * 0.16666666666666666)))
	else:
		tmp = x * (0.008333333333333333 * (x * (x * (x * x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.0)
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(x * 0.16666666666666666))));
	else
		tmp = Float64(x * Float64(0.008333333333333333 * Float64(x * Float64(x * Float64(x * x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.0)
		tmp = x * (1.0 + (x * (x * 0.16666666666666666)));
	else
		tmp = x * (0.008333333333333333 * (x * (x * (x * x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.0], N[(x * N[(1.0 + N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.008333333333333333 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5

   1. Initial program 39.4%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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 87.7%

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

   5. Simplified 87.7%

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

   if 5 < 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 80.4%

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

   5. Simplified 80.4%

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

   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 80.4%

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

   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. *-commutative N/A

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

       5. associate-*l* N/A

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

       6. metadata-eval N/A

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

       7. pow-sqr N/A

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

       8. associate-*r* N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. associate-*r* N/A

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

       12. *-commutative N/A

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

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

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

       14. *-commutative N/A

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

       15. associate-*l* N/A

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

       16. pow-sqr N/A

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

       17. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot {x}^{4}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{120}, \color{blue}{\left({x}^{4}\right)}\right)\right) \]

       19. metadata-eval N/A

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

       20. pow-sqr N/A

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

       21. unpow2 N/A

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

       22. associate-*l* N/A

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

       23. unpow2 N/A

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

       24. cube-mult N/A

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

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

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

       26. cube-mult N/A

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

       27. unpow2 N/A

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

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

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

       29. unpow2 N/A

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

       30. *-lowering-*.f64 80.4%

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

   11. Simplified 80.4%

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

Alternative 10: 91.0% accurate, 13.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.1%

   \[\frac{e^{x} - e^{-x}}{2} \]
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 90.5%

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

5. Simplified 90.5%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot 0.008333333333333333\right)\right)\right)} \]
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 90.5%

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

7. Applied egg-rr 90.5%

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

8. Final simplification 90.5%

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

Alternative 11: 68.2% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.5) x (* 0.16666666666666666 (* x (* x x)))))

C

double code(double x) {
	double tmp;
	if (x <= 2.5) {
		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 <= 2.5d0) 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 <= 2.5) {
		tmp = x;
	} else {
		tmp = 0.16666666666666666 * (x * (x * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.5:
		tmp = x
	else:
		tmp = 0.16666666666666666 * (x * (x * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.5)
		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 <= 2.5)
		tmp = x;
	else
		tmp = 0.16666666666666666 * (x * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.5], x, N[(0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.5

   1. Initial program 39.4%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 67.1%

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

   if 2.5 < 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 62.7%

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

   5. Simplified 62.7%

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

   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 62.7%

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

   8. Simplified 62.7%

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

Alternative 12: 84.6% accurate, 22.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.1%

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

3. Taylor expanded in x around 0

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

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

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

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

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 82.1%

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

5. Simplified 82.1%

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

Alternative 13: 52.3% accurate, 206.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

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

Java

public static double code(double x) {
	return x;
}

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 53.1%

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

3. Taylor expanded in x around 0

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

5. Simplified 53.1%

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

Reproduce

herbie shell --seed 2024156 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))