Hyperbolic sine

Percentage Accurate: 54.4% → 100.0%

Time: 10.1s

Alternatives: 16

Speedup: 22.9×

• 49.9% 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: 54.4% 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 56.4%

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

   1. sinh-def N/A

      \[\leadsto \sinh x \]

   2. sinh-lowering-sinh.f64 100.0%

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

4. Applied egg-rr 100.0%

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

Alternative 2: 75.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot 0.0001984126984126984\\ t_1 := x \cdot \left(x \cdot x\right)\\ t_2 := 0.008333333333333333 + t\_0\\ t_3 := \left(x \cdot x\right) \cdot t\_2\\ \mathbf{if}\;x \leq 2.9 \cdot 10^{+39}:\\ \;\;\;\;x + \frac{t\_1 \cdot \left(0.004629629629629629 + t\_1 \cdot \left(t\_1 \cdot \left(t\_2 \cdot \left(t\_2 \cdot t\_2\right)\right)\right)\right)}{0.027777777777777776 + t\_3 \cdot \left(t\_3 - 0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot t\_0\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) 0.0001984126984126984))
        (t_1 (* x (* x x)))
        (t_2 (+ 0.008333333333333333 t_0))
        (t_3 (* (* x x) t_2)))
   (if (<= x 2.9e+39)
     (+
      x
      (/
       (* t_1 (+ 0.004629629629629629 (* t_1 (* t_1 (* t_2 (* t_2 t_2))))))
       (+ 0.027777777777777776 (* t_3 (- t_3 0.16666666666666666)))))
     (* x (+ 1.0 (* (* x x) (+ 0.16666666666666666 (* x (* x t_0)))))))))

C

double code(double x) {
	double t_0 = (x * x) * 0.0001984126984126984;
	double t_1 = x * (x * x);
	double t_2 = 0.008333333333333333 + t_0;
	double t_3 = (x * x) * t_2;
	double tmp;
	if (x <= 2.9e+39) {
		tmp = x + ((t_1 * (0.004629629629629629 + (t_1 * (t_1 * (t_2 * (t_2 * t_2)))))) / (0.027777777777777776 + (t_3 * (t_3 - 0.16666666666666666))));
	} else {
		tmp = x * (1.0 + ((x * x) * (0.16666666666666666 + (x * (x * t_0)))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (x * x) * 0.0001984126984126984d0
    t_1 = x * (x * x)
    t_2 = 0.008333333333333333d0 + t_0
    t_3 = (x * x) * t_2
    if (x <= 2.9d+39) then
        tmp = x + ((t_1 * (0.004629629629629629d0 + (t_1 * (t_1 * (t_2 * (t_2 * t_2)))))) / (0.027777777777777776d0 + (t_3 * (t_3 - 0.16666666666666666d0))))
    else
        tmp = x * (1.0d0 + ((x * x) * (0.16666666666666666d0 + (x * (x * t_0)))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x * x) * 0.0001984126984126984;
	double t_1 = x * (x * x);
	double t_2 = 0.008333333333333333 + t_0;
	double t_3 = (x * x) * t_2;
	double tmp;
	if (x <= 2.9e+39) {
		tmp = x + ((t_1 * (0.004629629629629629 + (t_1 * (t_1 * (t_2 * (t_2 * t_2)))))) / (0.027777777777777776 + (t_3 * (t_3 - 0.16666666666666666))));
	} else {
		tmp = x * (1.0 + ((x * x) * (0.16666666666666666 + (x * (x * t_0)))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * x) * 0.0001984126984126984
	t_1 = x * (x * x)
	t_2 = 0.008333333333333333 + t_0
	t_3 = (x * x) * t_2
	tmp = 0
	if x <= 2.9e+39:
		tmp = x + ((t_1 * (0.004629629629629629 + (t_1 * (t_1 * (t_2 * (t_2 * t_2)))))) / (0.027777777777777776 + (t_3 * (t_3 - 0.16666666666666666))))
	else:
		tmp = x * (1.0 + ((x * x) * (0.16666666666666666 + (x * (x * t_0)))))
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * 0.0001984126984126984)
	t_1 = Float64(x * Float64(x * x))
	t_2 = Float64(0.008333333333333333 + t_0)
	t_3 = Float64(Float64(x * x) * t_2)
	tmp = 0.0
	if (x <= 2.9e+39)
		tmp = Float64(x + Float64(Float64(t_1 * Float64(0.004629629629629629 + Float64(t_1 * Float64(t_1 * Float64(t_2 * Float64(t_2 * t_2)))))) / Float64(0.027777777777777776 + Float64(t_3 * Float64(t_3 - 0.16666666666666666)))));
	else
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(0.16666666666666666 + Float64(x * Float64(x * t_0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * x) * 0.0001984126984126984;
	t_1 = x * (x * x);
	t_2 = 0.008333333333333333 + t_0;
	t_3 = (x * x) * t_2;
	tmp = 0.0;
	if (x <= 2.9e+39)
		tmp = x + ((t_1 * (0.004629629629629629 + (t_1 * (t_1 * (t_2 * (t_2 * t_2)))))) / (0.027777777777777776 + (t_3 * (t_3 - 0.16666666666666666))));
	else
		tmp = x * (1.0 + ((x * x) * (0.16666666666666666 + (x * (x * t_0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.008333333333333333 + t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x * x), $MachinePrecision] * t$95$2), $MachinePrecision]}, If[LessEqual[x, 2.9e+39], N[(x + N[(N[(t$95$1 * N[(0.004629629629629629 + N[(t$95$1 * N[(t$95$1 * N[(t$95$2 * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.027777777777777776 + N[(t$95$3 * N[(t$95$3 - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.16666666666666666 + N[(x * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.90000000000000029e39

   1. Initial program 43.9%

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

      1. sinh-def N/A

         \[\leadsto \sinh x \]

      2. sinh-lowering-sinh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

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

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

      15. *-lowering-*.f64 90.9%

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

   7. Simplified 90.9%

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

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

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

   9. Applied egg-rr 90.9%

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

       1. flip3-+ N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\frac{{\frac{1}{6}}^{3} + {\left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{1}{5040}\right)\right)}^{3}}{\frac{1}{6} \cdot \frac{1}{6} + \left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{1}{5040}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{1}{5040}\right)\right) - \frac{1}{6} \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{1}{5040}\right)\right)\right)} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), x\right) \]

       2. associate-*l/ N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\frac{\left({\frac{1}{6}}^{3} + {\left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{1}{5040}\right)\right)}^{3}\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)}{\frac{1}{6} \cdot \frac{1}{6} + \left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{1}{5040}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{1}{5040}\right)\right) - \frac{1}{6} \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{1}{5040}\right)\right)\right)}\right), x\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left({\frac{1}{6}}^{3} + {\left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{1}{5040}\right)\right)}^{3}\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), \left(\frac{1}{6} \cdot \frac{1}{6} + \left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{1}{5040}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{1}{5040}\right)\right) - \frac{1}{6} \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{1}{5040}\right)\right)\right)\right)\right), x\right) \]

   11. Applied egg-rr 67.9%

       \[\leadsto \color{blue}{\frac{\left(0.004629629629629629 + \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(0.008333333333333333 + \left(x \cdot x\right) \cdot 0.0001984126984126984\right) \cdot \left(\left(0.008333333333333333 + \left(x \cdot x\right) \cdot 0.0001984126984126984\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)}{0.027777777777777776 + \left(\left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot 0.0001984126984126984\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot 0.0001984126984126984\right) - 0.16666666666666666\right)}} + x \]

   if 2.90000000000000029e39 < x

   1. Initial program 100.0%

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

      1. sinh-def N/A

         \[\leadsto \sinh x \]

      2. sinh-lowering-sinh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

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

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

      15. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{1}{5040} \cdot {x}^{4}\right)}\right)\right)\right)\right) \]
   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. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.9 \cdot 10^{+39}:\\ \;\;\;\;x + \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.004629629629629629 + \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(0.008333333333333333 + \left(x \cdot x\right) \cdot 0.0001984126984126984\right) \cdot \left(\left(0.008333333333333333 + \left(x \cdot x\right) \cdot 0.0001984126984126984\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)}{0.027777777777777776 + \left(\left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot 0.0001984126984126984\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot 0.0001984126984126984\right) - 0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.3% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot 0.0001984126984126984\right)\\ \mathbf{if}\;x \leq 10^{+61}:\\ \;\;\;\;x + \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.027777777777777776 - t\_0 \cdot t\_0\right)}{0.16666666666666666 - t\_0}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.008333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (*
          (* x x)
          (+ 0.008333333333333333 (* (* x x) 0.0001984126984126984)))))
   (if (<= x 1e+61)
     (+
      x
      (/
       (* (* x (* x x)) (- 0.027777777777777776 (* t_0 t_0)))
       (- 0.16666666666666666 t_0)))
     (* x (* x (* x (* (* x x) 0.008333333333333333)))))))

C

double code(double x) {
	double t_0 = (x * x) * (0.008333333333333333 + ((x * x) * 0.0001984126984126984));
	double tmp;
	if (x <= 1e+61) {
		tmp = x + (((x * (x * x)) * (0.027777777777777776 - (t_0 * t_0))) / (0.16666666666666666 - t_0));
	} else {
		tmp = x * (x * (x * ((x * x) * 0.008333333333333333)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * x) * (0.008333333333333333d0 + ((x * x) * 0.0001984126984126984d0))
    if (x <= 1d+61) then
        tmp = x + (((x * (x * x)) * (0.027777777777777776d0 - (t_0 * t_0))) / (0.16666666666666666d0 - t_0))
    else
        tmp = x * (x * (x * ((x * x) * 0.008333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x * x) * (0.008333333333333333 + ((x * x) * 0.0001984126984126984));
	double tmp;
	if (x <= 1e+61) {
		tmp = x + (((x * (x * x)) * (0.027777777777777776 - (t_0 * t_0))) / (0.16666666666666666 - t_0));
	} else {
		tmp = x * (x * (x * ((x * x) * 0.008333333333333333)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * x) * (0.008333333333333333 + ((x * x) * 0.0001984126984126984))
	tmp = 0
	if x <= 1e+61:
		tmp = x + (((x * (x * x)) * (0.027777777777777776 - (t_0 * t_0))) / (0.16666666666666666 - t_0))
	else:
		tmp = x * (x * (x * ((x * x) * 0.008333333333333333)))
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * Float64(0.008333333333333333 + Float64(Float64(x * x) * 0.0001984126984126984)))
	tmp = 0.0
	if (x <= 1e+61)
		tmp = Float64(x + Float64(Float64(Float64(x * Float64(x * x)) * Float64(0.027777777777777776 - Float64(t_0 * t_0))) / Float64(0.16666666666666666 - t_0)));
	else
		tmp = Float64(x * Float64(x * Float64(x * Float64(Float64(x * x) * 0.008333333333333333))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * x) * (0.008333333333333333 + ((x * x) * 0.0001984126984126984));
	tmp = 0.0;
	if (x <= 1e+61)
		tmp = x + (((x * (x * x)) * (0.027777777777777776 - (t_0 * t_0))) / (0.16666666666666666 - t_0));
	else
		tmp = x * (x * (x * ((x * x) * 0.008333333333333333)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(0.008333333333333333 + N[(N[(x * x), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1e+61], N[(x + N[(N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(0.027777777777777776 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.16666666666666666 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 9.99999999999999949e60

   1. Initial program 45.8%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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 91.2%

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

   7. Simplified 91.2%

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

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

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

   9. Applied egg-rr 91.2%

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

       1. flip-+ N/A

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

       2. associate-*l/ N/A

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

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

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

   11. Applied egg-rr 70.5%

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

   if 9.99999999999999949e60 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

         \[\leadsto \left(\frac{1}{120} \cdot {x}^{\left(3 + 1\right)}\right) \cdot x \]

      5. pow-plus N/A

         \[\leadsto \left(\frac{1}{120} \cdot \left({x}^{3} \cdot x\right)\right) \cdot x \]

      6. associate-*l* N/A

         \[\leadsto \left(\left(\frac{1}{120} \cdot {x}^{3}\right) \cdot x\right) \cdot x \]

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. unpow2 N/A

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

      12. *-commutative N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. associate-*r* N/A

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

      19. unpow2 N/A

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

      20. unpow3 N/A

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

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

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

      22. unpow3 N/A

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

      23. unpow2 N/A

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

   8. Simplified 100.0%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+61}:\\ \;\;\;\;x + \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.027777777777777776 - \left(\left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot 0.0001984126984126984\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot 0.0001984126984126984\right)\right)\right)}{0.16666666666666666 - \left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot 0.0001984126984126984\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.008333333333333333\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot 0.008333333333333333\\ t_1 := 0.16666666666666666 + t\_0\\ t_2 := \left(x \cdot x\right) \cdot t\_1\\ \mathbf{if}\;x \leq 10^{+61}:\\ \;\;\;\;\frac{x \cdot \left(1 - x \cdot \left(\left(x \cdot t\_1\right) \cdot t\_2\right)\right)}{1 - t\_2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) 0.008333333333333333))
        (t_1 (+ 0.16666666666666666 t_0))
        (t_2 (* (* x x) t_1)))
   (if (<= x 1e+61)
     (/ (* x (- 1.0 (* x (* (* x t_1) t_2)))) (- 1.0 t_2))
     (* x (* x (* x t_0))))))

C

double code(double x) {
	double t_0 = (x * x) * 0.008333333333333333;
	double t_1 = 0.16666666666666666 + t_0;
	double t_2 = (x * x) * t_1;
	double tmp;
	if (x <= 1e+61) {
		tmp = (x * (1.0 - (x * ((x * t_1) * t_2)))) / (1.0 - t_2);
	} else {
		tmp = x * (x * (x * t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (x * x) * 0.008333333333333333d0
    t_1 = 0.16666666666666666d0 + t_0
    t_2 = (x * x) * t_1
    if (x <= 1d+61) then
        tmp = (x * (1.0d0 - (x * ((x * t_1) * t_2)))) / (1.0d0 - t_2)
    else
        tmp = x * (x * (x * t_0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x * x) * 0.008333333333333333;
	double t_1 = 0.16666666666666666 + t_0;
	double t_2 = (x * x) * t_1;
	double tmp;
	if (x <= 1e+61) {
		tmp = (x * (1.0 - (x * ((x * t_1) * t_2)))) / (1.0 - t_2);
	} else {
		tmp = x * (x * (x * t_0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * x) * 0.008333333333333333
	t_1 = 0.16666666666666666 + t_0
	t_2 = (x * x) * t_1
	tmp = 0
	if x <= 1e+61:
		tmp = (x * (1.0 - (x * ((x * t_1) * t_2)))) / (1.0 - t_2)
	else:
		tmp = x * (x * (x * t_0))
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * 0.008333333333333333)
	t_1 = Float64(0.16666666666666666 + t_0)
	t_2 = Float64(Float64(x * x) * t_1)
	tmp = 0.0
	if (x <= 1e+61)
		tmp = Float64(Float64(x * Float64(1.0 - Float64(x * Float64(Float64(x * t_1) * t_2)))) / Float64(1.0 - t_2));
	else
		tmp = Float64(x * Float64(x * Float64(x * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * x) * 0.008333333333333333;
	t_1 = 0.16666666666666666 + t_0;
	t_2 = (x * x) * t_1;
	tmp = 0.0;
	if (x <= 1e+61)
		tmp = (x * (1.0 - (x * ((x * t_1) * t_2)))) / (1.0 - t_2);
	else
		tmp = x * (x * (x * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]}, Block[{t$95$1 = N[(0.16666666666666666 + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * x), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[x, 1e+61], N[(N[(x * N[(1.0 - N[(x * N[(N[(x * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - t$95$2), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 9.99999999999999949e60

   1. Initial program 45.8%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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 85.1%

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

   5. Simplified 85.1%

      \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot 0.008333333333333333\right)\right)\right)} \]
   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 69.5%

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

   if 9.99999999999999949e60 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

         \[\leadsto \left(\frac{1}{120} \cdot {x}^{\left(3 + 1\right)}\right) \cdot x \]

      5. pow-plus N/A

         \[\leadsto \left(\frac{1}{120} \cdot \left({x}^{3} \cdot x\right)\right) \cdot x \]

      6. associate-*l* N/A

         \[\leadsto \left(\left(\frac{1}{120} \cdot {x}^{3}\right) \cdot x\right) \cdot x \]

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. unpow2 N/A

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

      12. *-commutative N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. associate-*r* N/A

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

      19. unpow2 N/A

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

      20. unpow3 N/A

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

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

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

      22. unpow3 N/A

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

      23. unpow2 N/A

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

   8. Simplified 100.0%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+61}:\\ \;\;\;\;\frac{x \cdot \left(1 - x \cdot \left(\left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot 0.008333333333333333\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot 0.008333333333333333\right)\right)\right)\right)}{1 - \left(x \cdot x\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot 0.008333333333333333\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.008333333333333333\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.4% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.4%

   \[\frac{e^{x} - e^{-x}}{2} \]
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 92.9%

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

7. Simplified 92.9%

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

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

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

9. Applied egg-rr 92.9%

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

    1. associate-*r* N/A

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

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

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

    3. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    11. *-lowering-*.f64 92.9%

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

11. Applied egg-rr 92.9%

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

12. Final simplification 92.9%

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

Alternative 6: 93.4% 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 56.4%

   \[\frac{e^{x} - e^{-x}}{2} \]
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 92.9%

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

7. Simplified 92.9%

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

Alternative 7: 93.3% accurate, 10.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	return x * (1.0 + ((x * x) * (0.16666666666666666 + (x * (x * ((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 * ((x * x) * 0.0001984126984126984d0))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.4%

   \[\frac{e^{x} - e^{-x}}{2} \]
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 92.9%

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

7. Simplified 92.9%

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

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. *-commutative N/A

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

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

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

   10. *-commutative N/A

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

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

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

   12. unpow2 N/A

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

   13. *-lowering-*.f64 92.6%

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

10. Simplified 92.6%

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

Alternative 8: 87.4% accurate, 11.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.3)
   (+ x (* (* x (* 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 + ((x * (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 + ((x * (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 + ((x * (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 + ((x * (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(Float64(x * Float64(x * 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 + ((x * (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[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $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 41.3%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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 94.9%

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

   7. Simplified 94.9%

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

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

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

   9. Applied egg-rr 94.9%

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

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\frac{1}{6} \cdot {x}^{3}\right)}, x\right) \]
   11. Step-by-step derivation

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 88.5%

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

   12. Simplified 88.5%

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

   if 3.2999999999999998 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 77.2%

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

   5. Simplified 77.2%

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

   6. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

      7. associate-*r* N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

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

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

      17. *-commutative N/A

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

      18. +-commutative N/A

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

      19. distribute-rgt-in N/A

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

   8. Simplified 77.2%

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

4. Final simplification 85.6%

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

Alternative 9: 93.0% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.4%

   \[\frac{e^{x} - e^{-x}}{2} \]
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 92.9%

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

7. Simplified 92.9%

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

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

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

9. Applied egg-rr 92.9%

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

10. Taylor expanded in x around inf

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

    1. metadata-eval N/A

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

    2. pow-sqr N/A

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

    3. associate-*l* N/A

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

    4. unpow2 N/A

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

    5. associate-*r* N/A

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

    6. *-commutative N/A

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

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

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

    8. *-commutative N/A

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

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

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

    10. *-commutative N/A

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

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

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

    12. unpow2 N/A

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

    13. *-lowering-*.f64 92.3%

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

12. Simplified 92.3%

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

13. Final simplification 92.3%

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

Alternative 10: 87.4% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 5

   1. Initial program 41.3%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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 94.9%

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

   7. Simplified 94.9%

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

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

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

   9. Applied egg-rr 94.9%

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

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\frac{1}{6} \cdot {x}^{3}\right)}, x\right) \]
   11. Step-by-step derivation

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 88.5%

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

   12. Simplified 88.5%

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

   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 77.2%

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

   5. Simplified 77.2%

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

   6. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

         \[\leadsto \left(\frac{1}{120} \cdot {x}^{\left(3 + 1\right)}\right) \cdot x \]

      5. pow-plus N/A

         \[\leadsto \left(\frac{1}{120} \cdot \left({x}^{3} \cdot x\right)\right) \cdot x \]

      6. associate-*l* N/A

         \[\leadsto \left(\left(\frac{1}{120} \cdot {x}^{3}\right) \cdot x\right) \cdot x \]

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. unpow2 N/A

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

      12. *-commutative N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. associate-*r* N/A

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

      19. unpow2 N/A

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

      20. unpow3 N/A

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

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

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

      22. unpow3 N/A

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

      23. unpow2 N/A

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

   8. Simplified 77.2%

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

4. Final simplification 85.6%

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

Alternative 11: 90.5% accurate, 13.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.4%

   \[\frac{e^{x} - e^{-x}}{2} \]
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 88.0%

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

5. Simplified 88.0%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

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

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

   5. associate-*r* N/A

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

   6. associate-*r* N/A

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

   7. cube-unmult N/A

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

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

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

   9. cube-unmult N/A

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

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

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

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

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

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

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

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

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

   14. *-lowering-*.f64 88.1%

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

7. Applied egg-rr 88.1%

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

8. Final simplification 88.1%

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

Alternative 12: 90.5% 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 56.4%

   \[\frac{e^{x} - e^{-x}}{2} \]
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 88.0%

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

5. Simplified 88.0%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

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

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

   4. *-lowering-*.f64 88.0%

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

7. Applied egg-rr 88.0%

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

8. Final simplification 88.0%

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

Alternative 13: 68.1% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.4) x (* (* x (* x x)) 0.16666666666666666)))

C

double code(double x) {
	double tmp;
	if (x <= 2.4) {
		tmp = x;
	} else {
		tmp = (x * (x * x)) * 0.16666666666666666;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.4d0) then
        tmp = x
    else
        tmp = (x * (x * x)) * 0.16666666666666666d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.4) {
		tmp = x;
	} else {
		tmp = (x * (x * x)) * 0.16666666666666666;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.4:
		tmp = x
	else:
		tmp = (x * (x * x)) * 0.16666666666666666
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.4)
		tmp = x;
	else
		tmp = Float64(Float64(x * Float64(x * x)) * 0.16666666666666666);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.4)
		tmp = x;
	else
		tmp = (x * (x * x)) * 0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.4], x, N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.39999999999999991

   1. Initial program 41.3%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 66.3%

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

   if 2.39999999999999991 < 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 59.8%

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

   5. Simplified 59.8%

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

   6. Taylor expanded in x around inf

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 59.8%

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

   8. Simplified 59.8%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.16666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 84.2% accurate, 22.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.4%

   \[\frac{e^{x} - e^{-x}}{2} \]
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 92.9%

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

7. Simplified 92.9%

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

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

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

9. Applied egg-rr 92.9%

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

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\frac{1}{6} \cdot {x}^{3}\right)}, x\right) \]
11. Step-by-step derivation

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

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

    2. cube-mult N/A

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

    3. unpow2 N/A

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

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

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

    5. unpow2 N/A

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

    6. *-lowering-*.f64 81.1%

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

12. Simplified 81.1%

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

13. Final simplification 81.1%

    \[\leadsto x + \left(x \cdot \left(x \cdot x\right)\right) \cdot 0.16666666666666666 \]
14. Add Preprocessing

Alternative 15: 84.2% 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 56.4%

   \[\frac{e^{x} - e^{-x}}{2} \]
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 81.1%

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

5. Simplified 81.1%

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

Alternative 16: 52.0% 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 56.4%

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

3. Taylor expanded in x around 0

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

5. Simplified 50.4%

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

Reproduce

herbie shell --seed 2024158 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))