Hyperbolic sine

Percentage Accurate: 54.2% → 100.0%

Time: 10.4s

Alternatives: 12

Speedup: 22.9×

• 49.4% 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: 54.2% 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 51.6%

   \[\frac{e^{x} - e^{-x}}{2} \]
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: 93.5% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(0.16666666666666666 + Float64(x * Float64(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[(x * N[(x * N[(0.008333333333333333 + N[(x * N[(x * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.6%

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

3. Taylor expanded in x around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

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

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

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

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

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

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

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

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

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

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

   12. *-commutative N/A

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

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

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

   14. unpow2 N/A

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

   15. *-lowering-*.f64 96.1%

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

5. Simplified 96.1%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

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

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

   4. *-lowering-*.f64 96.1%

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

7. Applied egg-rr 96.1%

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

8. Final simplification 96.1%

   \[\leadsto x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot 0.0001984126984126984\right)\right)\right)\right)\right) \]
9. Add Preprocessing

Alternative 3: 92.0% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.5:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(x \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot 0.008333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.0001984126984126984 \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 7.5)
   (+
    x
    (* (* x x) (* x (+ 0.16666666666666666 (* x (* x 0.008333333333333333))))))
   (* x (* x (* 0.0001984126984126984 (* x (* x (* x (* x x)))))))))

C

double code(double x) {
	double tmp;
	if (x <= 7.5) {
		tmp = x + ((x * x) * (x * (0.16666666666666666 + (x * (x * 0.008333333333333333)))));
	} else {
		tmp = x * (x * (0.0001984126984126984 * (x * (x * (x * (x * x))))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 7.5d0) then
        tmp = x + ((x * x) * (x * (0.16666666666666666d0 + (x * (x * 0.008333333333333333d0)))))
    else
        tmp = x * (x * (0.0001984126984126984d0 * (x * (x * (x * (x * x))))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 7.5) {
		tmp = x + ((x * x) * (x * (0.16666666666666666 + (x * (x * 0.008333333333333333)))));
	} else {
		tmp = x * (x * (0.0001984126984126984 * (x * (x * (x * (x * x))))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 7.5:
		tmp = x + ((x * x) * (x * (0.16666666666666666 + (x * (x * 0.008333333333333333)))))
	else:
		tmp = x * (x * (0.0001984126984126984 * (x * (x * (x * (x * x))))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 7.5)
		tmp = Float64(x + Float64(Float64(x * x) * Float64(x * Float64(0.16666666666666666 + Float64(x * Float64(x * 0.008333333333333333))))));
	else
		tmp = Float64(x * Float64(x * Float64(0.0001984126984126984 * Float64(x * Float64(x * Float64(x * Float64(x * x)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 7.5)
		tmp = x + ((x * x) * (x * (0.16666666666666666 + (x * (x * 0.008333333333333333)))));
	else
		tmp = x * (x * (0.0001984126984126984 * (x * (x * (x * (x * x))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 7.5], N[(x + N[(N[(x * x), $MachinePrecision] * N[(x * N[(0.16666666666666666 + N[(x * N[(x * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(0.0001984126984126984 * N[(x * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.5

   1. Initial program 37.7%

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 95.4%

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

   5. Simplified 95.4%

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

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

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

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

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

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

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

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

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

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

      10. associate-*l* N/A

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

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

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

      12. *-lowering-*.f64 95.4%

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

   7. Applied egg-rr 95.4%

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

   if 7.5 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 95.1%

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

   5. Simplified 95.1%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-lowering-*.f64 95.1%

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

   7. Applied egg-rr 95.1%

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

   8. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. metadata-eval N/A

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

      4. pow-plus N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

      19. pow-sqr N/A

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

      20. metadata-eval N/A

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

      21. associate-*l* N/A

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

      22. pow-plus N/A

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

      23. metadata-eval N/A

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

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

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

      25. metadata-eval N/A

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

   10. Simplified 95.1%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.5:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(x \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot 0.008333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.0001984126984126984 \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.0% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.5:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot 0.008333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.0001984126984126984 \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 7.5)
   (*
    x
    (+
     1.0
     (* x (* x (+ 0.16666666666666666 (* (* x x) 0.008333333333333333))))))
   (* x (* x (* 0.0001984126984126984 (* x (* x (* x (* x x)))))))))

C

double code(double x) {
	double tmp;
	if (x <= 7.5) {
		tmp = x * (1.0 + (x * (x * (0.16666666666666666 + ((x * x) * 0.008333333333333333)))));
	} else {
		tmp = x * (x * (0.0001984126984126984 * (x * (x * (x * (x * x))))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 7.5d0) then
        tmp = x * (1.0d0 + (x * (x * (0.16666666666666666d0 + ((x * x) * 0.008333333333333333d0)))))
    else
        tmp = x * (x * (0.0001984126984126984d0 * (x * (x * (x * (x * x))))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 7.5) {
		tmp = x * (1.0 + (x * (x * (0.16666666666666666 + ((x * x) * 0.008333333333333333)))));
	} else {
		tmp = x * (x * (0.0001984126984126984 * (x * (x * (x * (x * x))))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 7.5:
		tmp = x * (1.0 + (x * (x * (0.16666666666666666 + ((x * x) * 0.008333333333333333)))))
	else:
		tmp = x * (x * (0.0001984126984126984 * (x * (x * (x * (x * x))))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 7.5)
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(x * Float64(0.16666666666666666 + Float64(Float64(x * x) * 0.008333333333333333))))));
	else
		tmp = Float64(x * Float64(x * Float64(0.0001984126984126984 * Float64(x * Float64(x * Float64(x * Float64(x * x)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 7.5)
		tmp = x * (1.0 + (x * (x * (0.16666666666666666 + ((x * x) * 0.008333333333333333)))));
	else
		tmp = x * (x * (0.0001984126984126984 * (x * (x * (x * (x * x))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 7.5], N[(x * N[(1.0 + N[(x * N[(x * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(0.0001984126984126984 * N[(x * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.5

   1. Initial program 37.7%

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 95.4%

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

   5. Simplified 95.4%

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

   if 7.5 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 95.1%

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

   5. Simplified 95.1%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-lowering-*.f64 95.1%

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

   7. Applied egg-rr 95.1%

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

   8. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. metadata-eval N/A

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

      4. pow-plus N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

      19. pow-sqr N/A

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

      20. metadata-eval N/A

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

      21. associate-*l* N/A

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

      22. pow-plus N/A

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

      23. metadata-eval N/A

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

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

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

      25. metadata-eval N/A

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

   10. Simplified 95.1%

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

Alternative 5: 88.9% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.7:\\ \;\;\;\;\frac{x \cdot \left(x \cdot \left(x \cdot 0.3333333333333333\right)\right) + x \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.0001984126984126984 \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.7)
   (/ (+ (* x (* x (* x 0.3333333333333333))) (* x 2.0)) 2.0)
   (* x (* x (* 0.0001984126984126984 (* x (* x (* x (* x x)))))))))

C

double code(double x) {
	double tmp;
	if (x <= 5.7) {
		tmp = ((x * (x * (x * 0.3333333333333333))) + (x * 2.0)) / 2.0;
	} else {
		tmp = x * (x * (0.0001984126984126984 * (x * (x * (x * (x * x))))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5.7d0) then
        tmp = ((x * (x * (x * 0.3333333333333333d0))) + (x * 2.0d0)) / 2.0d0
    else
        tmp = x * (x * (0.0001984126984126984d0 * (x * (x * (x * (x * x))))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.7) {
		tmp = ((x * (x * (x * 0.3333333333333333))) + (x * 2.0)) / 2.0;
	} else {
		tmp = x * (x * (0.0001984126984126984 * (x * (x * (x * (x * x))))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.7:
		tmp = ((x * (x * (x * 0.3333333333333333))) + (x * 2.0)) / 2.0
	else:
		tmp = x * (x * (0.0001984126984126984 * (x * (x * (x * (x * x))))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.7)
		tmp = Float64(Float64(Float64(x * Float64(x * Float64(x * 0.3333333333333333))) + Float64(x * 2.0)) / 2.0);
	else
		tmp = Float64(x * Float64(x * Float64(0.0001984126984126984 * Float64(x * Float64(x * Float64(x * Float64(x * x)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.7)
		tmp = ((x * (x * (x * 0.3333333333333333))) + (x * 2.0)) / 2.0;
	else
		tmp = x * (x * (0.0001984126984126984 * (x * (x * (x * (x * x))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.7], N[(N[(N[(x * N[(x * N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(x * N[(x * N[(0.0001984126984126984 * N[(x * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.70000000000000018

   1. Initial program 37.7%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 91.6%

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

   5. Simplified 91.6%

      \[\leadsto \frac{\color{blue}{x \cdot \left(2 + x \cdot \left(x \cdot 0.3333333333333333\right)\right)}}{2} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 91.6%

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

   7. Applied egg-rr 91.6%

      \[\leadsto \frac{\color{blue}{x \cdot \left(x \cdot \left(x \cdot 0.3333333333333333\right)\right) + x \cdot 2}}{2} \]

   if 5.70000000000000018 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 95.1%

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

   5. Simplified 95.1%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-lowering-*.f64 95.1%

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

   7. Applied egg-rr 95.1%

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

   8. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. metadata-eval N/A

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

      4. pow-plus N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

      19. pow-sqr N/A

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

      20. metadata-eval N/A

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

      21. associate-*l* N/A

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

      22. pow-plus N/A

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

      23. metadata-eval N/A

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

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

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

      25. metadata-eval N/A

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

   10. Simplified 95.1%

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

Alternative 6: 93.4% accurate, 10.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.6%

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

3. Taylor expanded in x around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

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

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

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

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

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

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

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

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

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

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

   12. *-commutative N/A

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

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

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

   14. unpow2 N/A

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

   15. *-lowering-*.f64 96.1%

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

5. Simplified 96.1%

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

6. Taylor expanded in x around inf

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

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

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

   2. metadata-eval N/A

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

   3. pow-sqr N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. unpow2 N/A

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

   7. cube-mult N/A

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

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

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

   9. cube-mult N/A

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

   10. unpow2 N/A

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

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

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

   12. unpow2 N/A

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

   13. *-lowering-*.f64 96.0%

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

8. Simplified 96.0%

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

Alternative 7: 87.5% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.3:\\ \;\;\;\;\frac{x \cdot \left(x \cdot \left(x \cdot 0.3333333333333333\right)\right) + x \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot 0.008333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.3)
   (/ (+ (* x (* x (* x 0.3333333333333333))) (* x 2.0)) 2.0)
   (* x (* x (* x (+ 0.16666666666666666 (* (* x x) 0.008333333333333333)))))))

C

double code(double x) {
	double tmp;
	if (x <= 3.3) {
		tmp = ((x * (x * (x * 0.3333333333333333))) + (x * 2.0)) / 2.0;
	} 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 * 0.3333333333333333d0))) + (x * 2.0d0)) / 2.0d0
    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 * 0.3333333333333333))) + (x * 2.0)) / 2.0;
	} 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 * 0.3333333333333333))) + (x * 2.0)) / 2.0
	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(Float64(Float64(x * Float64(x * Float64(x * 0.3333333333333333))) + Float64(x * 2.0)) / 2.0);
	else
		tmp = Float64(x * Float64(x * Float64(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 * 0.3333333333333333))) + (x * 2.0)) / 2.0;
	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[(N[(N[(x * N[(x * N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(x * N[(x * N[(x * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.2999999999999998

   1. Initial program 37.7%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 91.6%

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

   5. Simplified 91.6%

      \[\leadsto \frac{\color{blue}{x \cdot \left(2 + x \cdot \left(x \cdot 0.3333333333333333\right)\right)}}{2} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 91.6%

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

   7. Applied egg-rr 91.6%

      \[\leadsto \frac{\color{blue}{x \cdot \left(x \cdot \left(x \cdot 0.3333333333333333\right)\right) + x \cdot 2}}{2} \]

   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. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 90.2%

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

   5. Simplified 90.2%

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

   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. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. fma-define N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{fma}\left(\frac{1}{6}, \color{blue}{\frac{1}{{x}^{2}} \cdot {x}^{4}}, \frac{1}{120} \cdot {x}^{4}\right)\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{fma}\left(\frac{1}{6}, \frac{1 \cdot {x}^{4}}{\color{blue}{{x}^{2}}}, \frac{1}{120} \cdot {x}^{4}\right)\right)\right) \]

      6. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{fma}\left(\frac{1}{6}, \frac{{x}^{4}}{{\color{blue}{x}}^{2}}, \frac{1}{120} \cdot {x}^{4}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{fma}\left(\frac{1}{6}, \frac{{x}^{\left(2 \cdot 2\right)}}{{x}^{2}}, \frac{1}{120} \cdot {x}^{4}\right)\right)\right) \]

      8. pow-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{fma}\left(\frac{1}{6}, \frac{{x}^{2} \cdot {x}^{2}}{{\color{blue}{x}}^{2}}, \frac{1}{120} \cdot {x}^{4}\right)\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{fma}\left(\frac{1}{6}, {x}^{2} \cdot \color{blue}{\frac{{x}^{2}}{{x}^{2}}}, \frac{1}{120} \cdot {x}^{4}\right)\right)\right) \]

      10. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{fma}\left(\frac{1}{6}, {x}^{2} \cdot 1, \frac{1}{120} \cdot {x}^{4}\right)\right)\right) \]

      11. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{fma}\left(\frac{1}{6}, {x}^{\color{blue}{2}}, \frac{1}{120} \cdot {x}^{4}\right)\right)\right) \]

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. associate-*l* N/A

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

      15. fma-define N/A

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

      16. distribute-rgt-in N/A

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

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

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

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

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

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

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

   8. Simplified 90.2%

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

Alternative 8: 87.5% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.3d0) then
        tmp = x * (1.0d0 + (x * (x * 0.16666666666666666d0)))
    else
        tmp = x * (x * (x * (0.16666666666666666d0 + ((x * x) * 0.008333333333333333d0))))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 3.3], N[(x * N[(1.0 + N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(x * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.2999999999999998

   1. Initial program 37.7%

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

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

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

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

      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. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 91.6%

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

   7. Simplified 91.6%

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

   if 3.2999999999999998 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 90.2%

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

   5. Simplified 90.2%

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

   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. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. fma-define N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{fma}\left(\frac{1}{6}, \color{blue}{\frac{1}{{x}^{2}} \cdot {x}^{4}}, \frac{1}{120} \cdot {x}^{4}\right)\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{fma}\left(\frac{1}{6}, \frac{1 \cdot {x}^{4}}{\color{blue}{{x}^{2}}}, \frac{1}{120} \cdot {x}^{4}\right)\right)\right) \]

      6. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{fma}\left(\frac{1}{6}, \frac{{x}^{4}}{{\color{blue}{x}}^{2}}, \frac{1}{120} \cdot {x}^{4}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{fma}\left(\frac{1}{6}, \frac{{x}^{\left(2 \cdot 2\right)}}{{x}^{2}}, \frac{1}{120} \cdot {x}^{4}\right)\right)\right) \]

      8. pow-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{fma}\left(\frac{1}{6}, \frac{{x}^{2} \cdot {x}^{2}}{{\color{blue}{x}}^{2}}, \frac{1}{120} \cdot {x}^{4}\right)\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{fma}\left(\frac{1}{6}, {x}^{2} \cdot \color{blue}{\frac{{x}^{2}}{{x}^{2}}}, \frac{1}{120} \cdot {x}^{4}\right)\right)\right) \]

      10. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{fma}\left(\frac{1}{6}, {x}^{2} \cdot 1, \frac{1}{120} \cdot {x}^{4}\right)\right)\right) \]

      11. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{fma}\left(\frac{1}{6}, {x}^{\color{blue}{2}}, \frac{1}{120} \cdot {x}^{4}\right)\right)\right) \]

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. associate-*l* N/A

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

      15. fma-define N/A

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

      16. distribute-rgt-in N/A

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

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

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

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

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

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

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

   8. Simplified 90.2%

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

Alternative 9: 87.5% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 5

   1. Initial program 37.7%

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

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

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

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

      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. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 91.6%

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

   7. Simplified 91.6%

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

   if 5 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 90.2%

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

   5. Simplified 90.2%

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

   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

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

      5. pow-sqr N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. pow-sqr N/A

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

      13. metadata-eval N/A

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

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

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

      15. metadata-eval N/A

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

      16. pow-sqr N/A

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

      19. unpow2 N/A

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

      20. cube-mult N/A

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

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

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

      22. cube-mult N/A

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

      23. unpow2 N/A

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

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

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

      25. unpow2 N/A

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

      26. *-lowering-*.f64 90.2%

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

   8. Simplified 90.2%

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

Alternative 10: 68.6% accurate, 17.1× speedup

Math

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

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

Derivation

1. Split input into 2 regimes

2. if x < 2.39999999999999991

   1. Initial program 37.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 69.9%

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

   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

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

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

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

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

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 78.6%

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

   5. Simplified 78.6%

      \[\leadsto \frac{\color{blue}{x \cdot \left(2 + x \cdot \left(x \cdot 0.3333333333333333\right)\right)}}{2} \]

   6. Taylor expanded in x around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

         \[\leadsto \frac{1}{6} \cdot \left({x}^{2} \cdot x\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\frac{1}{6} \cdot {x}^{2}\right) \cdot \color{blue}{x} \]

      4. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{6} \cdot {x}^{2}\right)} \]

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

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

      10. *-lowering-*.f64 78.6%

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

   8. Simplified 78.6%

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

Alternative 11: 84.5% 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 51.6%

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

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

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

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

   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. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

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

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

   7. *-lowering-*.f64 88.7%

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

7. Simplified 88.7%

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

Alternative 12: 52.3% accurate, 206.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

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

Java

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

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 51.6%

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

3. Taylor expanded in x around 0

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

5. Simplified 55.5%

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

Reproduce

herbie shell --seed 2024163 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))