Hyperbolic sine

Percentage Accurate: 54.2% → 100.0%

Time: 10.1s

Alternatives: 15

Speedup: 22.9×

• 49.7% 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 56.5%

   \[\frac{e^{x} - e^{-x}}{2} \]
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: 93.1% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.5%

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

   1. sinh-def N/A

      \[\leadsto \sinh x \]

   2. sinh-lowering-sinh.f64 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

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

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

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

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

   7. 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. unpow2 N/A

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

   14. associate-*l* N/A

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

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

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

   16. *-lowering-*.f64 92.4%

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

7. Simplified 92.4%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

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

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

9. Applied egg-rr 92.4%

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

10. Final simplification 92.4%

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

Alternative 3: 93.1% 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 56.5%

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

   1. sinh-def N/A

      \[\leadsto \sinh x \]

   2. sinh-lowering-sinh.f64 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

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

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

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

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

   7. 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. unpow2 N/A

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

   14. associate-*l* N/A

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

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

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

   16. *-lowering-*.f64 92.4%

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

7. Simplified 92.4%

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

Alternative 4: 93.0% accurate, 10.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.5%

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

   1. sinh-def N/A

      \[\leadsto \sinh x \]

   2. sinh-lowering-sinh.f64 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

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

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

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

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

   7. 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. unpow2 N/A

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

   14. associate-*l* N/A

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

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

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

   16. *-lowering-*.f64 92.4%

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

7. Simplified 92.4%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

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

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

9. Applied egg-rr 92.4%

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

10. Taylor expanded in x around inf

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

    1. metadata-eval N/A

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

    2. pow-plus N/A

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

    3. unpow3 N/A

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

    4. unpow2 N/A

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

    5. associate-*r* N/A

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

    6. unpow2 N/A

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

    7. associate-*l* N/A

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

    8. unpow2 N/A

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

    9. associate-*r* N/A

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

    10. *-commutative N/A

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

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

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

    12. *-commutative N/A

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

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

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

    14. *-commutative N/A

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

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

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

    16. unpow2 N/A

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

    17. *-lowering-*.f64 92.2%

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

12. Simplified 92.2%

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

13. Final simplification 92.2%

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

Alternative 5: 92.9% 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 56.5%

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

   1. sinh-def N/A

      \[\leadsto \sinh x \]

   2. sinh-lowering-sinh.f64 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

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

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

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

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

   7. 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. unpow2 N/A

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

   14. associate-*l* N/A

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

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

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

   16. *-lowering-*.f64 92.4%

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

7. Simplified 92.4%

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

8. Taylor expanded in x around inf

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

   1. *-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

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

   3. pow-plus N/A

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

   4. *-commutative N/A

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

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

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

   6. cube-mult N/A

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

   7. unpow2 N/A

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

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

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 92.2%

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

10. Simplified 92.2%

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

Alternative 6: 86.8% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 3.2999999999999998

   1. Initial program 39.8%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 87.8%

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

   5. Simplified 87.8%

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

      1. sub-neg N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-rgt-neg-out N/A

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

      5. unsub-neg N/A

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

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

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

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

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

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

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

      9. *-lowering-*.f64 87.8%

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

   7. Applied egg-rr 87.8%

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

   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. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 73.5%

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

   5. Simplified 73.5%

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

   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. distribute-rgt-in N/A

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

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot {x}^{\left(3 + 1\right)} + \left(\frac{1}{6} \cdot \frac{1}{\color{blue}{{x}^{2}}}\right) \cdot {x}^{4}\right)\right) \]

      3. pow-plus N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot \left({x}^{3} \cdot x\right) + \left(\frac{1}{6} \cdot \color{blue}{\frac{1}{{x}^{2}}}\right) \cdot {x}^{4}\right)\right) \]

      4. unpow3 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. associate-*l/ N/A

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

      12. associate-/l* N/A

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

      13. metadata-eval N/A

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

      14. pow-plus N/A

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

      15. unpow3 N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

      18. unpow2 N/A

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

      19. associate-/l* N/A

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

      20. *-lft-identity N/A

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

      21. associate-*l/ N/A

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

      22. lft-mult-inverse N/A

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

   8. Simplified 73.5%

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

Alternative 7: 92.7% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.5%

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

   1. sinh-def N/A

      \[\leadsto \sinh x \]

   2. sinh-lowering-sinh.f64 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

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

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

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

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

   7. 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. unpow2 N/A

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

   14. associate-*l* N/A

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

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

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

   16. *-lowering-*.f64 92.4%

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

7. Simplified 92.4%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

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

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

9. Applied egg-rr 92.4%

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

10. Taylor expanded in x around inf

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

    1. metadata-eval N/A

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

    2. pow-plus N/A

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

    3. metadata-eval N/A

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

    4. pow-plus N/A

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

    5. unpow3 N/A

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

    6. unpow2 N/A

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

    7. associate-*r* N/A

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

    8. unpow2 N/A

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

    9. associate-*l* N/A

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

    10. associate-*l* N/A

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

    11. *-commutative N/A

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

    12. associate-*l* N/A

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

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

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

    14. unpow2 N/A

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

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

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

    16. *-commutative N/A

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

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

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

    18. *-commutative N/A

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

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

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

    20. unpow2 N/A

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

    21. *-lowering-*.f64 91.9%

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

12. Simplified 91.9%

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

13. Final simplification 91.9%

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

Alternative 8: 86.8% accurate, 12.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 5.0) {
		tmp = x - (x * ((x * x) * -0.16666666666666666));
	} else {
		tmp = 0.008333333333333333 * (x * (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 - (x * ((x * x) * (-0.16666666666666666d0)))
    else
        tmp = 0.008333333333333333d0 * (x * (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 - (x * ((x * x) * -0.16666666666666666));
	} else {
		tmp = 0.008333333333333333 * (x * (x * (x * (x * x))));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 5

   1. Initial program 39.8%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 87.8%

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

   5. Simplified 87.8%

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

      1. sub-neg N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-rgt-neg-out N/A

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

      5. unsub-neg N/A

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

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

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

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

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

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

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

      9. *-lowering-*.f64 87.8%

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

   7. Applied egg-rr 87.8%

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

   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. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 73.5%

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

   5. Simplified 73.5%

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

   6. Taylor expanded in x around inf

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

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{120}, \color{blue}{\left({x}^{5}\right)}\right) \]

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

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

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

      5. metadata-eval N/A

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

      6. pow-plus N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. unpow2 N/A

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

      14. cube-mult N/A

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

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

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

      16. cube-mult N/A

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

      17. unpow2 N/A

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

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

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

      19. unpow2 N/A

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

      20. *-lowering-*.f64 73.5%

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

   8. Simplified 73.5%

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

4. Final simplification 83.8%

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

Alternative 9: 90.2% accurate, 13.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)} \]
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. *-lowering-*.f64 N/A

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

   4. unpow2 N/A

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

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

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

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

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

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

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

   11. *-commutative N/A

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

   12. *-lowering-*.f64 87.2%

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

5. Simplified 87.2%

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

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

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

   13. *-lowering-*.f64 87.2%

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

7. Applied egg-rr 87.2%

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

8. Final simplification 87.2%

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

Alternative 10: 90.2% accurate, 13.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)} \]
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. *-lowering-*.f64 N/A

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

   4. unpow2 N/A

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

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

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

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

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

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

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

   11. *-commutative N/A

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

   12. *-lowering-*.f64 87.2%

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

5. Simplified 87.2%

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

Alternative 11: 67.9% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot x\right)}{6}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 2.5) x (/ (* x (* x x)) 6.0)))

C

double code(double x) {
	double tmp;
	if (x <= 2.5) {
		tmp = x;
	} else {
		tmp = (x * (x * x)) / 6.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.5d0) then
        tmp = x
    else
        tmp = (x * (x * x)) / 6.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.5) {
		tmp = x;
	} else {
		tmp = (x * (x * x)) / 6.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.5:
		tmp = x
	else:
		tmp = (x * (x * x)) / 6.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.5)
		tmp = x;
	else
		tmp = Float64(Float64(x * Float64(x * x)) / 6.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.5)
		tmp = x;
	else
		tmp = (x * (x * x)) / 6.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.5], x, N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] / 6.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.5

   1. Initial program 39.8%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 67.2%

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

   if 2.5 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 61.3%

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

   5. Simplified 61.3%

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

   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 61.3%

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

   8. Simplified 61.3%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{\frac{2}{\color{blue}{\frac{1}{3}}}} \]

      5. div-inv N/A

         \[\leadsto \frac{x \cdot \left(x \cdot x\right)}{\color{blue}{\frac{2}{\frac{1}{3}}}} \]

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

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

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

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

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

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

      9. metadata-eval 61.3%

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

   10. Applied egg-rr 61.3%

       \[\leadsto \color{blue}{\frac{x \cdot \left(x \cdot x\right)}{6}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 67.9% accurate, 17.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2.5

   1. Initial program 39.8%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 67.2%

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

   if 2.5 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 61.3%

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

   5. Simplified 61.3%

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

   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 61.3%

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

   8. Simplified 61.3%

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

Alternative 13: 83.7% accurate, 22.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in x around 0

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

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

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

   2. metadata-eval N/A

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

   3. cancel-sign-sub-inv N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

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

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. unpow2 N/A

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

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

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

   12. unpow2 N/A

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

   13. *-lowering-*.f64 80.4%

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

5. Simplified 80.4%

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

   1. sub-neg N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. distribute-rgt-neg-out N/A

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

   5. unsub-neg N/A

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

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

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

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

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

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

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

   9. *-lowering-*.f64 80.4%

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

7. Applied egg-rr 80.4%

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

Alternative 14: 83.7% accurate, 22.9× speedup

Math

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

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

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in x around 0

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

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

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

   2. metadata-eval N/A

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

   3. cancel-sign-sub-inv N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

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

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. unpow2 N/A

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

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

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

   12. unpow2 N/A

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

   13. *-lowering-*.f64 80.4%

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

5. Simplified 80.4%

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

Alternative 15: 52.2% accurate, 206.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

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

Java

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

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in x around 0

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

5. Simplified 50.0%

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

Reproduce

herbie shell --seed 2024164 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))