exp2 (problem 3.3.7)

Percentage Accurate: 54.4% → 99.2%

Time: 15.7s

Alternatives: 9

Speedup: 68.7×

Specification

\[\left|x\right| \leq 710\]

Math

\[\begin{array}{l} \\ \left(e^{x} - 2\right) + e^{-x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(e^{x} - 2\right) + e^{-x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.08333333333333333 + \left(x \cdot x\right) \cdot \left(0.002777777777777778 + \left(x \cdot x\right) \cdot 4.96031746031746 \cdot 10^{-5}\right)\\ t_1 := \left(x \cdot x\right) \cdot t\_0\\ \frac{\left(x \cdot x\right) \cdot \left(1 - \left(x \cdot x\right) \cdot \left(t\_0 \cdot t\_1\right)\right)}{1 - t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (+
          0.08333333333333333
          (*
           (* x x)
           (+ 0.002777777777777778 (* (* x x) 4.96031746031746e-5)))))
        (t_1 (* (* x x) t_0)))
   (/ (* (* x x) (- 1.0 (* (* x x) (* t_0 t_1)))) (- 1.0 t_1))))

C

double code(double x) {
	double t_0 = 0.08333333333333333 + ((x * x) * (0.002777777777777778 + ((x * x) * 4.96031746031746e-5)));
	double t_1 = (x * x) * t_0;
	return ((x * x) * (1.0 - ((x * x) * (t_0 * t_1)))) / (1.0 - t_1);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    t_0 = 0.08333333333333333d0 + ((x * x) * (0.002777777777777778d0 + ((x * x) * 4.96031746031746d-5)))
    t_1 = (x * x) * t_0
    code = ((x * x) * (1.0d0 - ((x * x) * (t_0 * t_1)))) / (1.0d0 - t_1)
end function

Java

public static double code(double x) {
	double t_0 = 0.08333333333333333 + ((x * x) * (0.002777777777777778 + ((x * x) * 4.96031746031746e-5)));
	double t_1 = (x * x) * t_0;
	return ((x * x) * (1.0 - ((x * x) * (t_0 * t_1)))) / (1.0 - t_1);
}

Python

def code(x):
	t_0 = 0.08333333333333333 + ((x * x) * (0.002777777777777778 + ((x * x) * 4.96031746031746e-5)))
	t_1 = (x * x) * t_0
	return ((x * x) * (1.0 - ((x * x) * (t_0 * t_1)))) / (1.0 - t_1)

Julia

function code(x)
	t_0 = Float64(0.08333333333333333 + Float64(Float64(x * x) * Float64(0.002777777777777778 + Float64(Float64(x * x) * 4.96031746031746e-5))))
	t_1 = Float64(Float64(x * x) * t_0)
	return Float64(Float64(Float64(x * x) * Float64(1.0 - Float64(Float64(x * x) * Float64(t_0 * t_1)))) / Float64(1.0 - t_1))
end

MATLAB

function tmp = code(x)
	t_0 = 0.08333333333333333 + ((x * x) * (0.002777777777777778 + ((x * x) * 4.96031746031746e-5)));
	t_1 = (x * x) * t_0;
	tmp = ((x * x) * (1.0 - ((x * x) * (t_0 * t_1)))) / (1.0 - t_1);
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.08333333333333333 + N[(N[(x * x), $MachinePrecision] * N[(0.002777777777777778 + N[(N[(x * x), $MachinePrecision] * 4.96031746031746e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(N[(N[(x * x), $MachinePrecision] * N[(1.0 - N[(N[(x * x), $MachinePrecision] * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 54.2%

   \[\left(e^{x} - 2\right) + e^{-x} \]
2. Step-by-step derivation

   1. associate-+l- N/A

      \[\leadsto e^{x} - \color{blue}{\left(2 - e^{\mathsf{neg}\left(x\right)}\right)} \]

   2. sub-neg N/A

      \[\leadsto e^{x} + \color{blue}{\left(\mathsf{neg}\left(\left(2 - e^{\mathsf{neg}\left(x\right)}\right)\right)\right)} \]

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

      \[\leadsto \mathsf{+.f64}\left(\left(e^{x}\right), \color{blue}{\left(\mathsf{neg}\left(\left(2 - e^{\mathsf{neg}\left(x\right)}\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{\left(2 - e^{\mathsf{neg}\left(x\right)}\right)}\right)\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{neg}\left(\left(2 + \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(e^{\mathsf{neg}\left(x\right)}\right)\right) + 2\right)\right)\right)\right) \]

   7. distribute-neg-in N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right) \]

   8. remove-double-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(e^{\mathsf{neg}\left(x\right)} + \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\left(e^{\mathsf{neg}\left(x\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right) \]

   10. exp-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\left(\frac{1}{e^{x}}\right), \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(e^{x}\right)\right), \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right)\right) \]

   13. metadata-eval 54.1%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right), -2\right)\right) \]

3. Simplified 54.1%

   \[\leadsto \color{blue}{e^{x} + \left(\frac{1}{e^{x}} + -2\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)} \]
6. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)} \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)}\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{12}} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{360}} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{360}} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{360}, \color{blue}{\left(\frac{1}{20160} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{360}, \left({x}^{2} \cdot \color{blue}{\frac{1}{20160}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{360}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{20160}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

7. Simplified 99.2%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(0.08333333333333333 + \left(x \cdot x\right) \cdot \left(0.002777777777777778 + \left(x \cdot x\right) \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right)\right)\right)} \]
8. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{12} + \left(x \cdot x\right) \cdot \left(\frac{1}{360} + \left(x \cdot x\right) \cdot \frac{1}{20160}\right)\right)\right)\right)} \]

   2. flip-+ N/A

      \[\leadsto \left(x \cdot x\right) \cdot \frac{1 \cdot 1 - \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \left(x \cdot x\right) \cdot \left(\frac{1}{360} + \left(x \cdot x\right) \cdot \frac{1}{20160}\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \left(x \cdot x\right) \cdot \left(\frac{1}{360} + \left(x \cdot x\right) \cdot \frac{1}{20160}\right)\right)\right)\right)}{\color{blue}{1 - x \cdot \left(x \cdot \left(\frac{1}{12} + \left(x \cdot x\right) \cdot \left(\frac{1}{360} + \left(x \cdot x\right) \cdot \frac{1}{20160}\right)\right)\right)}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\left(x \cdot x\right) \cdot \left(1 \cdot 1 - \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \left(x \cdot x\right) \cdot \left(\frac{1}{360} + \left(x \cdot x\right) \cdot \frac{1}{20160}\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \left(x \cdot x\right) \cdot \left(\frac{1}{360} + \left(x \cdot x\right) \cdot \frac{1}{20160}\right)\right)\right)\right)\right)}{\color{blue}{1 - x \cdot \left(x \cdot \left(\frac{1}{12} + \left(x \cdot x\right) \cdot \left(\frac{1}{360} + \left(x \cdot x\right) \cdot \frac{1}{20160}\right)\right)\right)}} \]

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

      \[\leadsto \mathsf{/.f64}\left(\left(\left(x \cdot x\right) \cdot \left(1 \cdot 1 - \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \left(x \cdot x\right) \cdot \left(\frac{1}{360} + \left(x \cdot x\right) \cdot \frac{1}{20160}\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \left(x \cdot x\right) \cdot \left(\frac{1}{360} + \left(x \cdot x\right) \cdot \frac{1}{20160}\right)\right)\right)\right)\right)\right), \color{blue}{\left(1 - x \cdot \left(x \cdot \left(\frac{1}{12} + \left(x \cdot x\right) \cdot \left(\frac{1}{360} + \left(x \cdot x\right) \cdot \frac{1}{20160}\right)\right)\right)\right)}\right) \]

9. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\frac{\left(x \cdot x\right) \cdot \left(1 - \left(x \cdot x\right) \cdot \left(\left(0.08333333333333333 + \left(x \cdot x\right) \cdot \left(0.002777777777777778 + \left(x \cdot x\right) \cdot 4.96031746031746 \cdot 10^{-5}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.08333333333333333 + \left(x \cdot x\right) \cdot \left(0.002777777777777778 + \left(x \cdot x\right) \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right)\right)\right)}{1 - \left(x \cdot x\right) \cdot \left(0.08333333333333333 + \left(x \cdot x\right) \cdot \left(0.002777777777777778 + \left(x \cdot x\right) \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)}} \]
10. Add Preprocessing

Alternative 2: 99.2% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot \left(0.002777777777777778 + \left(x \cdot x\right) \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (* x x)
  (+
   1.0
   (*
    (* x x)
    (+
     0.08333333333333333
     (* x (* x (+ 0.002777777777777778 (* (* x x) 4.96031746031746e-5)))))))))

C

double code(double x) {
	return (x * x) * (1.0 + ((x * x) * (0.08333333333333333 + (x * (x * (0.002777777777777778 + ((x * x) * 4.96031746031746e-5)))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * x) * (1.0d0 + ((x * x) * (0.08333333333333333d0 + (x * (x * (0.002777777777777778d0 + ((x * x) * 4.96031746031746d-5)))))))
end function

Java

public static double code(double x) {
	return (x * x) * (1.0 + ((x * x) * (0.08333333333333333 + (x * (x * (0.002777777777777778 + ((x * x) * 4.96031746031746e-5)))))));
}

Python

def code(x):
	return (x * x) * (1.0 + ((x * x) * (0.08333333333333333 + (x * (x * (0.002777777777777778 + ((x * x) * 4.96031746031746e-5)))))))

Julia

function code(x)
	return Float64(Float64(x * x) * Float64(1.0 + Float64(Float64(x * x) * Float64(0.08333333333333333 + Float64(x * Float64(x * Float64(0.002777777777777778 + Float64(Float64(x * x) * 4.96031746031746e-5))))))))
end

MATLAB

function tmp = code(x)
	tmp = (x * x) * (1.0 + ((x * x) * (0.08333333333333333 + (x * (x * (0.002777777777777778 + ((x * x) * 4.96031746031746e-5)))))));
end

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.08333333333333333 + N[(x * N[(x * N[(0.002777777777777778 + N[(N[(x * x), $MachinePrecision] * 4.96031746031746e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.2%

   \[\left(e^{x} - 2\right) + e^{-x} \]
2. Step-by-step derivation

   1. associate-+l- N/A

      \[\leadsto e^{x} - \color{blue}{\left(2 - e^{\mathsf{neg}\left(x\right)}\right)} \]

   2. sub-neg N/A

      \[\leadsto e^{x} + \color{blue}{\left(\mathsf{neg}\left(\left(2 - e^{\mathsf{neg}\left(x\right)}\right)\right)\right)} \]

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

      \[\leadsto \mathsf{+.f64}\left(\left(e^{x}\right), \color{blue}{\left(\mathsf{neg}\left(\left(2 - e^{\mathsf{neg}\left(x\right)}\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{\left(2 - e^{\mathsf{neg}\left(x\right)}\right)}\right)\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{neg}\left(\left(2 + \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(e^{\mathsf{neg}\left(x\right)}\right)\right) + 2\right)\right)\right)\right) \]

   7. distribute-neg-in N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right) \]

   8. remove-double-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(e^{\mathsf{neg}\left(x\right)} + \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\left(e^{\mathsf{neg}\left(x\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right) \]

   10. exp-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\left(\frac{1}{e^{x}}\right), \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(e^{x}\right)\right), \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right)\right) \]

   13. metadata-eval 54.1%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right), -2\right)\right) \]

3. Simplified 54.1%

   \[\leadsto \color{blue}{e^{x} + \left(\frac{1}{e^{x}} + -2\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)} \]
6. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)} \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)}\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{12}} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{360}} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{360}} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{360}, \color{blue}{\left(\frac{1}{20160} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{360}, \left({x}^{2} \cdot \color{blue}{\frac{1}{20160}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{360}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{20160}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

7. Simplified 99.2%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(0.08333333333333333 + \left(x \cdot x\right) \cdot \left(0.002777777777777778 + \left(x \cdot x\right) \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right)\right)\right)} \]

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)} \]
9. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{12}} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{12}} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{12}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{360}} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{12}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{360}, \color{blue}{\left(\frac{1}{20160} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{360}, \left({x}^{2} \cdot \color{blue}{\frac{1}{20160}}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{360}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{20160}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{360}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{20160}\right)\right)\right)\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f64 99.2%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{360}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{20160}\right)\right)\right)\right)\right)\right)\right)\right) \]

10. Simplified 99.2%

    \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot \left(0.002777777777777778 + \left(x \cdot x\right) \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right)\right)} \]
11. Add Preprocessing

Alternative 3: 99.2% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x + x \cdot \left(\left(x \cdot x\right) \cdot \left(0.08333333333333333 + \left(x \cdot x\right) \cdot \left(0.002777777777777778 + \left(x \cdot x\right) \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (+
   x
   (*
    x
    (*
     (* x x)
     (+
      0.08333333333333333
      (*
       (* x x)
       (+ 0.002777777777777778 (* (* x x) 4.96031746031746e-5)))))))))

C

double code(double x) {
	return x * (x + (x * ((x * x) * (0.08333333333333333 + ((x * x) * (0.002777777777777778 + ((x * x) * 4.96031746031746e-5)))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x + (x * ((x * x) * (0.08333333333333333d0 + ((x * x) * (0.002777777777777778d0 + ((x * x) * 4.96031746031746d-5)))))))
end function

Java

public static double code(double x) {
	return x * (x + (x * ((x * x) * (0.08333333333333333 + ((x * x) * (0.002777777777777778 + ((x * x) * 4.96031746031746e-5)))))));
}

Python

def code(x):
	return x * (x + (x * ((x * x) * (0.08333333333333333 + ((x * x) * (0.002777777777777778 + ((x * x) * 4.96031746031746e-5)))))))

Julia

function code(x)
	return Float64(x * Float64(x + Float64(x * Float64(Float64(x * x) * Float64(0.08333333333333333 + Float64(Float64(x * x) * Float64(0.002777777777777778 + Float64(Float64(x * x) * 4.96031746031746e-5))))))))
end

MATLAB

function tmp = code(x)
	tmp = x * (x + (x * ((x * x) * (0.08333333333333333 + ((x * x) * (0.002777777777777778 + ((x * x) * 4.96031746031746e-5)))))));
end

Wolfram

code[x_] := N[(x * N[(x + N[(x * N[(N[(x * x), $MachinePrecision] * N[(0.08333333333333333 + N[(N[(x * x), $MachinePrecision] * N[(0.002777777777777778 + N[(N[(x * x), $MachinePrecision] * 4.96031746031746e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.2%

   \[\left(e^{x} - 2\right) + e^{-x} \]
2. Step-by-step derivation

   1. associate-+l- N/A

      \[\leadsto e^{x} - \color{blue}{\left(2 - e^{\mathsf{neg}\left(x\right)}\right)} \]

   2. sub-neg N/A

      \[\leadsto e^{x} + \color{blue}{\left(\mathsf{neg}\left(\left(2 - e^{\mathsf{neg}\left(x\right)}\right)\right)\right)} \]

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

      \[\leadsto \mathsf{+.f64}\left(\left(e^{x}\right), \color{blue}{\left(\mathsf{neg}\left(\left(2 - e^{\mathsf{neg}\left(x\right)}\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{\left(2 - e^{\mathsf{neg}\left(x\right)}\right)}\right)\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{neg}\left(\left(2 + \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(e^{\mathsf{neg}\left(x\right)}\right)\right) + 2\right)\right)\right)\right) \]

   7. distribute-neg-in N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right) \]

   8. remove-double-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(e^{\mathsf{neg}\left(x\right)} + \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\left(e^{\mathsf{neg}\left(x\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right) \]

   10. exp-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\left(\frac{1}{e^{x}}\right), \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(e^{x}\right)\right), \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right)\right) \]

   13. metadata-eval 54.1%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right), -2\right)\right) \]

3. Simplified 54.1%

   \[\leadsto \color{blue}{e^{x} + \left(\frac{1}{e^{x}} + -2\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)} \]
6. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)} \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)}\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{12}} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{360}} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{360}} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{360}, \color{blue}{\left(\frac{1}{20160} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{360}, \left({x}^{2} \cdot \color{blue}{\frac{1}{20160}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{360}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{20160}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

7. Simplified 99.2%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(0.08333333333333333 + \left(x \cdot x\right) \cdot \left(0.002777777777777778 + \left(x \cdot x\right) \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right)\right)\right)} \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \left(x \cdot x\right) \cdot \left(\frac{1}{360} + \left(x \cdot x\right) \cdot \frac{1}{20160}\right)\right)\right) + \color{blue}{1}\right)\right)\right) \]

   2. distribute-rgt-in N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\left(x \cdot \left(x \cdot \left(\frac{1}{12} + \left(x \cdot x\right) \cdot \left(\frac{1}{360} + \left(x \cdot x\right) \cdot \frac{1}{20160}\right)\right)\right)\right) \cdot x + \color{blue}{1 \cdot x}\right)\right) \]

   3. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\left(x \cdot \left(x \cdot \left(\frac{1}{12} + \left(x \cdot x\right) \cdot \left(\frac{1}{360} + \left(x \cdot x\right) \cdot \frac{1}{20160}\right)\right)\right)\right) \cdot x + x\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(x \cdot \left(x \cdot \left(\frac{1}{12} + \left(x \cdot x\right) \cdot \left(\frac{1}{360} + \left(x \cdot x\right) \cdot \frac{1}{20160}\right)\right)\right)\right) \cdot x\right), \color{blue}{x}\right)\right) \]

9. Applied egg-rr 99.2%

   \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\left(x \cdot x\right) \cdot \left(0.08333333333333333 + \left(x \cdot x\right) \cdot \left(0.002777777777777778 + \left(x \cdot x\right) \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right) + x\right)} \]

10. Final simplification 99.2%

    \[\leadsto x \cdot \left(x + x \cdot \left(\left(x \cdot x\right) \cdot \left(0.08333333333333333 + \left(x \cdot x\right) \cdot \left(0.002777777777777778 + \left(x \cdot x\right) \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right)\right) \]
11. Add Preprocessing

Alternative 4: 99.2% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(0.08333333333333333 + \left(x \cdot x\right) \cdot \left(0.002777777777777778 + \left(x \cdot x\right) \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (*
   x
   (+
    1.0
    (*
     x
     (*
      x
      (+
       0.08333333333333333
       (*
        (* x x)
        (+ 0.002777777777777778 (* (* x x) 4.96031746031746e-5))))))))))

C

double code(double x) {
	return x * (x * (1.0 + (x * (x * (0.08333333333333333 + ((x * x) * (0.002777777777777778 + ((x * x) * 4.96031746031746e-5))))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x * (1.0d0 + (x * (x * (0.08333333333333333d0 + ((x * x) * (0.002777777777777778d0 + ((x * x) * 4.96031746031746d-5))))))))
end function

Java

public static double code(double x) {
	return x * (x * (1.0 + (x * (x * (0.08333333333333333 + ((x * x) * (0.002777777777777778 + ((x * x) * 4.96031746031746e-5))))))));
}

Python

def code(x):
	return x * (x * (1.0 + (x * (x * (0.08333333333333333 + ((x * x) * (0.002777777777777778 + ((x * x) * 4.96031746031746e-5))))))))

Julia

function code(x)
	return Float64(x * Float64(x * Float64(1.0 + Float64(x * Float64(x * Float64(0.08333333333333333 + Float64(Float64(x * x) * Float64(0.002777777777777778 + Float64(Float64(x * x) * 4.96031746031746e-5)))))))))
end

MATLAB

function tmp = code(x)
	tmp = x * (x * (1.0 + (x * (x * (0.08333333333333333 + ((x * x) * (0.002777777777777778 + ((x * x) * 4.96031746031746e-5))))))));
end

Wolfram

code[x_] := N[(x * N[(x * N[(1.0 + N[(x * N[(x * N[(0.08333333333333333 + N[(N[(x * x), $MachinePrecision] * N[(0.002777777777777778 + N[(N[(x * x), $MachinePrecision] * 4.96031746031746e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.2%

   \[\left(e^{x} - 2\right) + e^{-x} \]
2. Step-by-step derivation

   1. associate-+l- N/A

      \[\leadsto e^{x} - \color{blue}{\left(2 - e^{\mathsf{neg}\left(x\right)}\right)} \]

   2. sub-neg N/A

      \[\leadsto e^{x} + \color{blue}{\left(\mathsf{neg}\left(\left(2 - e^{\mathsf{neg}\left(x\right)}\right)\right)\right)} \]

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

      \[\leadsto \mathsf{+.f64}\left(\left(e^{x}\right), \color{blue}{\left(\mathsf{neg}\left(\left(2 - e^{\mathsf{neg}\left(x\right)}\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{\left(2 - e^{\mathsf{neg}\left(x\right)}\right)}\right)\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{neg}\left(\left(2 + \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(e^{\mathsf{neg}\left(x\right)}\right)\right) + 2\right)\right)\right)\right) \]

   7. distribute-neg-in N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right) \]

   8. remove-double-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(e^{\mathsf{neg}\left(x\right)} + \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\left(e^{\mathsf{neg}\left(x\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right) \]

   10. exp-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\left(\frac{1}{e^{x}}\right), \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(e^{x}\right)\right), \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right)\right) \]

   13. metadata-eval 54.1%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right), -2\right)\right) \]

3. Simplified 54.1%

   \[\leadsto \color{blue}{e^{x} + \left(\frac{1}{e^{x}} + -2\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)} \]
6. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)} \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)}\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{12}} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{360}} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{360}} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{360}, \color{blue}{\left(\frac{1}{20160} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{360}, \left({x}^{2} \cdot \color{blue}{\frac{1}{20160}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{360}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{20160}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

7. Simplified 99.2%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(0.08333333333333333 + \left(x \cdot x\right) \cdot \left(0.002777777777777778 + \left(x \cdot x\right) \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right)\right)\right)} \]
8. Add Preprocessing

Alternative 5: 99.1% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot 0.002777777777777778\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (* x x)
  (+
   1.0
   (* (* x x) (+ 0.08333333333333333 (* x (* x 0.002777777777777778)))))))

C

double code(double x) {
	return (x * x) * (1.0 + ((x * x) * (0.08333333333333333 + (x * (x * 0.002777777777777778)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * x) * (1.0d0 + ((x * x) * (0.08333333333333333d0 + (x * (x * 0.002777777777777778d0)))))
end function

Java

public static double code(double x) {
	return (x * x) * (1.0 + ((x * x) * (0.08333333333333333 + (x * (x * 0.002777777777777778)))));
}

Python

def code(x):
	return (x * x) * (1.0 + ((x * x) * (0.08333333333333333 + (x * (x * 0.002777777777777778)))))

Julia

function code(x)
	return Float64(Float64(x * x) * Float64(1.0 + Float64(Float64(x * x) * Float64(0.08333333333333333 + Float64(x * Float64(x * 0.002777777777777778))))))
end

MATLAB

function tmp = code(x)
	tmp = (x * x) * (1.0 + ((x * x) * (0.08333333333333333 + (x * (x * 0.002777777777777778)))));
end

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.08333333333333333 + N[(x * N[(x * 0.002777777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.2%

   \[\left(e^{x} - 2\right) + e^{-x} \]
2. Step-by-step derivation

   1. associate-+l- N/A

      \[\leadsto e^{x} - \color{blue}{\left(2 - e^{\mathsf{neg}\left(x\right)}\right)} \]

   2. sub-neg N/A

      \[\leadsto e^{x} + \color{blue}{\left(\mathsf{neg}\left(\left(2 - e^{\mathsf{neg}\left(x\right)}\right)\right)\right)} \]

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

      \[\leadsto \mathsf{+.f64}\left(\left(e^{x}\right), \color{blue}{\left(\mathsf{neg}\left(\left(2 - e^{\mathsf{neg}\left(x\right)}\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{\left(2 - e^{\mathsf{neg}\left(x\right)}\right)}\right)\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{neg}\left(\left(2 + \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(e^{\mathsf{neg}\left(x\right)}\right)\right) + 2\right)\right)\right)\right) \]

   7. distribute-neg-in N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right) \]

   8. remove-double-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(e^{\mathsf{neg}\left(x\right)} + \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\left(e^{\mathsf{neg}\left(x\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right) \]

   10. exp-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\left(\frac{1}{e^{x}}\right), \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(e^{x}\right)\right), \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right)\right) \]

   13. metadata-eval 54.1%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right), -2\right)\right) \]

3. Simplified 54.1%

   \[\leadsto \color{blue}{e^{x} + \left(\frac{1}{e^{x}} + -2\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)} \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)}\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{12}} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{12}} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\frac{1}{360} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{12}, \left({x}^{2} \cdot \color{blue}{\frac{1}{360}}\right)\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{12}, \left(\left(x \cdot x\right) \cdot \frac{1}{360}\right)\right)\right)\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{12}, \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{360}\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \frac{1}{360}\right)}\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f64 99.1%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{360}}\right)\right)\right)\right)\right)\right) \]

7. Simplified 99.1%

   \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.08333333333333333 + x \cdot \left(x \cdot 0.002777777777777778\right)\right)\right)} \]
8. Add Preprocessing

Alternative 6: 98.9% accurate, 18.7× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot 0.08333333333333333\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (* x x) (+ 1.0 (* (* x x) 0.08333333333333333))))

C

double code(double x) {
	return (x * x) * (1.0 + ((x * x) * 0.08333333333333333));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * x) * (1.0d0 + ((x * x) * 0.08333333333333333d0))
end function

Java

public static double code(double x) {
	return (x * x) * (1.0 + ((x * x) * 0.08333333333333333));
}

Python

def code(x):
	return (x * x) * (1.0 + ((x * x) * 0.08333333333333333))

Julia

function code(x)
	return Float64(Float64(x * x) * Float64(1.0 + Float64(Float64(x * x) * 0.08333333333333333)))
end

MATLAB

function tmp = code(x)
	tmp = (x * x) * (1.0 + ((x * x) * 0.08333333333333333));
end

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.2%

   \[\left(e^{x} - 2\right) + e^{-x} \]
2. Step-by-step derivation

   1. associate-+l- N/A

      \[\leadsto e^{x} - \color{blue}{\left(2 - e^{\mathsf{neg}\left(x\right)}\right)} \]

   2. sub-neg N/A

      \[\leadsto e^{x} + \color{blue}{\left(\mathsf{neg}\left(\left(2 - e^{\mathsf{neg}\left(x\right)}\right)\right)\right)} \]

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

      \[\leadsto \mathsf{+.f64}\left(\left(e^{x}\right), \color{blue}{\left(\mathsf{neg}\left(\left(2 - e^{\mathsf{neg}\left(x\right)}\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{\left(2 - e^{\mathsf{neg}\left(x\right)}\right)}\right)\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{neg}\left(\left(2 + \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(e^{\mathsf{neg}\left(x\right)}\right)\right) + 2\right)\right)\right)\right) \]

   7. distribute-neg-in N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right) \]

   8. remove-double-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(e^{\mathsf{neg}\left(x\right)} + \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\left(e^{\mathsf{neg}\left(x\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right) \]

   10. exp-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\left(\frac{1}{e^{x}}\right), \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(e^{x}\right)\right), \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right)\right) \]

   13. metadata-eval 54.1%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right), -2\right)\right) \]

3. Simplified 54.1%

   \[\leadsto \color{blue}{e^{x} + \left(\frac{1}{e^{x}} + -2\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right)} \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(1 + \frac{1}{12} \cdot {x}^{2}\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{1} + \frac{1}{12} \cdot {x}^{2}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{1} + \frac{1}{12} \cdot {x}^{2}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{12} \cdot {x}^{2}\right)}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \color{blue}{\frac{1}{12}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{12}}\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{12}\right)\right)\right) \]

   8. *-lowering-*.f64 99.1%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{12}\right)\right)\right) \]

7. Simplified 99.1%

   \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot 0.08333333333333333\right)} \]
8. Add Preprocessing

Alternative 7: 98.9% accurate, 18.7× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x + x \cdot \left(x \cdot \left(x \cdot 0.08333333333333333\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (+ x (* x (* x (* x 0.08333333333333333))))))

C

double code(double x) {
	return x * (x + (x * (x * (x * 0.08333333333333333))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x + (x * (x * (x * 0.08333333333333333d0))))
end function

Java

public static double code(double x) {
	return x * (x + (x * (x * (x * 0.08333333333333333))));
}

Python

def code(x):
	return x * (x + (x * (x * (x * 0.08333333333333333))))

Julia

function code(x)
	return Float64(x * Float64(x + Float64(x * Float64(x * Float64(x * 0.08333333333333333)))))
end

MATLAB

function tmp = code(x)
	tmp = x * (x + (x * (x * (x * 0.08333333333333333))));
end

Wolfram

code[x_] := N[(x * N[(x + N[(x * N[(x * N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.2%

   \[\left(e^{x} - 2\right) + e^{-x} \]
2. Step-by-step derivation

   1. associate-+l- N/A

      \[\leadsto e^{x} - \color{blue}{\left(2 - e^{\mathsf{neg}\left(x\right)}\right)} \]

   2. sub-neg N/A

      \[\leadsto e^{x} + \color{blue}{\left(\mathsf{neg}\left(\left(2 - e^{\mathsf{neg}\left(x\right)}\right)\right)\right)} \]

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

      \[\leadsto \mathsf{+.f64}\left(\left(e^{x}\right), \color{blue}{\left(\mathsf{neg}\left(\left(2 - e^{\mathsf{neg}\left(x\right)}\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{\left(2 - e^{\mathsf{neg}\left(x\right)}\right)}\right)\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{neg}\left(\left(2 + \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(e^{\mathsf{neg}\left(x\right)}\right)\right) + 2\right)\right)\right)\right) \]

   7. distribute-neg-in N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right) \]

   8. remove-double-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(e^{\mathsf{neg}\left(x\right)} + \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\left(e^{\mathsf{neg}\left(x\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right) \]

   10. exp-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\left(\frac{1}{e^{x}}\right), \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(e^{x}\right)\right), \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right)\right) \]

   13. metadata-eval 54.1%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right), -2\right)\right) \]

3. Simplified 54.1%

   \[\leadsto \color{blue}{e^{x} + \left(\frac{1}{e^{x}} + -2\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)} \]
6. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)} \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)}\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{12}} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{360}} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{360}} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{360}, \color{blue}{\left(\frac{1}{20160} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{360}, \left({x}^{2} \cdot \color{blue}{\frac{1}{20160}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{360}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{20160}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

7. Simplified 99.2%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(0.08333333333333333 + \left(x \cdot x\right) \cdot \left(0.002777777777777778 + \left(x \cdot x\right) \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right)\right)\right)} \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{12} \cdot x\right)}\right)\right)\right)\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{1}{12}}\right)\right)\right)\right)\right) \]

   2. *-lowering-*.f64 99.0%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{12}}\right)\right)\right)\right)\right) \]

10. Simplified 99.0%

    \[\leadsto x \cdot \left(x \cdot \left(1 + x \cdot \color{blue}{\left(x \cdot 0.08333333333333333\right)}\right)\right) \]
11. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{12}\right) + \color{blue}{1}\right)\right)\right) \]

    2. distribute-rgt-in N/A

       \[\leadsto \mathsf{*.f64}\left(x, \left(\left(x \cdot \left(x \cdot \frac{1}{12}\right)\right) \cdot x + \color{blue}{1 \cdot x}\right)\right) \]

    3. *-lft-identity N/A

       \[\leadsto \mathsf{*.f64}\left(x, \left(\left(x \cdot \left(x \cdot \frac{1}{12}\right)\right) \cdot x + x\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(x \cdot \left(x \cdot \frac{1}{12}\right)\right) \cdot x\right), \color{blue}{x}\right)\right) \]

    5. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{12}\right)\right)\right), x\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \frac{1}{12}\right)\right)\right), x\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{12}\right)\right)\right), x\right)\right) \]

    8. *-lowering-*.f64 99.0%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{12}\right)\right)\right), x\right)\right) \]

12. Applied egg-rr 99.0%

    \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot 0.08333333333333333\right)\right) + x\right)} \]

13. Final simplification 99.0%

    \[\leadsto x \cdot \left(x + x \cdot \left(x \cdot \left(x \cdot 0.08333333333333333\right)\right)\right) \]
14. Add Preprocessing

Alternative 8: 98.9% accurate, 18.7× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot 0.08333333333333333\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (* x (+ 1.0 (* x (* x 0.08333333333333333))))))

C

double code(double x) {
	return x * (x * (1.0 + (x * (x * 0.08333333333333333))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x * (1.0d0 + (x * (x * 0.08333333333333333d0))))
end function

Java

public static double code(double x) {
	return x * (x * (1.0 + (x * (x * 0.08333333333333333))));
}

Python

def code(x):
	return x * (x * (1.0 + (x * (x * 0.08333333333333333))))

Julia

function code(x)
	return Float64(x * Float64(x * Float64(1.0 + Float64(x * Float64(x * 0.08333333333333333)))))
end

MATLAB

function tmp = code(x)
	tmp = x * (x * (1.0 + (x * (x * 0.08333333333333333))));
end

Wolfram

code[x_] := N[(x * N[(x * N[(1.0 + N[(x * N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.2%

   \[\left(e^{x} - 2\right) + e^{-x} \]
2. Step-by-step derivation

   1. associate-+l- N/A

      \[\leadsto e^{x} - \color{blue}{\left(2 - e^{\mathsf{neg}\left(x\right)}\right)} \]

   2. sub-neg N/A

      \[\leadsto e^{x} + \color{blue}{\left(\mathsf{neg}\left(\left(2 - e^{\mathsf{neg}\left(x\right)}\right)\right)\right)} \]

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

      \[\leadsto \mathsf{+.f64}\left(\left(e^{x}\right), \color{blue}{\left(\mathsf{neg}\left(\left(2 - e^{\mathsf{neg}\left(x\right)}\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{\left(2 - e^{\mathsf{neg}\left(x\right)}\right)}\right)\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{neg}\left(\left(2 + \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(e^{\mathsf{neg}\left(x\right)}\right)\right) + 2\right)\right)\right)\right) \]

   7. distribute-neg-in N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right) \]

   8. remove-double-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(e^{\mathsf{neg}\left(x\right)} + \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\left(e^{\mathsf{neg}\left(x\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right) \]

   10. exp-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\left(\frac{1}{e^{x}}\right), \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(e^{x}\right)\right), \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right)\right) \]

   13. metadata-eval 54.1%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right), -2\right)\right) \]

3. Simplified 54.1%

   \[\leadsto \color{blue}{e^{x} + \left(\frac{1}{e^{x}} + -2\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)} \]
6. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)} \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)}\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{12}} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{360}} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{360}} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{360}, \color{blue}{\left(\frac{1}{20160} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{360}, \left({x}^{2} \cdot \color{blue}{\frac{1}{20160}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{360}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{20160}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

7. Simplified 99.2%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(0.08333333333333333 + \left(x \cdot x\right) \cdot \left(0.002777777777777778 + \left(x \cdot x\right) \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right)\right)\right)} \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{12} \cdot x\right)}\right)\right)\right)\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{1}{12}}\right)\right)\right)\right)\right) \]

   2. *-lowering-*.f64 99.0%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{12}}\right)\right)\right)\right)\right) \]

10. Simplified 99.0%

    \[\leadsto x \cdot \left(x \cdot \left(1 + x \cdot \color{blue}{\left(x \cdot 0.08333333333333333\right)}\right)\right) \]
11. Add Preprocessing

Alternative 9: 98.4% accurate, 68.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return x * x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.2%

   \[\left(e^{x} - 2\right) + e^{-x} \]
2. Step-by-step derivation

   1. associate-+l- N/A

      \[\leadsto e^{x} - \color{blue}{\left(2 - e^{\mathsf{neg}\left(x\right)}\right)} \]

   2. sub-neg N/A

      \[\leadsto e^{x} + \color{blue}{\left(\mathsf{neg}\left(\left(2 - e^{\mathsf{neg}\left(x\right)}\right)\right)\right)} \]

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

      \[\leadsto \mathsf{+.f64}\left(\left(e^{x}\right), \color{blue}{\left(\mathsf{neg}\left(\left(2 - e^{\mathsf{neg}\left(x\right)}\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{neg}\left(\color{blue}{\left(2 - e^{\mathsf{neg}\left(x\right)}\right)}\right)\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{neg}\left(\left(2 + \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right)\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(e^{\mathsf{neg}\left(x\right)}\right)\right) + 2\right)\right)\right)\right) \]

   7. distribute-neg-in N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(e^{\mathsf{neg}\left(x\right)}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right) \]

   8. remove-double-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \left(e^{\mathsf{neg}\left(x\right)} + \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\left(e^{\mathsf{neg}\left(x\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right) \]

   10. exp-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\left(\frac{1}{e^{x}}\right), \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(e^{x}\right)\right), \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right)\right) \]

   13. metadata-eval 54.1%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right), -2\right)\right) \]

3. Simplified 54.1%

   \[\leadsto \color{blue}{e^{x} + \left(\frac{1}{e^{x}} + -2\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2}} \]
6. Step-by-step derivation

   1. unpow2 N/A

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

   2. *-lowering-*.f64 98.7%

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

7. Simplified 98.7%

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

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sinh \left(\frac{x}{2}\right)\\ 4 \cdot \left(t\_0 \cdot t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sinh (/ x 2.0)))) (* 4.0 (* t_0 t_0))))

C

double code(double x) {
	double t_0 = sinh((x / 2.0));
	return 4.0 * (t_0 * t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sinh((x / 2.0d0))
    code = 4.0d0 * (t_0 * t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.sinh((x / 2.0));
	return 4.0 * (t_0 * t_0);
}

Python

def code(x):
	t_0 = math.sinh((x / 2.0))
	return 4.0 * (t_0 * t_0)

Julia

function code(x)
	t_0 = sinh(Float64(x / 2.0))
	return Float64(4.0 * Float64(t_0 * t_0))
end

MATLAB

function tmp = code(x)
	t_0 = sinh((x / 2.0));
	tmp = 4.0 * (t_0 * t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sinh[N[(x / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024186 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64
  :pre (<= (fabs x) 710.0)

  :alt
  (! :herbie-platform default (* 4 (* (sinh (/ x 2)) (sinh (/ x 2)))))

  (+ (- (exp x) 2.0) (exp (- x))))