Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 7.4s

Alternatives: 11

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\cosh x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (cosh x)))

C

double code(double x) {
	return 1.0 / cosh(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / cosh(x)
end function

Java

public static double code(double x) {
	return 1.0 / Math.cosh(x);
}

Python

def code(x):
	return 1.0 / math.cosh(x)

Julia

function code(x)
	return Float64(1.0 / cosh(x))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / cosh(x);
end

Wolfram

code[x_] := N[(1.0 / N[Cosh[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. clear-num N/A

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

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

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

   3. cosh-def N/A

      \[\leadsto \mathsf{/.f64}\left(1, \cosh x\right) \]

   4. cosh-lowering-cosh.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cosh.f64}\left(x\right)\right) \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{1}{\cosh x}} \]
5. Add Preprocessing

Alternative 2: 94.9% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \frac{32}{\left(8 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(4 - x \cdot \left(x \cdot \left(x \cdot x + -2\right)\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  32.0
  (*
   (+ 8.0 (* (* x x) (* x (* x (* x x)))))
   (- 4.0 (* x (* x (+ (* x x) -2.0)))))))

C

double code(double x) {
	return 32.0 / ((8.0 + ((x * x) * (x * (x * (x * x))))) * (4.0 - (x * (x * ((x * x) + -2.0)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 32.0d0 / ((8.0d0 + ((x * x) * (x * (x * (x * x))))) * (4.0d0 - (x * (x * ((x * x) + (-2.0d0))))))
end function

Java

public static double code(double x) {
	return 32.0 / ((8.0 + ((x * x) * (x * (x * (x * x))))) * (4.0 - (x * (x * ((x * x) + -2.0)))));
}

Python

def code(x):
	return 32.0 / ((8.0 + ((x * x) * (x * (x * (x * x))))) * (4.0 - (x * (x * ((x * x) + -2.0)))))

Julia

function code(x)
	return Float64(32.0 / Float64(Float64(8.0 + Float64(Float64(x * x) * Float64(x * Float64(x * Float64(x * x))))) * Float64(4.0 - Float64(x * Float64(x * Float64(Float64(x * x) + -2.0))))))
end

MATLAB

function tmp = code(x)
	tmp = 32.0 / ((8.0 + ((x * x) * (x * (x * (x * x))))) * (4.0 - (x * (x * ((x * x) + -2.0)))));
end

Wolfram

code[x_] := N[(32.0 / N[(N[(8.0 + N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(4.0 - N[(x * N[(x * N[(N[(x * x), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

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

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

   2. unpow2 N/A

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

   3. *-lowering-*.f64 77.7%

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

5. Simplified 77.7%

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

   1. flip3-+ N/A

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

   2. associate-/r/ N/A

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

   3. flip-+ N/A

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

   4. frac-times N/A

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

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

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

7. Applied egg-rr 52.4%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(16 - \left(x \cdot x\right) \cdot \left(\left(x \cdot x + -2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x + -2\right)\right)\right)\right)\right)}{\left(8 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(4 - x \cdot \left(x \cdot \left(x \cdot x + -2\right)\right)\right)}} \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{32}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), -2\right)\right)\right)\right)\right)\right) \]
9. Step-by-step derivation

10. Simplified 96.9%

    \[\leadsto \frac{\color{blue}{32}}{\left(8 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(4 - x \cdot \left(x \cdot \left(x \cdot x + -2\right)\right)\right)} \]
11. Add Preprocessing

Alternative 3: 91.9% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \frac{2}{2 + \left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.08333333333333333 + \left(x \cdot x\right) \cdot 0.002777777777777778\right)\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  2.0
  (+
   2.0
   (*
    (* x x)
    (+
     1.0
     (* x (* x (+ 0.08333333333333333 (* (* x x) 0.002777777777777778)))))))))

C

double code(double x) {
	return 2.0 / (2.0 + ((x * x) * (1.0 + (x * (x * (0.08333333333333333 + ((x * x) * 0.002777777777777778)))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / (2.0d0 + ((x * x) * (1.0d0 + (x * (x * (0.08333333333333333d0 + ((x * x) * 0.002777777777777778d0)))))))
end function

Java

public static double code(double x) {
	return 2.0 / (2.0 + ((x * x) * (1.0 + (x * (x * (0.08333333333333333 + ((x * x) * 0.002777777777777778)))))));
}

Python

def code(x):
	return 2.0 / (2.0 + ((x * x) * (1.0 + (x * (x * (0.08333333333333333 + ((x * x) * 0.002777777777777778)))))))

Julia

function code(x)
	return Float64(2.0 / Float64(2.0 + Float64(Float64(x * x) * Float64(1.0 + Float64(x * Float64(x * Float64(0.08333333333333333 + Float64(Float64(x * x) * 0.002777777777777778))))))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / (2.0 + ((x * x) * (1.0 + (x * (x * (0.08333333333333333 + ((x * x) * 0.002777777777777778)))))));
end

Wolfram

code[x_] := N[(2.0 / N[(2.0 + N[(N[(x * x), $MachinePrecision] * N[(1.0 + N[(x * N[(x * N[(0.08333333333333333 + N[(N[(x * x), $MachinePrecision] * 0.002777777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(2 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)}\right) \]
4. Step-by-step derivation

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

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

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

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

   3. unpow2 N/A

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

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

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

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

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

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

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

   10. *-commutative N/A

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

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

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

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

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

   13. *-commutative N/A

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f64 92.6%

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

5. Simplified 92.6%

   \[\leadsto \frac{2}{\color{blue}{2 + \left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.08333333333333333 + \left(x \cdot x\right) \cdot 0.002777777777777778\right)\right)\right)}} \]
6. Add Preprocessing

Alternative 4: 91.8% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \frac{2}{2 + \left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.002777777777777778\right)\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  2.0
  (+ 2.0 (* (* x x) (+ 1.0 (* x (* x (* (* x x) 0.002777777777777778))))))))

C

double code(double x) {
	return 2.0 / (2.0 + ((x * x) * (1.0 + (x * (x * ((x * x) * 0.002777777777777778))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / (2.0d0 + ((x * x) * (1.0d0 + (x * (x * ((x * x) * 0.002777777777777778d0))))))
end function

Java

public static double code(double x) {
	return 2.0 / (2.0 + ((x * x) * (1.0 + (x * (x * ((x * x) * 0.002777777777777778))))));
}

Python

def code(x):
	return 2.0 / (2.0 + ((x * x) * (1.0 + (x * (x * ((x * x) * 0.002777777777777778))))))

Julia

function code(x)
	return Float64(2.0 / Float64(2.0 + Float64(Float64(x * x) * Float64(1.0 + Float64(x * Float64(x * Float64(Float64(x * x) * 0.002777777777777778)))))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / (2.0 + ((x * x) * (1.0 + (x * (x * ((x * x) * 0.002777777777777778))))));
end

Wolfram

code[x_] := N[(2.0 / N[(2.0 + N[(N[(x * x), $MachinePrecision] * N[(1.0 + N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.002777777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(2 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)\right)}\right) \]
4. Step-by-step derivation

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

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

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

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

   3. unpow2 N/A

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

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

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

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

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

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

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

   10. *-commutative N/A

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

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

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

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

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

   13. *-commutative N/A

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f64 92.6%

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

5. Simplified 92.6%

   \[\leadsto \frac{2}{\color{blue}{2 + \left(x \cdot x\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.08333333333333333 + \left(x \cdot x\right) \cdot 0.002777777777777778\right)\right)\right)}} \]

6. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{360} \cdot {x}^{3}\right)}\right)\right)\right)\right)\right) \]
7. Step-by-step derivation

   1. unpow3 N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. *-commutative N/A

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 92.6%

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

8. Simplified 92.6%

   \[\leadsto \frac{2}{2 + \left(x \cdot x\right) \cdot \left(1 + x \cdot \color{blue}{\left(x \cdot \left(\left(x \cdot x\right) \cdot 0.002777777777777778\right)\right)}\right)} \]
9. Add Preprocessing

Alternative 5: 87.6% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 1.0 (+ 1.0 (* x (* x (+ 0.5 (* (* x x) 0.041666666666666664)))))))

C

double code(double x) {
	return 1.0 / (1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (1.0d0 + (x * (x * (0.5d0 + ((x * x) * 0.041666666666666664d0)))))
end function

Java

public static double code(double x) {
	return 1.0 / (1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664)))));
}

Python

def code(x):
	return 1.0 / (1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664)))))

Julia

function code(x)
	return Float64(1.0 / Float64(1.0 + Float64(x * Float64(x * Float64(0.5 + Float64(Float64(x * x) * 0.041666666666666664))))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664)))));
end

Wolfram

code[x_] := N[(1.0 / N[(1.0 + N[(x * N[(x * N[(0.5 + N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. clear-num N/A

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

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

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

   3. cosh-def N/A

      \[\leadsto \mathsf{/.f64}\left(1, \cosh x\right) \]

   4. cosh-lowering-cosh.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{cosh.f64}\left(x\right)\right) \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{1}{\cosh x}} \]

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}\right) \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot x\right)}\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 88.9%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

7. Simplified 88.9%

   \[\leadsto \frac{1}{\color{blue}{1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)}} \]
8. Add Preprocessing

Alternative 6: 81.9% accurate, 14.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 700:\\ \;\;\;\;\frac{2}{x \cdot x + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 700.0) (/ 2.0 (+ (* x x) 2.0)) (/ -4.0 (* x (* x (* x x))))))

C

double code(double x) {
	double tmp;
	if (x <= 700.0) {
		tmp = 2.0 / ((x * x) + 2.0);
	} else {
		tmp = -4.0 / (x * (x * (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 700.0d0) then
        tmp = 2.0d0 / ((x * x) + 2.0d0)
    else
        tmp = (-4.0d0) / (x * (x * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 700.0) {
		tmp = 2.0 / ((x * x) + 2.0);
	} else {
		tmp = -4.0 / (x * (x * (x * x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 700.0:
		tmp = 2.0 / ((x * x) + 2.0)
	else:
		tmp = -4.0 / (x * (x * (x * x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 700.0)
		tmp = Float64(2.0 / Float64(Float64(x * x) + 2.0));
	else
		tmp = Float64(-4.0 / Float64(x * Float64(x * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 700.0)
		tmp = 2.0 / ((x * x) + 2.0);
	else
		tmp = -4.0 / (x * (x * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 700.0], N[(2.0 / N[(N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(-4.0 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 700

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 83.5%

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

   5. Simplified 83.5%

      \[\leadsto \frac{2}{\color{blue}{2 + x \cdot x}} \]

   if 700 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 58.2%

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

   5. Simplified 58.2%

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

      1. flip-+ N/A

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

      2. associate-/r/ N/A

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

      3. associate-*l/ N/A

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

      4. sub-neg N/A

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

      5. distribute-rgt-in N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. sub-neg N/A

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

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

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

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(4, \left(\mathsf{neg}\left(2 \cdot \left(x \cdot x\right)\right)\right)\right), \left(\color{blue}{2} \cdot 2 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(4, \left(\mathsf{neg}\left(\left(x \cdot x\right) \cdot 2\right)\right)\right), \left(2 \cdot 2 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \]

      13. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(4, \left(\left(x \cdot x\right) \cdot \left(\mathsf{neg}\left(2\right)\right)\right)\right), \left(2 \cdot \color{blue}{2} - \left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\mathsf{neg}\left(2\right)\right)\right)\right), \left(2 \cdot \color{blue}{2} - \left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\mathsf{neg}\left(2\right)\right)\right)\right), \left(2 \cdot 2 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), -2\right)\right), \left(2 \cdot 2 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), -2\right)\right), \mathsf{\_.f64}\left(\left(2 \cdot 2\right), \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), -2\right)\right), \mathsf{\_.f64}\left(4, \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot x\right)\right)\right)\right) \]

      19. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), -2\right)\right), \mathsf{\_.f64}\left(4, \left(x \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right) \]

      20. cube-unmult N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), -2\right)\right), \mathsf{\_.f64}\left(4, \left(x \cdot {x}^{\color{blue}{3}}\right)\right)\right) \]

   7. Applied egg-rr 26.1%

      \[\leadsto \color{blue}{\frac{4 + \left(x \cdot x\right) \cdot -2}{4 - x \cdot \left(x \cdot \left(x \cdot x\right)\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{4}, \mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 82.4%

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

   11. Taylor expanded in x around inf

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

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

          \[\leadsto \mathsf{/.f64}\left(-4, \color{blue}{\left({x}^{4}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(-4, \left({x}^{\left(3 + \color{blue}{1}\right)}\right)\right) \]

       3. pow-plus N/A

          \[\leadsto \mathsf{/.f64}\left(-4, \left({x}^{3} \cdot \color{blue}{x}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-4, \left(x \cdot \color{blue}{{x}^{3}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{3}\right)}\right)\right) \]

       6. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

       7. unpow2 N/A

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

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

          \[\leadsto \mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 82.4%

           \[\leadsto \mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

   13. Simplified 82.4%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 700:\\ \;\;\;\;\frac{2}{x \cdot x + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.6% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.25:\\ \;\;\;\;1 + x \cdot \left(x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.25) (+ 1.0 (* x (* x -0.5))) (/ 2.0 (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.25) {
		tmp = 1.0 + (x * (x * -0.5));
	} else {
		tmp = 2.0 / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.25d0) then
        tmp = 1.0d0 + (x * (x * (-0.5d0)))
    else
        tmp = 2.0d0 / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.25) {
		tmp = 1.0 + (x * (x * -0.5));
	} else {
		tmp = 2.0 / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.25:
		tmp = 1.0 + (x * (x * -0.5))
	else:
		tmp = 2.0 / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.25)
		tmp = Float64(1.0 + Float64(x * Float64(x * -0.5)));
	else
		tmp = Float64(2.0 / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.25)
		tmp = 1.0 + (x * (x * -0.5));
	else
		tmp = 2.0 / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.25], N[(1.0 + N[(x * N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 66.6%

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

   5. Simplified 66.6%

      \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot -0.5\right)} \]

   if 1.25 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 58.2%

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

   5. Simplified 58.2%

      \[\leadsto \frac{2}{\color{blue}{2 + x \cdot x}} \]

   6. Taylor expanded in x around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 58.2%

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

   8. Simplified 58.2%

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

Alternative 8: 87.2% accurate, 18.7× speedup

Math

\[\begin{array}{l} \\ \frac{4}{4 - x \cdot \left(x \cdot \left(x \cdot x\right)\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 4.0 (- 4.0 (* x (* x (* x x))))))

C

double code(double x) {
	return 4.0 / (4.0 - (x * (x * (x * x))));
}

Fortran

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

Java

public static double code(double x) {
	return 4.0 / (4.0 - (x * (x * (x * x))));
}

Python

def code(x):
	return 4.0 / (4.0 - (x * (x * (x * x))))

Julia

function code(x)
	return Float64(4.0 / Float64(4.0 - Float64(x * Float64(x * Float64(x * x)))))
end

MATLAB

function tmp = code(x)
	tmp = 4.0 / (4.0 - (x * (x * (x * x))));
end

Wolfram

code[x_] := N[(4.0 / N[(4.0 - N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

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

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

   2. unpow2 N/A

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

   3. *-lowering-*.f64 77.7%

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

5. Simplified 77.7%

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

   1. flip-+ N/A

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

   2. associate-/r/ N/A

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

   3. associate-*l/ N/A

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

   4. sub-neg N/A

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

   5. distribute-rgt-in N/A

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

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

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

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

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

   8. *-commutative N/A

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

   9. sub-neg N/A

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

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

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

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(4, \left(\mathsf{neg}\left(2 \cdot \left(x \cdot x\right)\right)\right)\right), \left(\color{blue}{2} \cdot 2 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(4, \left(\mathsf{neg}\left(\left(x \cdot x\right) \cdot 2\right)\right)\right), \left(2 \cdot 2 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \]

   13. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(4, \left(\left(x \cdot x\right) \cdot \left(\mathsf{neg}\left(2\right)\right)\right)\right), \left(2 \cdot \color{blue}{2} - \left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\mathsf{neg}\left(2\right)\right)\right)\right), \left(2 \cdot \color{blue}{2} - \left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\mathsf{neg}\left(2\right)\right)\right)\right), \left(2 \cdot 2 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), -2\right)\right), \left(2 \cdot 2 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), -2\right)\right), \mathsf{\_.f64}\left(\left(2 \cdot 2\right), \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), -2\right)\right), \mathsf{\_.f64}\left(4, \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot x\right)\right)\right)\right) \]

   19. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), -2\right)\right), \mathsf{\_.f64}\left(4, \left(x \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right) \]

   20. cube-unmult N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), -2\right)\right), \mathsf{\_.f64}\left(4, \left(x \cdot {x}^{\color{blue}{3}}\right)\right)\right) \]

7. Applied egg-rr 63.3%

   \[\leadsto \color{blue}{\frac{4 + \left(x \cdot x\right) \cdot -2}{4 - x \cdot \left(x \cdot \left(x \cdot x\right)\right)}} \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{4}, \mathsf{\_.f64}\left(4, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]
9. Step-by-step derivation

10. Simplified 88.9%

    \[\leadsto \frac{\color{blue}{4}}{4 - x \cdot \left(x \cdot \left(x \cdot x\right)\right)} \]
11. Add Preprocessing

Alternative 9: 62.6% accurate, 20.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.4) 1.0 (/ 2.0 (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.4) {
		tmp = 1.0;
	} else {
		tmp = 2.0 / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.4d0) then
        tmp = 1.0d0
    else
        tmp = 2.0d0 / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.4) {
		tmp = 1.0;
	} else {
		tmp = 2.0 / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.4:
		tmp = 1.0
	else:
		tmp = 2.0 / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.4)
		tmp = 1.0;
	else
		tmp = Float64(2.0 / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.4)
		tmp = 1.0;
	else
		tmp = 2.0 / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.4], 1.0, N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 66.6%

      \[\leadsto \color{blue}{1} \]

   if 1.3999999999999999 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 58.2%

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

   5. Simplified 58.2%

      \[\leadsto \frac{2}{\color{blue}{2 + x \cdot x}} \]

   6. Taylor expanded in x around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 58.2%

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

   8. Simplified 58.2%

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

Alternative 10: 75.8% accurate, 29.4× speedup

Math

\[\begin{array}{l} \\ \frac{2}{x \cdot x + 2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (+ (* x x) 2.0)))

C

double code(double x) {
	return 2.0 / ((x * x) + 2.0);
}

Fortran

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

Java

public static double code(double x) {
	return 2.0 / ((x * x) + 2.0);
}

Python

def code(x):
	return 2.0 / ((x * x) + 2.0)

Julia

function code(x)
	return Float64(2.0 / Float64(Float64(x * x) + 2.0))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / ((x * x) + 2.0);
end

Wolfram

code[x_] := N[(2.0 / N[(N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

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

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

   2. unpow2 N/A

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

   3. *-lowering-*.f64 77.7%

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

5. Simplified 77.7%

   \[\leadsto \frac{2}{\color{blue}{2 + x \cdot x}} \]

6. Final simplification 77.7%

   \[\leadsto \frac{2}{x \cdot x + 2} \]
7. Add Preprocessing

Alternative 11: 50.4% accurate, 206.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Simplified 52.0%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024155 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))