Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 8.0s

Alternatives: 15

Speedup: 1.9×

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, 1.9× 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. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-+.f64 N/A

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

   4. lift-exp.f64 N/A

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

   5. lift-exp.f64 N/A

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

   6. lift-neg.f64 N/A

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

   7. cosh-def N/A

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

   8. lower-/.f64 N/A

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

   9. lower-cosh.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 91.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} + e^{-x} \leq 4:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.08333333333333333, x \cdot \left(x \cdot x\right), x\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.002777777777777778, 0.08333333333333333\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (exp x) (exp (- x))) 4.0)
   (/ 2.0 (fma x (fma 0.08333333333333333 (* x (* x x)) x) 2.0))
   (/
    2.0
    (*
     x
     (*
      x
      (* x (* x (fma x (* x 0.002777777777777778) 0.08333333333333333))))))))

C

double code(double x) {
	double tmp;
	if ((exp(x) + exp(-x)) <= 4.0) {
		tmp = 2.0 / fma(x, fma(0.08333333333333333, (x * (x * x)), x), 2.0);
	} else {
		tmp = 2.0 / (x * (x * (x * (x * fma(x, (x * 0.002777777777777778), 0.08333333333333333)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) + exp(Float64(-x))) <= 4.0)
		tmp = Float64(2.0 / fma(x, fma(0.08333333333333333, Float64(x * Float64(x * x)), x), 2.0));
	else
		tmp = Float64(2.0 / Float64(x * Float64(x * Float64(x * Float64(x * fma(x, Float64(x * 0.002777777777777778), 0.08333333333333333))))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 4.0], N[(2.0 / N[(x * N[(0.08333333333333333 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(x * N[(x * N[(x * N[(x * N[(x * N[(x * 0.002777777777777778), $MachinePrecision] + 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 4

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft1-in N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{1}{12}, x \cdot \color{blue}{\left(x \cdot x\right)}, x\right), 2\right)} \]

      15. lower-*.f64 99.6

          \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.08333333333333333, x \cdot \color{blue}{\left(x \cdot x\right)}, x\right), 2\right)} \]

   5. Applied rewrites 99.6%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.08333333333333333, x \cdot \left(x \cdot x\right), x\right), 2\right)}} \]

   if 4 < (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{360}, \frac{1}{12}\right), 1\right), 2\right)} \]

      13. lower-*.f64 85.5

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

   5. Applied rewrites 85.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 85.5%

      \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, 0.002777777777777778, 0.08333333333333333\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 85.5%

       \[\leadsto \frac{2}{\left(\left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.002777777777777778, 0.08333333333333333\right)\right)\right) \cdot x\right) \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} + e^{-x} \leq 4:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.08333333333333333, x \cdot \left(x \cdot x\right), x\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.002777777777777778, 0.08333333333333333\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;e^{x} + e^{-x} \leq 4:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.08333333333333333, t\_0, x\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(x \cdot x\right) \cdot 0.002777777777777778\right) \cdot \left(x \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= (+ (exp x) (exp (- x))) 4.0)
     (/ 2.0 (fma x (fma 0.08333333333333333 t_0 x) 2.0))
     (/ 2.0 (* (* (* x x) 0.002777777777777778) (* x t_0))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double tmp;
	if ((exp(x) + exp(-x)) <= 4.0) {
		tmp = 2.0 / fma(x, fma(0.08333333333333333, t_0, x), 2.0);
	} else {
		tmp = 2.0 / (((x * x) * 0.002777777777777778) * (x * t_0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (Float64(exp(x) + exp(Float64(-x))) <= 4.0)
		tmp = Float64(2.0 / fma(x, fma(0.08333333333333333, t_0, x), 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(x * x) * 0.002777777777777778) * Float64(x * t_0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 4.0], N[(2.0 / N[(x * N[(0.08333333333333333 * t$95$0 + x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(x * x), $MachinePrecision] * 0.002777777777777778), $MachinePrecision] * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 4

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft1-in N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{1}{12}, x \cdot \color{blue}{\left(x \cdot x\right)}, x\right), 2\right)} \]

      15. lower-*.f64 99.6

          \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.08333333333333333, x \cdot \color{blue}{\left(x \cdot x\right)}, x\right), 2\right)} \]

   5. Applied rewrites 99.6%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.08333333333333333, x \cdot \left(x \cdot x\right), x\right), 2\right)}} \]

   if 4 < (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{360}, \frac{1}{12}\right), 1\right), 2\right)} \]

      13. lower-*.f64 85.5

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

   5. Applied rewrites 85.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 85.5%

      \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, 0.002777777777777778, 0.08333333333333333\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}} \]

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 85.5%

       \[\leadsto \frac{2}{\left(\left(x \cdot x\right) \cdot 0.002777777777777778\right) \cdot \left(x \cdot \left(\color{blue}{x} \cdot \left(x \cdot x\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 76.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \mathsf{fma}\left(x, x \cdot 0.002777777777777778, 0.08333333333333333\right)\\ \mathbf{if}\;x \leq 2 \cdot 10^{+77}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, t\_0 \cdot t\_0, -1\right), \frac{1}{\mathsf{fma}\left(x, t\_0, -1\right)}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.08333333333333333\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (fma x (* x 0.002777777777777778) 0.08333333333333333))))
   (if (<= x 2e+77)
     (/
      2.0
      (fma
       (* (* x x) (fma (* x x) (* t_0 t_0) -1.0))
       (/ 1.0 (fma x t_0 -1.0))
       2.0))
     (/ 2.0 (* x (* x (* (* x x) 0.08333333333333333)))))))

C

double code(double x) {
	double t_0 = x * fma(x, (x * 0.002777777777777778), 0.08333333333333333);
	double tmp;
	if (x <= 2e+77) {
		tmp = 2.0 / fma(((x * x) * fma((x * x), (t_0 * t_0), -1.0)), (1.0 / fma(x, t_0, -1.0)), 2.0);
	} else {
		tmp = 2.0 / (x * (x * ((x * x) * 0.08333333333333333)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(x * fma(x, Float64(x * 0.002777777777777778), 0.08333333333333333))
	tmp = 0.0
	if (x <= 2e+77)
		tmp = Float64(2.0 / fma(Float64(Float64(x * x) * fma(Float64(x * x), Float64(t_0 * t_0), -1.0)), Float64(1.0 / fma(x, t_0, -1.0)), 2.0));
	else
		tmp = Float64(2.0 / Float64(x * Float64(x * Float64(Float64(x * x) * 0.08333333333333333))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * N[(x * 0.002777777777777778), $MachinePrecision] + 0.08333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2e+77], N[(2.0 / N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999999999997e77

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 75.9

         \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]

   5. Applied rewrites 75.9%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{360}, \frac{1}{12}\right)}, 1\right), 2\right)} \]

      14. lower-*.f64 89.8

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

   8. Applied rewrites 89.8%

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

   10. Applied rewrites 67.6%

       \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.002777777777777778, 0.08333333333333333\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.002777777777777778, 0.08333333333333333\right)\right), -1\right) \cdot \left(x \cdot x\right), \color{blue}{\frac{1}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.002777777777777778, 0.08333333333333333\right), -1\right)}}, 2\right)} \]

   if 1.99999999999999997e77 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft1-in N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{1}{12}, x \cdot \color{blue}{\left(x \cdot x\right)}, x\right), 2\right)} \]

      15. lower-*.f64 100.0

          \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.08333333333333333, x \cdot \color{blue}{\left(x \cdot x\right)}, x\right), 2\right)} \]

   5. Applied rewrites 100.0%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.08333333333333333, x \cdot \left(x \cdot x\right), x\right), 2\right)}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{2}{\frac{1}{12} \cdot \color{blue}{{x}^{4}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto \frac{2}{x \cdot \color{blue}{\left(x \cdot \left(\left(x \cdot x\right) \cdot 0.08333333333333333\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+77}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.002777777777777778, 0.08333333333333333\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.002777777777777778, 0.08333333333333333\right)\right), -1\right), \frac{1}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.002777777777777778, 0.08333333333333333\right), -1\right)}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.08333333333333333\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot 0.08333333333333333\\ t_1 := \mathsf{fma}\left(x, t\_0, x\right)\\ \mathbf{if}\;x \leq 2.15 \cdot 10^{+77}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(x, t\_1 \cdot \left(x \cdot t\_1\right), -4\right)}{\mathsf{fma}\left(x, t\_1, -2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot \left(x \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) 0.08333333333333333)) (t_1 (fma x t_0 x)))
   (if (<= x 2.15e+77)
     (/ 2.0 (/ (fma x (* t_1 (* x t_1)) -4.0) (fma x t_1 -2.0)))
     (/ 2.0 (* x (* x t_0))))))

C

double code(double x) {
	double t_0 = (x * x) * 0.08333333333333333;
	double t_1 = fma(x, t_0, x);
	double tmp;
	if (x <= 2.15e+77) {
		tmp = 2.0 / (fma(x, (t_1 * (x * t_1)), -4.0) / fma(x, t_1, -2.0));
	} else {
		tmp = 2.0 / (x * (x * t_0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * 0.08333333333333333)
	t_1 = fma(x, t_0, x)
	tmp = 0.0
	if (x <= 2.15e+77)
		tmp = Float64(2.0 / Float64(fma(x, Float64(t_1 * Float64(x * t_1)), -4.0) / fma(x, t_1, -2.0)));
	else
		tmp = Float64(2.0 / Float64(x * Float64(x * t_0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0 + x), $MachinePrecision]}, If[LessEqual[x, 2.15e+77], N[(2.0 / N[(N[(x * N[(t$95$1 * N[(x * t$95$1), $MachinePrecision]), $MachinePrecision] + -4.0), $MachinePrecision] / N[(x * t$95$1 + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(x * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.14999999999999996e77

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft1-in N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{1}{12}, x \cdot \color{blue}{\left(x \cdot x\right)}, x\right), 2\right)} \]

      15. lower-*.f64 85.1

          \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.08333333333333333, x \cdot \color{blue}{\left(x \cdot x\right)}, x\right), 2\right)} \]

   5. Applied rewrites 85.1%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.08333333333333333, x \cdot \left(x \cdot x\right), x\right), 2\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 67.0%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.08333333333333333, x\right) \cdot \left(x \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.08333333333333333, x\right)\right), -4\right)}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.08333333333333333, x\right), -2\right)}}} \]

   if 2.14999999999999996e77 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft1-in N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{1}{12}, x \cdot \color{blue}{\left(x \cdot x\right)}, x\right), 2\right)} \]

      15. lower-*.f64 100.0

          \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.08333333333333333, x \cdot \color{blue}{\left(x \cdot x\right)}, x\right), 2\right)} \]

   5. Applied rewrites 100.0%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.08333333333333333, x \cdot \left(x \cdot x\right), x\right), 2\right)}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{2}{\frac{1}{12} \cdot \color{blue}{{x}^{4}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto \frac{2}{x \cdot \color{blue}{\left(x \cdot \left(\left(x \cdot x\right) \cdot 0.08333333333333333\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 92.0% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. lower-fma.f64 74.6

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]

5. Applied rewrites 74.6%

   \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

   13. lower-fma.f64 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{360}, \frac{1}{12}\right)}, 1\right), 2\right)} \]

   14. lower-*.f64 92.1

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

8. Applied rewrites 92.1%

   \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.002777777777777778, 0.08333333333333333\right), 1\right), 2\right)}} \]
9. Add Preprocessing

Alternative 7: 91.9% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. lower-fma.f64 74.6

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]

5. Applied rewrites 74.6%

   \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

   13. lower-fma.f64 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{360}, \frac{1}{12}\right)}, 1\right), 2\right)} \]

   14. lower-*.f64 92.1

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

8. Applied rewrites 92.1%

   \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.002777777777777778, 0.08333333333333333\right), 1\right), 2\right)}} \]
9. Step-by-step derivation

10. Applied rewrites 92.1%

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

11. Taylor expanded in x around 0

    \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot \left(x \cdot x\right), x \cdot \frac{1}{360}, 1\right), 2\right)} \]
12. Step-by-step derivation

13. Applied rewrites 91.9%

    \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot \left(x \cdot x\right), x \cdot 0.002777777777777778, 1\right), 2\right)} \]
14. Add Preprocessing

Alternative 8: 91.6% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.002777777777777778, 0.08333333333333333\right)\right), 2\right)} \end{array} \]

FPCore

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

C

double code(double x) {
	return 2.0 / fma((x * x), (x * (x * fma((x * x), 0.002777777777777778, 0.08333333333333333))), 2.0);
}

Julia

function code(x)
	return Float64(2.0 / fma(Float64(x * x), Float64(x * Float64(x * fma(Float64(x * x), 0.002777777777777778, 0.08333333333333333))), 2.0))
end

Wolfram

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

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. unpow2 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{360}, \frac{1}{12}\right), 1\right), 2\right)} \]

   13. lower-*.f64 92.1

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

5. Applied rewrites 92.1%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 91.6%

   \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, 0.002777777777777778, 0.08333333333333333\right)\right)}, 2\right)} \]
9. Add Preprocessing

Alternative 9: 88.0% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), 0.08333333333333333, \mathsf{fma}\left(x, x, 2\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 2.0 (fma (* x (* x (* x x))) 0.08333333333333333 (fma x x 2.0))))

C

double code(double x) {
	return 2.0 / fma((x * (x * (x * x))), 0.08333333333333333, fma(x, x, 2.0));
}

Julia

function code(x)
	return Float64(2.0 / fma(Float64(x * Float64(x * Float64(x * x))), 0.08333333333333333, fma(x, x, 2.0)))
end

Wolfram

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

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. distribute-lft1-in N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{1}{12}, x \cdot \color{blue}{\left(x \cdot x\right)}, x\right), 2\right)} \]

   15. lower-*.f64 88.4

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.08333333333333333, x \cdot \color{blue}{\left(x \cdot x\right)}, x\right), 2\right)} \]

5. Applied rewrites 88.4%

   \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.08333333333333333, x \cdot \left(x \cdot x\right), x\right), 2\right)}} \]
6. Step-by-step derivation

7. Applied rewrites 88.7%

   \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), \color{blue}{0.08333333333333333}, \mathsf{fma}\left(x, x, 2\right)\right)} \]
8. Add Preprocessing

Alternative 10: 69.8% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.20833333333333334, -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{0.08333333333333333 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (fma (* x x) (fma (* x x) 0.20833333333333334 -0.5) 1.0)
   (/ 2.0 (* 0.08333333333333333 (* x (* x (* x x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = fma((x * x), fma((x * x), 0.20833333333333334, -0.5), 1.0);
	} else {
		tmp = 2.0 / (0.08333333333333333 * (x * (x * (x * x))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = fma(Float64(x * x), fma(Float64(x * x), 0.20833333333333334, -0.5), 1.0);
	else
		tmp = Float64(2.0 / Float64(0.08333333333333333 * Float64(x * Float64(x * Float64(x * x)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.20833333333333334 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(2.0 / N[(0.08333333333333333 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{5}{24} \cdot {x}^{2} - \frac{1}{2}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{5}{24} \cdot {x}^{2} - \frac{1}{2}\right) + 1} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{5}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{5}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{5}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{5}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{5}{24}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \frac{5}{24} + \color{blue}{\frac{-1}{2}}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{5}{24}, \frac{-1}{2}\right)}, 1\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{5}{24}, \frac{-1}{2}\right), 1\right) \]

      10. lower-*.f64 63.1

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.20833333333333334, -0.5\right), 1\right) \]

   5. Applied rewrites 63.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.20833333333333334, -0.5\right), 1\right)} \]

   if 1.8500000000000001 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft1-in N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{1}{12}, x \cdot \color{blue}{\left(x \cdot x\right)}, x\right), 2\right)} \]

      15. lower-*.f64 83.3

          \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.08333333333333333, x \cdot \color{blue}{\left(x \cdot x\right)}, x\right), 2\right)} \]

   5. Applied rewrites 83.3%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.08333333333333333, x \cdot \left(x \cdot x\right), x\right), 2\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 83.3%

      \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot \left(x \cdot 0.08333333333333333\right), \color{blue}{x \cdot x}, \mathsf{fma}\left(x, x, 2\right)\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{2}{\frac{1}{12} \cdot \color{blue}{{x}^{4}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 83.3%

       \[\leadsto \frac{2}{0.08333333333333333 \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 69.8% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.20833333333333334, -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.08333333333333333\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (fma (* x x) (fma (* x x) 0.20833333333333334 -0.5) 1.0)
   (/ 2.0 (* x (* x (* (* x x) 0.08333333333333333))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = fma((x * x), fma((x * x), 0.20833333333333334, -0.5), 1.0);
	} else {
		tmp = 2.0 / (x * (x * ((x * x) * 0.08333333333333333)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = fma(Float64(x * x), fma(Float64(x * x), 0.20833333333333334, -0.5), 1.0);
	else
		tmp = Float64(2.0 / Float64(x * Float64(x * Float64(Float64(x * x) * 0.08333333333333333))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.20833333333333334 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(2.0 / N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{5}{24} \cdot {x}^{2} - \frac{1}{2}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{5}{24} \cdot {x}^{2} - \frac{1}{2}\right) + 1} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{5}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{5}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{5}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{5}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{5}{24}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \frac{5}{24} + \color{blue}{\frac{-1}{2}}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{5}{24}, \frac{-1}{2}\right)}, 1\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{5}{24}, \frac{-1}{2}\right), 1\right) \]

      10. lower-*.f64 63.1

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.20833333333333334, -0.5\right), 1\right) \]

   5. Applied rewrites 63.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.20833333333333334, -0.5\right), 1\right)} \]

   if 1.8500000000000001 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft1-in N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{1}{12}, x \cdot \color{blue}{\left(x \cdot x\right)}, x\right), 2\right)} \]

      15. lower-*.f64 83.3

          \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.08333333333333333, x \cdot \color{blue}{\left(x \cdot x\right)}, x\right), 2\right)} \]

   5. Applied rewrites 83.3%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.08333333333333333, x \cdot \left(x \cdot x\right), x\right), 2\right)}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{2}{\frac{1}{12} \cdot \color{blue}{{x}^{4}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 83.3%

      \[\leadsto \frac{2}{x \cdot \color{blue}{\left(x \cdot \left(\left(x \cdot x\right) \cdot 0.08333333333333333\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 88.0% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.08333333333333333, x \cdot \left(x \cdot x\right), x\right), 2\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 2.0 (fma x (fma 0.08333333333333333 (* x (* x x)) x) 2.0)))

C

double code(double x) {
	return 2.0 / fma(x, fma(0.08333333333333333, (x * (x * x)), x), 2.0);
}

Julia

function code(x)
	return Float64(2.0 / fma(x, fma(0.08333333333333333, Float64(x * Float64(x * x)), x), 2.0))
end

Wolfram

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

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. distribute-lft1-in N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{1}{12}, x \cdot \color{blue}{\left(x \cdot x\right)}, x\right), 2\right)} \]

   15. lower-*.f64 88.4

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.08333333333333333, x \cdot \color{blue}{\left(x \cdot x\right)}, x\right), 2\right)} \]

5. Applied rewrites 88.4%

   \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.08333333333333333, x \cdot \left(x \cdot x\right), x\right), 2\right)}} \]
6. Add Preprocessing

Alternative 13: 64.1% accurate, 9.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot {x}^{2} + 1} \]

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 63.0

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

   5. Applied rewrites 63.0%

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

   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

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 58.2

         \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]

   5. Applied rewrites 58.2%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 58.2%

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

Alternative 14: 76.4% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(x, x, 2\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (fma x x 2.0)))

C

double code(double x) {
	return 2.0 / fma(x, x, 2.0);
}

Julia

function code(x)
	return Float64(2.0 / fma(x, x, 2.0))
end

Wolfram

code[x_] := N[(2.0 / N[(x * x + 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

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. lower-fma.f64 74.6

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]

5. Applied rewrites 74.6%

   \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]
6. Add Preprocessing

Alternative 15: 51.6% accurate, 217.0× 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. Applied rewrites 47.1%

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

Reproduce

herbie shell --seed 2024234 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))