Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 7.7s

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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{-x} + e^{x} \leq 4:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.08472222222222223, x \cdot x, 0.20833333333333334\right), x \cdot x, -0.5\right), x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right) \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (exp (- x)) (exp x)) 4.0)
   (fma
    (fma (fma -0.08472222222222223 (* x x) 0.20833333333333334) (* x x) -0.5)
    (* x x)
    1.0)
   (/
    2.0
    (*
     (* (* x x) x)
     (* (fma 0.002777777777777778 (* x x) 0.08333333333333333) x)))))

C

double code(double x) {
	double tmp;
	if ((exp(-x) + exp(x)) <= 4.0) {
		tmp = fma(fma(fma(-0.08472222222222223, (x * x), 0.20833333333333334), (x * x), -0.5), (x * x), 1.0);
	} else {
		tmp = 2.0 / (((x * x) * x) * (fma(0.002777777777777778, (x * x), 0.08333333333333333) * x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(Float64(-x)) + exp(x)) <= 4.0)
		tmp = fma(fma(fma(-0.08472222222222223, Float64(x * x), 0.20833333333333334), Float64(x * x), -0.5), Float64(x * x), 1.0);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(x * x) * x) * Float64(fma(0.002777777777777778, Float64(x * x), 0.08333333333333333) * x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[(-x)], $MachinePrecision] + N[Exp[x], $MachinePrecision]), $MachinePrecision], 4.0], N[(N[(N[(-0.08472222222222223 * N[(x * x), $MachinePrecision] + 0.20833333333333334), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[(2.0 / N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(N[(0.002777777777777778 * N[(x * x), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * x), $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

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{5}{24} + \frac{-61}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{5}{24} + \frac{-61}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{5}{24} + \frac{-61}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot {x}^{2}} + 1 \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{5}{24} + \frac{-61}{720} \cdot {x}^{2}\right) - \frac{1}{2}, {x}^{2}, 1\right)} \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{5}{24} + \frac{-61}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, {x}^{2}, 1\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{5}{24} + \frac{-61}{720} \cdot {x}^{2}\right) \cdot {x}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {x}^{2}, 1\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{5}{24} + \frac{-61}{720} \cdot {x}^{2}\right) \cdot {x}^{2} + \color{blue}{\frac{-1}{2}}, {x}^{2}, 1\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{5}{24} + \frac{-61}{720} \cdot {x}^{2}, {x}^{2}, \frac{-1}{2}\right)}, {x}^{2}, 1\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-61}{720} \cdot {x}^{2} + \frac{5}{24}}, {x}^{2}, \frac{-1}{2}\right), {x}^{2}, 1\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-61}{720}, {x}^{2}, \frac{5}{24}\right)}, {x}^{2}, \frac{-1}{2}\right), {x}^{2}, 1\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-61}{720}, \color{blue}{x \cdot x}, \frac{5}{24}\right), {x}^{2}, \frac{-1}{2}\right), {x}^{2}, 1\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-61}{720}, \color{blue}{x \cdot x}, \frac{5}{24}\right), {x}^{2}, \frac{-1}{2}\right), {x}^{2}, 1\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-61}{720}, x \cdot x, \frac{5}{24}\right), \color{blue}{x \cdot x}, \frac{-1}{2}\right), {x}^{2}, 1\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-61}{720}, x \cdot x, \frac{5}{24}\right), \color{blue}{x \cdot x}, \frac{-1}{2}\right), {x}^{2}, 1\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-61}{720}, x \cdot x, \frac{5}{24}\right), x \cdot x, \frac{-1}{2}\right), \color{blue}{x \cdot x}, 1\right) \]

      15. lower-*.f64 99.8

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.08472222222222223, x \cdot x, 0.20833333333333334\right), x \cdot x, -0.5\right), \color{blue}{x \cdot x}, 1\right) \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.08472222222222223, x \cdot x, 0.20833333333333334\right), x \cdot x, -0.5\right), x \cdot x, 1\right)} \]

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 82.4

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

   5. Applied rewrites 82.4%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{2}{{x}^{6} \cdot \color{blue}{\left(\frac{1}{360} + \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right)}} \]

   8. Applied rewrites 82.4%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} + e^{x} \leq 4:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.08472222222222223, x \cdot x, 0.20833333333333334\right), x \cdot x, -0.5\right), x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right) \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{-x} + e^{x} \leq 4:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.08472222222222223, x \cdot x, 0.20833333333333334\right), x \cdot x, -0.5\right), x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(x \cdot x\right) \cdot 0.002777777777777778\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (exp (- x)) (exp x)) 4.0)
   (fma
    (fma (fma -0.08472222222222223 (* x x) 0.20833333333333334) (* x x) -0.5)
    (* x x)
    1.0)
   (/ 2.0 (* (* (* (* x x) 0.002777777777777778) x) (* (* x x) x)))))

C

double code(double x) {
	double tmp;
	if ((exp(-x) + exp(x)) <= 4.0) {
		tmp = fma(fma(fma(-0.08472222222222223, (x * x), 0.20833333333333334), (x * x), -0.5), (x * x), 1.0);
	} else {
		tmp = 2.0 / ((((x * x) * 0.002777777777777778) * x) * ((x * x) * x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(Float64(-x)) + exp(x)) <= 4.0)
		tmp = fma(fma(fma(-0.08472222222222223, Float64(x * x), 0.20833333333333334), Float64(x * x), -0.5), Float64(x * x), 1.0);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(x * x) * 0.002777777777777778) * x) * Float64(Float64(x * x) * x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[(-x)], $MachinePrecision] + N[Exp[x], $MachinePrecision]), $MachinePrecision], 4.0], N[(N[(N[(-0.08472222222222223 * N[(x * x), $MachinePrecision] + 0.20833333333333334), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(x * x), $MachinePrecision] * 0.002777777777777778), $MachinePrecision] * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * x), $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

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{5}{24} + \frac{-61}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{5}{24} + \frac{-61}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{5}{24} + \frac{-61}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot {x}^{2}} + 1 \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{5}{24} + \frac{-61}{720} \cdot {x}^{2}\right) - \frac{1}{2}, {x}^{2}, 1\right)} \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{5}{24} + \frac{-61}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, {x}^{2}, 1\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{5}{24} + \frac{-61}{720} \cdot {x}^{2}\right) \cdot {x}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {x}^{2}, 1\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{5}{24} + \frac{-61}{720} \cdot {x}^{2}\right) \cdot {x}^{2} + \color{blue}{\frac{-1}{2}}, {x}^{2}, 1\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{5}{24} + \frac{-61}{720} \cdot {x}^{2}, {x}^{2}, \frac{-1}{2}\right)}, {x}^{2}, 1\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-61}{720} \cdot {x}^{2} + \frac{5}{24}}, {x}^{2}, \frac{-1}{2}\right), {x}^{2}, 1\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-61}{720}, {x}^{2}, \frac{5}{24}\right)}, {x}^{2}, \frac{-1}{2}\right), {x}^{2}, 1\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-61}{720}, \color{blue}{x \cdot x}, \frac{5}{24}\right), {x}^{2}, \frac{-1}{2}\right), {x}^{2}, 1\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-61}{720}, \color{blue}{x \cdot x}, \frac{5}{24}\right), {x}^{2}, \frac{-1}{2}\right), {x}^{2}, 1\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-61}{720}, x \cdot x, \frac{5}{24}\right), \color{blue}{x \cdot x}, \frac{-1}{2}\right), {x}^{2}, 1\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-61}{720}, x \cdot x, \frac{5}{24}\right), \color{blue}{x \cdot x}, \frac{-1}{2}\right), {x}^{2}, 1\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-61}{720}, x \cdot x, \frac{5}{24}\right), x \cdot x, \frac{-1}{2}\right), \color{blue}{x \cdot x}, 1\right) \]

      15. lower-*.f64 99.8

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.08472222222222223, x \cdot x, 0.20833333333333334\right), x \cdot x, -0.5\right), \color{blue}{x \cdot x}, 1\right) \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.08472222222222223, x \cdot x, 0.20833333333333334\right), x \cdot x, -0.5\right), x \cdot x, 1\right)} \]

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 82.4

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

   5. Applied rewrites 82.4%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{2}{\frac{1}{360} \cdot \color{blue}{{x}^{6}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 82.4%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} + e^{x} \leq 4:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.08472222222222223, x \cdot x, 0.20833333333333334\right), x \cdot x, -0.5\right), x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(x \cdot x\right) \cdot 0.002777777777777778\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{-x} + e^{x} \leq 4:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (exp (- x)) (exp x)) 4.0) (fma (* x x) -0.5 1.0) (/ 2.0 (* x x))))

C

double code(double x) {
	double tmp;
	if ((exp(-x) + exp(x)) <= 4.0) {
		tmp = fma((x * x), -0.5, 1.0);
	} else {
		tmp = 2.0 / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(Float64(-x)) + exp(x)) <= 4.0)
		tmp = fma(Float64(x * x), -0.5, 1.0);
	else
		tmp = Float64(2.0 / Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[(-x)], $MachinePrecision] + N[Exp[x], $MachinePrecision]), $MachinePrecision], 4.0], N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision], N[(2.0 / N[(x * x), $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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.6

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

   5. Applied rewrites 99.6%

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

   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

      \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2}}} \]
   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 51.2

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

   5. Applied rewrites 51.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 51.2%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} + e^{x} \leq 4:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right)\\ t_1 := t\_0 \cdot x\\ \mathbf{if}\;x \leq 2 \cdot 10^{+77}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1 \cdot t\_1, x \cdot x, -1\right) \cdot \left(x \cdot x\right), \frac{1}{\mathsf{fma}\left(t\_0, x \cdot x, -1\right)}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right) \cdot x, x, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma 0.002777777777777778 (* x x) 0.08333333333333333))
        (t_1 (* t_0 x)))
   (if (<= x 2e+77)
     (/
      2.0
      (fma
       (* (fma (* t_1 t_1) (* x x) -1.0) (* x x))
       (/ 1.0 (fma t_0 (* x x) -1.0))
       2.0))
     (/ 2.0 (fma (* (fma 0.08333333333333333 (* x x) 1.0) x) x 2.0)))))

C

double code(double x) {
	double t_0 = fma(0.002777777777777778, (x * x), 0.08333333333333333);
	double t_1 = t_0 * x;
	double tmp;
	if (x <= 2e+77) {
		tmp = 2.0 / fma((fma((t_1 * t_1), (x * x), -1.0) * (x * x)), (1.0 / fma(t_0, (x * x), -1.0)), 2.0);
	} else {
		tmp = 2.0 / fma((fma(0.08333333333333333, (x * x), 1.0) * x), x, 2.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(0.002777777777777778, Float64(x * x), 0.08333333333333333)
	t_1 = Float64(t_0 * x)
	tmp = 0.0
	if (x <= 2e+77)
		tmp = Float64(2.0 / fma(Float64(fma(Float64(t_1 * t_1), Float64(x * x), -1.0) * Float64(x * x)), Float64(1.0 / fma(t_0, Float64(x * x), -1.0)), 2.0));
	else
		tmp = Float64(2.0 / fma(Float64(fma(0.08333333333333333, Float64(x * x), 1.0) * x), x, 2.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.002777777777777778 * N[(x * x), $MachinePrecision] + 0.08333333333333333), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * x), $MachinePrecision]}, If[LessEqual[x, 2e+77], N[(2.0 / N[(N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] * N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(t$95$0 * N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(0.08333333333333333 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * x + 2.0), $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

      \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 89.4

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

   5. Applied rewrites 89.4%

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

   7. Applied rewrites 72.1%

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

   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

      \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 100.0%

       \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right) \cdot x, x, 2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 75.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

      \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 89.4

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

   5. Applied rewrites 89.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 89.1%

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

   10. Applied rewrites 71.8%

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

   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

      \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 100.0%

       \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right) \cdot x, x, 2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 75.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right) \cdot x\\ t_1 := t\_0 \cdot x\\ \mathbf{if}\;x \leq 2 \cdot 10^{+77}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(t\_1, t\_1, -1\right) \cdot x}{\mathsf{fma}\left(t\_0, x, -1\right)}, x, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right) \cdot x, x, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fma 0.002777777777777778 (* x x) 0.08333333333333333) x))
        (t_1 (* t_0 x)))
   (if (<= x 2e+77)
     (/ 2.0 (fma (/ (* (fma t_1 t_1 -1.0) x) (fma t_0 x -1.0)) x 2.0))
     (/ 2.0 (fma (* (fma 0.08333333333333333 (* x x) 1.0) x) x 2.0)))))

C

double code(double x) {
	double t_0 = fma(0.002777777777777778, (x * x), 0.08333333333333333) * x;
	double t_1 = t_0 * x;
	double tmp;
	if (x <= 2e+77) {
		tmp = 2.0 / fma(((fma(t_1, t_1, -1.0) * x) / fma(t_0, x, -1.0)), x, 2.0);
	} else {
		tmp = 2.0 / fma((fma(0.08333333333333333, (x * x), 1.0) * x), x, 2.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(fma(0.002777777777777778, Float64(x * x), 0.08333333333333333) * x)
	t_1 = Float64(t_0 * x)
	tmp = 0.0
	if (x <= 2e+77)
		tmp = Float64(2.0 / fma(Float64(Float64(fma(t_1, t_1, -1.0) * x) / fma(t_0, x, -1.0)), x, 2.0));
	else
		tmp = Float64(2.0 / fma(Float64(fma(0.08333333333333333, Float64(x * x), 1.0) * x), x, 2.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(0.002777777777777778 * N[(x * x), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * x), $MachinePrecision]}, If[LessEqual[x, 2e+77], N[(2.0 / N[(N[(N[(N[(t$95$1 * t$95$1 + -1.0), $MachinePrecision] * x), $MachinePrecision] / N[(t$95$0 * x + -1.0), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(0.08333333333333333 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * x + 2.0), $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

      \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 89.4

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

   5. Applied rewrites 89.4%

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

   7. Applied rewrites 89.4%

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

   9. Applied rewrites 89.4%

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

   11. Applied rewrites 70.7%

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

   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

      \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 100.0%

       \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right) \cdot x, x, 2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 81.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right) \cdot x\\ \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(t\_0 \cdot t\_0, x \cdot x, -1\right)}{\mathsf{fma}\left(0.08333333333333333, x \cdot x, -1\right)}, x \cdot x, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fma 0.002777777777777778 (* x x) 0.08333333333333333) x)))
   (if (<= x 1.35e+154)
     (/
      2.0
      (fma
       (/
        (fma (* t_0 t_0) (* x x) -1.0)
        (fma 0.08333333333333333 (* x x) -1.0))
       (* x x)
       2.0))
     (/ 2.0 (* x x)))))

C

double code(double x) {
	double t_0 = fma(0.002777777777777778, (x * x), 0.08333333333333333) * x;
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 2.0 / fma((fma((t_0 * t_0), (x * x), -1.0) / fma(0.08333333333333333, (x * x), -1.0)), (x * x), 2.0);
	} else {
		tmp = 2.0 / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(fma(0.002777777777777778, Float64(x * x), 0.08333333333333333) * x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(2.0 / fma(Float64(fma(Float64(t_0 * t_0), Float64(x * x), -1.0) / fma(0.08333333333333333, Float64(x * x), -1.0)), Float64(x * x), 2.0));
	else
		tmp = Float64(2.0 / Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(0.002777777777777778 * N[(x * x), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, 1.35e+154], N[(2.0 / N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision] / N[(0.08333333333333333 * N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 90.0

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

   5. Applied rewrites 90.0%

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

   7. Applied rewrites 66.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 78.6%

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

   if 1.35000000000000003e154 < 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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

Alternative 9: 72.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ t_1 := t\_0 \cdot x\\ \mathbf{if}\;x \leq 2.4 \cdot 10^{+51}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(t\_1, t\_1, -16\right)}{\mathsf{fma}\left(x, x, -2\right) \cdot \mathsf{fma}\left(t\_0, x, 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(x \cdot x\right) \cdot 0.002777777777777778\right) \cdot x\right) \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) x)) (t_1 (* t_0 x)))
   (if (<= x 2.4e+51)
     (/ 2.0 (/ (fma t_1 t_1 -16.0) (* (fma x x -2.0) (fma t_0 x 4.0))))
     (/ 2.0 (* (* (* (* x x) 0.002777777777777778) x) t_0)))))

C

double code(double x) {
	double t_0 = (x * x) * x;
	double t_1 = t_0 * x;
	double tmp;
	if (x <= 2.4e+51) {
		tmp = 2.0 / (fma(t_1, t_1, -16.0) / (fma(x, x, -2.0) * fma(t_0, x, 4.0)));
	} else {
		tmp = 2.0 / ((((x * x) * 0.002777777777777778) * x) * t_0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * x)
	t_1 = Float64(t_0 * x)
	tmp = 0.0
	if (x <= 2.4e+51)
		tmp = Float64(2.0 / Float64(fma(t_1, t_1, -16.0) / Float64(fma(x, x, -2.0) * fma(t_0, x, 4.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(x * x) * 0.002777777777777778) * x) * t_0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * x), $MachinePrecision]}, If[LessEqual[x, 2.4e+51], N[(2.0 / N[(N[(t$95$1 * t$95$1 + -16.0), $MachinePrecision] / N[(N[(x * x + -2.0), $MachinePrecision] * N[(t$95$0 * x + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(x * x), $MachinePrecision] * 0.002777777777777778), $MachinePrecision] * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.3999999999999999e51

   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 77.7

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

   5. Applied rewrites 77.7%

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

   7. Applied rewrites 64.9%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x, \left(\left(x \cdot x\right) \cdot x\right) \cdot x, -16\right) \cdot 1}{\color{blue}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, x, 4\right) \cdot \mathsf{fma}\left(x, x, -2\right)}}} \]

   if 2.3999999999999999e51 < 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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{2}{\frac{1}{360} \cdot \color{blue}{{x}^{6}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{+51}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x, \left(\left(x \cdot x\right) \cdot x\right) \cdot x, -16\right)}{\mathsf{fma}\left(x, x, -2\right) \cdot \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, x, 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(x \cdot x\right) \cdot 0.002777777777777778\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 92.0% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}} \]
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. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 91.2

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

5. Applied rewrites 91.2%

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

7. Applied rewrites 91.2%

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

9. Applied rewrites 91.2%

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

Alternative 11: 91.8% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}} \]
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. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 91.2

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

5. Applied rewrites 91.2%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 91.0%

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

9. Final simplification 91.0%

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

Alternative 12: 91.5% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}} \]
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. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 91.2

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

5. Applied rewrites 91.2%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 90.9%

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

9. Final simplification 90.9%

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

Alternative 13: 88.1% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}} \]
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. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 91.2

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

5. Applied rewrites 91.2%

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

7. Applied rewrites 91.2%

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

9. Applied rewrites 91.2%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 85.9%

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

Alternative 14: 76.1% 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 75.4

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

5. Applied rewrites 75.4%

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

Alternative 15: 50.6% 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 51.3%

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

Reproduce

herbie shell --seed 2024235 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))