Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 8.4s

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: 88.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{e^{x} + e^{-x}} \leq 10^{-199}:\\ \;\;\;\;\frac{2}{x \cdot \mathsf{fma}\left(x, x \cdot \left(x \cdot 0.08333333333333333\right), x\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.20833333333333334, -0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ 2.0 (+ (exp x) (exp (- x)))) 1e-199)
   (/ 2.0 (* x (fma x (* x (* x 0.08333333333333333)) x)))
   (fma x (* x (fma (* x x) 0.20833333333333334 -0.5)) 1.0)))

C

double code(double x) {
	double tmp;
	if ((2.0 / (exp(x) + exp(-x))) <= 1e-199) {
		tmp = 2.0 / (x * fma(x, (x * (x * 0.08333333333333333)), x));
	} else {
		tmp = fma(x, (x * fma((x * x), 0.20833333333333334, -0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(2.0 / Float64(exp(x) + exp(Float64(-x)))) <= 1e-199)
		tmp = Float64(2.0 / Float64(x * fma(x, Float64(x * Float64(x * 0.08333333333333333)), x)));
	else
		tmp = fma(x, Float64(x * fma(Float64(x * x), 0.20833333333333334, -0.5)), 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(2.0 / N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-199], N[(2.0 / N[(x * N[(x * N[(x * N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.20833333333333334 + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))) < 9.99999999999999982e-200

   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 56.9

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

   5. Applied rewrites 56.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 71.5

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

   8. Applied rewrites 71.5%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 71.5%

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

   if 9.99999999999999982e-200 < (/.f64 #s(literal 2 binary64) (+.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

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 99.1

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

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.20833333333333334, -0.5\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 92.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} + e^{-x} \leq 4:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.08333333333333333, 1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (exp x) (exp (- x))) 4.0)
   (/ 2.0 (fma (* x x) (fma x (* x 0.08333333333333333) 1.0) 2.0))
   (/
    1.0
    (*
     x
     (*
      (* x x)
      (* x (fma x (* x 0.001388888888888889) 0.041666666666666664)))))))

C

double code(double x) {
	double tmp;
	if ((exp(x) + exp(-x)) <= 4.0) {
		tmp = 2.0 / fma((x * x), fma(x, (x * 0.08333333333333333), 1.0), 2.0);
	} else {
		tmp = 1.0 / (x * ((x * x) * (x * fma(x, (x * 0.001388888888888889), 0.041666666666666664))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) + exp(Float64(-x))) <= 4.0)
		tmp = Float64(2.0 / fma(Float64(x * x), fma(x, Float64(x * 0.08333333333333333), 1.0), 2.0));
	else
		tmp = Float64(1.0 / Float64(x * Float64(Float64(x * x) * Float64(x * fma(x, Float64(x * 0.001388888888888889), 0.041666666666666664)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 4.0], N[(2.0 / N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.08333333333333333), $MachinePrecision] + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2}}} \]
   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 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.3

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

   8. Applied rewrites 99.3%

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

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

   1. Initial program 100.0%

      \[\frac{2}{e^{x} + e^{-x}} \]
   2. Add Preprocessing
   3. 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. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 78.2

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \]

   7. Applied rewrites 78.2%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{{x}^{6} \cdot \color{blue}{\left(\frac{1}{720} + \frac{1}{24} \cdot \frac{1}{{x}^{2}}\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 78.2%

       \[\leadsto \frac{1}{\mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 78.2%

       \[\leadsto \frac{1}{\left(\left(x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right)\right) \cdot \left(x \cdot x\right)\right) \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} + e^{-x} \leq 4:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.08333333333333333, 1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{e^{x} + e^{-x}} \leq 10^{-199}:\\ \;\;\;\;\frac{1}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.041666666666666664}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.20833333333333334, -0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ 2.0 (+ (exp x) (exp (- x)))) 1e-199)
   (/ 1.0 (* (* x (* x (* x x))) 0.041666666666666664))
   (fma x (* x (fma (* x x) 0.20833333333333334 -0.5)) 1.0)))

C

double code(double x) {
	double tmp;
	if ((2.0 / (exp(x) + exp(-x))) <= 1e-199) {
		tmp = 1.0 / ((x * (x * (x * x))) * 0.041666666666666664);
	} else {
		tmp = fma(x, (x * fma((x * x), 0.20833333333333334, -0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(2.0 / Float64(exp(x) + exp(Float64(-x)))) <= 1e-199)
		tmp = Float64(1.0 / Float64(Float64(x * Float64(x * Float64(x * x))) * 0.041666666666666664));
	else
		tmp = fma(x, Float64(x * fma(Float64(x * x), 0.20833333333333334, -0.5)), 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(2.0 / N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-199], N[(1.0 / N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.20833333333333334 + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))) < 9.99999999999999982e-200

   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. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 78.2

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \]

   7. Applied rewrites 78.2%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{{x}^{6} \cdot \color{blue}{\left(\frac{1}{720} + \frac{1}{24} \cdot \frac{1}{{x}^{2}}\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 78.2%

       \[\leadsto \frac{1}{\mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{1}{\frac{1}{24} \cdot {x}^{\color{blue}{4}}} \]
   12. Step-by-step derivation

   13. Applied rewrites 71.5%

       \[\leadsto \frac{1}{0.041666666666666664 \cdot \left(x \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}\right)} \]

   if 9.99999999999999982e-200 < (/.f64 #s(literal 2 binary64) (+.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

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 99.1

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

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.20833333333333334, -0.5\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{e^{x} + e^{-x}} \leq 10^{-199}:\\ \;\;\;\;\frac{1}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.041666666666666664}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.20833333333333334, -0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} + e^{-x} \leq 4:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.08333333333333333, 1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (exp x) (exp (- x))) 4.0)
   (/ 2.0 (fma (* x x) (fma x (* x 0.08333333333333333) 1.0) 2.0))
   (/ 1.0 (* (* x x) (* x (* (* x x) (* x 0.001388888888888889)))))))

C

double code(double x) {
	double tmp;
	if ((exp(x) + exp(-x)) <= 4.0) {
		tmp = 2.0 / fma((x * x), fma(x, (x * 0.08333333333333333), 1.0), 2.0);
	} else {
		tmp = 1.0 / ((x * x) * (x * ((x * x) * (x * 0.001388888888888889))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) + exp(Float64(-x))) <= 4.0)
		tmp = Float64(2.0 / fma(Float64(x * x), fma(x, Float64(x * 0.08333333333333333), 1.0), 2.0));
	else
		tmp = Float64(1.0 / Float64(Float64(x * x) * Float64(x * Float64(Float64(x * x) * Float64(x * 0.001388888888888889)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 4.0], N[(2.0 / N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.08333333333333333), $MachinePrecision] + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2}}} \]
   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 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.3

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

   8. Applied rewrites 99.3%

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

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

   1. Initial program 100.0%

      \[\frac{2}{e^{x} + e^{-x}} \]
   2. Add Preprocessing
   3. 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. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 78.2

          \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \]

   7. Applied rewrites 78.2%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\frac{1}{720} \cdot \color{blue}{{x}^{6}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 78.2%

       \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 76.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (+.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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 99.0

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

   5. Applied rewrites 99.0%

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

   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

      \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2}}} \]
   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 56.9

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

   5. Applied rewrites 56.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 56.9%

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

Alternative 7: 94.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{0.001736111111111111 - \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 1.9290123456790124 \cdot 10^{-6}}{0.041666666666666664}, 0.5\right), 1\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  1.0
  (fma
   (* x x)
   (fma
    x
    (*
     x
     (/
      (- 0.001736111111111111 (* (* x (* x (* x x))) 1.9290123456790124e-6))
      0.041666666666666664))
    0.5)
   1.0)))

C

double code(double x) {
	return 1.0 / fma((x * x), fma(x, (x * ((0.001736111111111111 - ((x * (x * (x * x))) * 1.9290123456790124e-6)) / 0.041666666666666664)), 0.5), 1.0);
}

Julia

function code(x)
	return Float64(1.0 / fma(Float64(x * x), fma(x, Float64(x * Float64(Float64(0.001736111111111111 - Float64(Float64(x * Float64(x * Float64(x * x))) * 1.9290123456790124e-6)) / 0.041666666666666664)), 0.5), 1.0))
end

Wolfram

code[x_] := N[(1.0 / N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(0.001736111111111111 - N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.9290123456790124e-6), $MachinePrecision]), $MachinePrecision] / 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $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. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 89.7

       \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \]

7. Applied rewrites 89.7%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}} \]
8. Step-by-step derivation

9. Applied rewrites 64.7%

   \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{0.001736111111111111 - \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 1.9290123456790124 \cdot 10^{-6}}{\color{blue}{0.041666666666666664 - \left(x \cdot x\right) \cdot 0.001388888888888889}}, 0.5\right), 1\right)} \]

10. Taylor expanded in x around 0

    \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{\frac{1}{576} - \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{518400}}{\frac{1}{24}}, \frac{1}{2}\right), 1\right)} \]
11. Step-by-step derivation

12. Applied rewrites 91.9%

    \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{0.001736111111111111 - \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 1.9290123456790124 \cdot 10^{-6}}{0.041666666666666664}, 0.5\right), 1\right)} \]
13. Add Preprocessing

Alternative 8: 92.3% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  1.0
  (fma
   (* x x)
   (fma x (* x (fma (* x x) 0.001388888888888889 0.041666666666666664)) 0.5)
   1.0)))

C

double code(double x) {
	return 1.0 / fma((x * x), fma(x, (x * fma((x * x), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0);
}

Julia

function code(x)
	return Float64(1.0 / fma(Float64(x * x), fma(x, Float64(x * fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0))
end

Wolfram

code[x_] := N[(1.0 / N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $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. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 89.7

       \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \]

7. Applied rewrites 89.7%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}} \]
8. Add Preprocessing

Alternative 9: 91.9% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right)\right), 1\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  1.0
  (fma
   (* x x)
   (* x (* x (fma x (* x 0.001388888888888889) 0.041666666666666664)))
   1.0)))

C

double code(double x) {
	return 1.0 / fma((x * x), (x * (x * fma(x, (x * 0.001388888888888889), 0.041666666666666664))), 1.0);
}

Julia

function code(x)
	return Float64(1.0 / fma(Float64(x * x), Float64(x * Float64(x * fma(x, Float64(x * 0.001388888888888889), 0.041666666666666664))), 1.0))
end

Wolfram

code[x_] := N[(1.0 / N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $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. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 89.7

       \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \]

7. Applied rewrites 89.7%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}} \]

8. Taylor expanded in x around inf

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

10. Applied rewrites 89.1%

    \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right)\right)}, 1\right)} \]
11. Add Preprocessing

Alternative 10: 91.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(x \cdot x, \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.001388888888888889, 1\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 1.0 (fma (* x x) (* (* x (* x (* x x))) 0.001388888888888889) 1.0)))

C

double code(double x) {
	return 1.0 / fma((x * x), ((x * (x * (x * x))) * 0.001388888888888889), 1.0);
}

Julia

function code(x)
	return Float64(1.0 / fma(Float64(x * x), Float64(Float64(x * Float64(x * Float64(x * x))) * 0.001388888888888889), 1.0))
end

Wolfram

code[x_] := N[(1.0 / N[(N[(x * x), $MachinePrecision] * N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] + 1.0), $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. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 89.7

       \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \]

7. Applied rewrites 89.7%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}} \]
8. Step-by-step derivation

9. Applied rewrites 89.7%

   \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.001388888888888889, \color{blue}{x}, x \cdot 0.041666666666666664\right), 0.5\right), 1\right)} \]

10. Taylor expanded in x around inf

    \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \frac{1}{720} \cdot \color{blue}{{x}^{4}}, 1\right)} \]
11. Step-by-step derivation

12. Applied rewrites 89.1%

    \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, 0.001388888888888889 \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}, 1\right)} \]

13. Final simplification 89.1%

    \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.001388888888888889, 1\right)} \]
14. Add Preprocessing

Alternative 11: 88.5% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. lower-fma.f64 79.8

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

5. Applied rewrites 79.8%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 86.6

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

8. Applied rewrites 86.6%

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

Alternative 12: 88.5% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(1.0 / N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $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. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 89.7

       \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \]

7. Applied rewrites 89.7%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}} \]

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 86.6

       \[\leadsto \frac{1}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.041666666666666664, 0.5\right), 1\right)} \]

10. Applied rewrites 86.6%

    \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}} \]
11. Add Preprocessing

Alternative 13: 88.2% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2}}} \]
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 79.8

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

5. Applied rewrites 79.8%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 86.6

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

8. Applied rewrites 86.6%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 86.0%

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

Alternative 14: 76.8% 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 79.8

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

5. Applied rewrites 79.8%

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

Alternative 15: 51.6% accurate, 217.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 54.8%

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

Reproduce

herbie shell --seed 2024223 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))