Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 11.1s

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, 0.7× speedup

Math

\[\begin{array}{l} \\ {\left({\cosh x}^{-0.5}\right)}^{2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (pow (pow (cosh x) -0.5) 2.0))

C

double code(double x) {
	return pow(pow(cosh(x), -0.5), 2.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (cosh(x) ** (-0.5d0)) ** 2.0d0
end function

Java

public static double code(double x) {
	return Math.pow(Math.pow(Math.cosh(x), -0.5), 2.0);
}

Python

def code(x):
	return math.pow(math.pow(math.cosh(x), -0.5), 2.0)

Julia

function code(x)
	return (cosh(x) ^ -0.5) ^ 2.0
end

MATLAB

function tmp = code(x)
	tmp = (cosh(x) ^ -0.5) ^ 2.0;
end

Wolfram

code[x_] := N[Power[N[Power[N[Cosh[x], $MachinePrecision], -0.5], $MachinePrecision], 2.0], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. clear-num N/A

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

   2. cosh-def N/A

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

   3. inv-pow N/A

      \[\leadsto {\cosh x}^{\color{blue}{-1}} \]

   4. sqr-pow N/A

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

   5. pow2 N/A

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

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

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

   7. metadata-eval N/A

      \[\leadsto \mathsf{pow.f64}\left(\left({\cosh x}^{\frac{-1}{2}}\right), 2\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{pow.f64}\left(\left({\cosh x}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), 2\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{pow.f64}\left(\left({\cosh x}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), 2\right) \]

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

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\cosh x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), 2\right) \]

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

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{cosh.f64}\left(x\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), 2\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{cosh.f64}\left(x\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), 2\right) \]

   13. metadata-eval 100.0%

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{cosh.f64}\left(x\right), \frac{-1}{2}\right), 2\right) \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{{\left({\cosh x}^{-0.5}\right)}^{2}} \]
5. Add Preprocessing

Alternative 2: 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. clear-num N/A

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

   2. cosh-def N/A

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

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

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

   4. cosh-lowering-cosh.f64 100.0%

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

4. Applied egg-rr 100.0%

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

Alternative 3: 94.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, \mathsf{fma}\left(x, x, 0\right), 0\right)\\ \mathbf{if}\;x \leq 4 \cdot 10^{+30}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(t\_0, t\_0, 0\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.002777777777777778, 0.08333333333333333\right), 1\right)}{\mathsf{fma}\left(x, t\_0, 0\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma x (fma x x 0.0) 0.0)))
   (if (<= x 4e+30)
     (/
      1.0
      (fma
       (* x x)
       (fma
        x
        (* x (fma (* x x) 0.001388888888888889 0.041666666666666664))
        0.5)
       1.0))
     (if (<= x 1.15e+77)
       (/
        2.0
        (/
         (*
          (fma t_0 t_0 0.0)
          (fma
           x
           (* x (fma (fma x x 0.0) 0.002777777777777778 0.08333333333333333))
           1.0))
         (fma x t_0 0.0)))
       (/ 24.0 (* x (* x (* x x))))))))

C

double code(double x) {
	double t_0 = fma(x, fma(x, x, 0.0), 0.0);
	double tmp;
	if (x <= 4e+30) {
		tmp = 1.0 / fma((x * x), fma(x, (x * fma((x * x), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0);
	} else if (x <= 1.15e+77) {
		tmp = 2.0 / ((fma(t_0, t_0, 0.0) * fma(x, (x * fma(fma(x, x, 0.0), 0.002777777777777778, 0.08333333333333333)), 1.0)) / fma(x, t_0, 0.0));
	} else {
		tmp = 24.0 / (x * (x * (x * x)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(x, fma(x, x, 0.0), 0.0)
	tmp = 0.0
	if (x <= 4e+30)
		tmp = Float64(1.0 / fma(Float64(x * x), fma(x, Float64(x * fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0));
	elseif (x <= 1.15e+77)
		tmp = Float64(2.0 / Float64(Float64(fma(t_0, t_0, 0.0) * fma(x, Float64(x * fma(fma(x, x, 0.0), 0.002777777777777778, 0.08333333333333333)), 1.0)) / fma(x, t_0, 0.0)));
	else
		tmp = Float64(24.0 / Float64(x * Float64(x * Float64(x * x))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x + 0.0), $MachinePrecision] + 0.0), $MachinePrecision]}, If[LessEqual[x, 4e+30], 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], If[LessEqual[x, 1.15e+77], N[(2.0 / N[(N[(N[(t$95$0 * t$95$0 + 0.0), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x + 0.0), $MachinePrecision] * 0.002777777777777778 + 0.08333333333333333), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x * t$95$0 + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(24.0 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 4.0000000000000001e30

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. cosh-def N/A

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

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

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

      4. cosh-lowering-cosh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + \color{blue}{1}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}, 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right), 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right), 1\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \color{blue}{\frac{1}{2}}\right), 1\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}\right), 1\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + \frac{1}{2}\right), 1\right)\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}, \frac{1}{2}\right), 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right), \frac{1}{2}\right), 1\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{720} \cdot {x}^{2} + \color{blue}{\frac{1}{24}}\right)\right), \frac{1}{2}\right), 1\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{720} + \frac{1}{24}\right)\right), \frac{1}{2}\right), 1\right)\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}, \frac{1}{24}\right)\right), \frac{1}{2}\right), 1\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left(x \cdot x\right), \frac{1}{720}, \frac{1}{24}\right)\right), \frac{1}{2}\right), 1\right)\right) \]

      14. *-lowering-*.f64 91.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}, \frac{1}{24}\right)\right), \frac{1}{2}\right), 1\right)\right) \]

   7. Simplified 91.5%

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

   if 4.0000000000000001e30 < x < 1.14999999999999997e77

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. +-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(\left({x}^{2} + 0\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right), 2\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(\left(x \cdot x + 0\right), \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(x, x, 0\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right), 2\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(x, x, 0\right), \left({x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) + \color{blue}{1}\right), 2\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(x, x, 0\right), \left(\left(x \cdot x\right) \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) + 1\right), 2\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(x, x, 0\right), \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) + 1\right), 2\right)\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(x, x, 0\right), \mathsf{fma.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}, 1\right), 2\right)\right) \]

      10. +-rgt-identity N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(x, x, 0\right), \mathsf{fma.f64}\left(x, \left(x \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) + \color{blue}{0}\right), 1\right), 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(x, x, 0\right), \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)}, 0\right), 1\right), 2\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(x, x, 0\right), \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(\frac{1}{360} \cdot {x}^{2} + \color{blue}{\frac{1}{12}}\right), 0\right), 1\right), 2\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(x, x, 0\right), \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \frac{1}{360} + \frac{1}{12}\right), 0\right), 1\right), 2\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(x, x, 0\right), \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(\left(x \cdot x\right) \cdot \frac{1}{360} + \frac{1}{12}\right), 0\right), 1\right), 2\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(x, x, 0\right), \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \left(x \cdot \frac{1}{360}\right) + \frac{1}{12}\right), 0\right), 1\right), 2\right)\right) \]

      16. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(x, x, 0\right), \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(x \cdot \frac{1}{360}\right)}, \frac{1}{12}\right), 0\right), 1\right), 2\right)\right) \]

      17. +-rgt-identity N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(x, x, 0\right), \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \frac{1}{360} + \color{blue}{0}\right), \frac{1}{12}\right), 0\right), 1\right), 2\right)\right) \]

      18. accelerator-lowering-fma.f64 50.1%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(x, x, 0\right), \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\frac{1}{360}}, 0\right), \frac{1}{12}\right), 0\right), 1\right), 2\right)\right) \]

   5. Simplified 50.1%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.002777777777777778, 0\right), 0.08333333333333333\right), 0\right), 1\right), 2\right)}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({x}^{6} \cdot \left(\frac{1}{360} + \left(\frac{1}{12} \cdot \frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right)\right)\right)}\right) \]

   8. Simplified 50.1%

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

   10. Applied egg-rr 100.0%

       \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, x, 0\right), 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, x, 0\right), 0\right), 0\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.002777777777777778, 0.08333333333333333\right), 1\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x, 0\right), 0\right), 0\right)}}} \]

   if 1.14999999999999997e77 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. +-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \left(x \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right) + \color{blue}{0}\right), 2\right)\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(1 + \frac{1}{12} \cdot {x}^{2}\right)}, 0\right), 2\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(\frac{1}{12} \cdot {x}^{2} + \color{blue}{1}\right), 0\right), 2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \frac{1}{12} + 1\right), 0\right), 2\right)\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{12}}, 1\right), 0\right), 2\right)\right) \]

      10. +-rgt-identity N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(\left({x}^{2} + 0\right), \frac{1}{12}, 1\right), 0\right), 2\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(\left(x \cdot x + 0\right), \frac{1}{12}, 1\right), 0\right), 2\right)\right) \]

      12. accelerator-lowering-fma.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(x, x, 0\right), \frac{1}{12}, 1\right), 0\right), 2\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.08333333333333333, 1\right), 0\right), 2\right)}} \]

   6. Taylor expanded in x around inf

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

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

         \[\leadsto \mathsf{/.f64}\left(24, \color{blue}{\left({x}^{4}\right)}\right) \]

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(24, \left(x \cdot {x}^{\color{blue}{3}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(24, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{3}\right)}\right)\right) \]

      9. cube-mult N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 93.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+26}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.002777777777777778, 0.08333333333333333\right), 1\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x, 0\right), 0\right), 0\right)}{\mathsf{fma}\left(x, x, 0\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1e+26)
   (/
    1.0
    (fma
     (* x x)
     (fma x (* x (fma (* x x) 0.001388888888888889 0.041666666666666664)) 0.5)
     1.0))
   (if (<= x 1.35e+154)
     (/
      2.0
      (/
       (*
        (fma
         x
         (* x (fma (fma x x 0.0) 0.002777777777777778 0.08333333333333333))
         1.0)
        (fma x (fma x (fma x x 0.0) 0.0) 0.0))
       (fma x x 0.0)))
     (/ 2.0 (* x x)))))

C

double code(double x) {
	double tmp;
	if (x <= 1e+26) {
		tmp = 1.0 / fma((x * x), fma(x, (x * fma((x * x), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0);
	} else if (x <= 1.35e+154) {
		tmp = 2.0 / ((fma(x, (x * fma(fma(x, x, 0.0), 0.002777777777777778, 0.08333333333333333)), 1.0) * fma(x, fma(x, fma(x, x, 0.0), 0.0), 0.0)) / fma(x, x, 0.0));
	} else {
		tmp = 2.0 / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e+26)
		tmp = Float64(1.0 / fma(Float64(x * x), fma(x, Float64(x * fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0));
	elseif (x <= 1.35e+154)
		tmp = Float64(2.0 / Float64(Float64(fma(x, Float64(x * fma(fma(x, x, 0.0), 0.002777777777777778, 0.08333333333333333)), 1.0) * fma(x, fma(x, fma(x, x, 0.0), 0.0), 0.0)) / fma(x, x, 0.0)));
	else
		tmp = Float64(2.0 / Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1e+26], 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], If[LessEqual[x, 1.35e+154], N[(2.0 / N[(N[(N[(x * N[(x * N[(N[(x * x + 0.0), $MachinePrecision] * 0.002777777777777778 + 0.08333333333333333), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(x * N[(x * x + 0.0), $MachinePrecision] + 0.0), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision] / N[(x * x + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.00000000000000005e26

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. cosh-def N/A

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

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

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

      4. cosh-lowering-cosh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + \color{blue}{1}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}, 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right), 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right), 1\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \color{blue}{\frac{1}{2}}\right), 1\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}\right), 1\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + \frac{1}{2}\right), 1\right)\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}, \frac{1}{2}\right), 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right), \frac{1}{2}\right), 1\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{720} \cdot {x}^{2} + \color{blue}{\frac{1}{24}}\right)\right), \frac{1}{2}\right), 1\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{720} + \frac{1}{24}\right)\right), \frac{1}{2}\right), 1\right)\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}, \frac{1}{24}\right)\right), \frac{1}{2}\right), 1\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left(x \cdot x\right), \frac{1}{720}, \frac{1}{24}\right)\right), \frac{1}{2}\right), 1\right)\right) \]

      14. *-lowering-*.f64 92.4%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}, \frac{1}{24}\right)\right), \frac{1}{2}\right), 1\right)\right) \]

   7. Simplified 92.4%

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

   if 1.00000000000000005e26 < x < 1.35000000000000003e154

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. +-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(\left({x}^{2} + 0\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right), 2\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(\left(x \cdot x + 0\right), \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(x, x, 0\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right), 2\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(x, x, 0\right), \left({x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) + \color{blue}{1}\right), 2\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(x, x, 0\right), \left(\left(x \cdot x\right) \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) + 1\right), 2\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(x, x, 0\right), \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) + 1\right), 2\right)\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(x, x, 0\right), \mathsf{fma.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}, 1\right), 2\right)\right) \]

      10. +-rgt-identity N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(x, x, 0\right), \mathsf{fma.f64}\left(x, \left(x \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) + \color{blue}{0}\right), 1\right), 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(x, x, 0\right), \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)}, 0\right), 1\right), 2\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(x, x, 0\right), \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(\frac{1}{360} \cdot {x}^{2} + \color{blue}{\frac{1}{12}}\right), 0\right), 1\right), 2\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(x, x, 0\right), \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \frac{1}{360} + \frac{1}{12}\right), 0\right), 1\right), 2\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(x, x, 0\right), \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(\left(x \cdot x\right) \cdot \frac{1}{360} + \frac{1}{12}\right), 0\right), 1\right), 2\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(x, x, 0\right), \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \left(x \cdot \frac{1}{360}\right) + \frac{1}{12}\right), 0\right), 1\right), 2\right)\right) \]

      16. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(x, x, 0\right), \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(x \cdot \frac{1}{360}\right)}, \frac{1}{12}\right), 0\right), 1\right), 2\right)\right) \]

      17. +-rgt-identity N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(x, x, 0\right), \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \frac{1}{360} + \color{blue}{0}\right), \frac{1}{12}\right), 0\right), 1\right), 2\right)\right) \]

      18. accelerator-lowering-fma.f64 76.9%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(x, x, 0\right), \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\frac{1}{360}}, 0\right), \frac{1}{12}\right), 0\right), 1\right), 2\right)\right) \]

   5. Simplified 76.9%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.002777777777777778, 0\right), 0.08333333333333333\right), 0\right), 1\right), 2\right)}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({x}^{6} \cdot \left(\frac{1}{360} + \left(\frac{1}{12} \cdot \frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right)\right)\right)}\right) \]

   8. Simplified 76.9%

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

      1. +-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(x \cdot x + 0\right) \cdot \left(\color{blue}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{360} + \frac{1}{12}\right)} + 1\right)\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 0 \cdot 0}{x \cdot x - 0} \cdot \left(\color{blue}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{360} + \frac{1}{12}\right)} + 1\right)\right)\right) \]

      3. --rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 0 \cdot 0}{x \cdot x} \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \frac{1}{360} + \frac{1}{12}\right)} + 1\right)\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 0 \cdot 0\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{360} + \frac{1}{12}\right) + 1\right)}{\color{blue}{x \cdot x}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 0 \cdot 0\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{360} + \frac{1}{12}\right) + 1\right)\right), \color{blue}{\left(x \cdot x\right)}\right)\right) \]

   10. Applied egg-rr 88.2%

       \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x, 0\right), 0\right), 0\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.002777777777777778, 0.08333333333333333\right), 1\right)}{\mathsf{fma}\left(x, x, 0\right)}}} \]

   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

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. accelerator-lowering-fma.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+26}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.002777777777777778, 0.08333333333333333\right), 1\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x, 0\right), 0\right), 0\right)}{\mathsf{fma}\left(x, x, 0\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.3% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.5:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, x, 0\right), 0\right), x \cdot 0.08333333333333333, \mathsf{fma}\left(x, x, 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.5)
   (/
    2.0
    (fma (fma x (fma x x 0.0) 0.0) (* x 0.08333333333333333) (fma x x 2.0)))
   (/
    1.0
    (*
     (* x x)
     (* (* x x) (fma (* x x) 0.001388888888888889 0.041666666666666664))))))

C

double code(double x) {
	double tmp;
	if (x <= 4.5) {
		tmp = 2.0 / fma(fma(x, fma(x, x, 0.0), 0.0), (x * 0.08333333333333333), fma(x, x, 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 (x <= 4.5)
		tmp = Float64(2.0 / fma(fma(x, fma(x, x, 0.0), 0.0), Float64(x * 0.08333333333333333), fma(x, x, 2.0)));
	else
		tmp = Float64(1.0 / Float64(Float64(x * x) * Float64(Float64(x * x) * fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 4.5], N[(2.0 / N[(N[(x * N[(x * x + 0.0), $MachinePrecision] + 0.0), $MachinePrecision] * N[(x * 0.08333333333333333), $MachinePrecision] + N[(x * x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. +-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \left(x \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right) + \color{blue}{0}\right), 2\right)\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(1 + \frac{1}{12} \cdot {x}^{2}\right)}, 0\right), 2\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(\frac{1}{12} \cdot {x}^{2} + \color{blue}{1}\right), 0\right), 2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \frac{1}{12} + 1\right), 0\right), 2\right)\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{12}}, 1\right), 0\right), 2\right)\right) \]

      10. +-rgt-identity N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(\left({x}^{2} + 0\right), \frac{1}{12}, 1\right), 0\right), 2\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(\left(x \cdot x + 0\right), \frac{1}{12}, 1\right), 0\right), 2\right)\right) \]

      12. accelerator-lowering-fma.f64 93.1%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(x, x, 0\right), \frac{1}{12}, 1\right), 0\right), 2\right)\right) \]

   5. Simplified 93.1%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.08333333333333333, 1\right), 0\right), 2\right)}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{12} + 1\right)\right) + 2\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(x \cdot \left(\left(\left(x \cdot x + 0\right) \cdot \frac{1}{12}\right) \cdot x + 1 \cdot x\right) + 2\right)\right) \]

      3. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(x \cdot \left(\left(\left(x \cdot x + 0\right) \cdot \frac{1}{12}\right) \cdot x + x\right) + 2\right)\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(x \cdot \left(\left(\left(x \cdot x + 0\right) \cdot \frac{1}{12}\right) \cdot x\right) + x \cdot x\right) + 2\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(x \cdot \left(\left(\left(x \cdot x + 0\right) \cdot \frac{1}{12}\right) \cdot x\right) + \color{blue}{\left(x \cdot x + 2\right)}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{12}\right)\right) + \left(x \cdot \color{blue}{x} + 2\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{12}\right) + \left(\color{blue}{x \cdot x} + 2\right)\right)\right) \]

      8. +-rgt-identity N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. pow3 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \left({x}^{3} \cdot \left(x \cdot \frac{1}{12}\right) + \left(\color{blue}{x} \cdot x + 2\right)\right)\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(\left({x}^{3}\right), \color{blue}{\left(x \cdot \frac{1}{12}\right)}, \left(x \cdot x + 2\right)\right)\right) \]

   7. Applied egg-rr 93.1%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, x, 0\right), 0\right), x \cdot 0.08333333333333333, \mathsf{fma}\left(x, x, 2\right)\right)}} \]

   if 4.5 < x

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. cosh-def N/A

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

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

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

      4. cosh-lowering-cosh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + \color{blue}{1}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}, 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right), 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right), 1\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \color{blue}{\frac{1}{2}}\right), 1\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}\right), 1\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + \frac{1}{2}\right), 1\right)\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}, \frac{1}{2}\right), 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right), \frac{1}{2}\right), 1\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{720} \cdot {x}^{2} + \color{blue}{\frac{1}{24}}\right)\right), \frac{1}{2}\right), 1\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{720} + \frac{1}{24}\right)\right), \frac{1}{2}\right), 1\right)\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}, \frac{1}{24}\right)\right), \frac{1}{2}\right), 1\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left(x \cdot x\right), \frac{1}{720}, \frac{1}{24}\right)\right), \frac{1}{2}\right), 1\right)\right) \]

      14. *-lowering-*.f64 82.3%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}, \frac{1}{24}\right)\right), \frac{1}{2}\right), 1\right)\right) \]

   7. Simplified 82.3%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left({x}^{6} \cdot \left(\frac{1}{720} + \frac{1}{24} \cdot \frac{1}{{x}^{2}}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({x}^{6} \cdot \left(\frac{1}{24} \cdot \frac{1}{{x}^{2}} + \color{blue}{\frac{1}{720}}\right)\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{1}{24} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{6} + \color{blue}{\frac{1}{720} \cdot {x}^{6}}\right)\right) \]

   10. Simplified 82.3%

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

Alternative 6: 90.3% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.2:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, x, 0\right), 0\right), x \cdot 0.08333333333333333, \mathsf{fma}\left(x, x, 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{720}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 6.2)
   (/
    2.0
    (fma (fma x (fma x x 0.0) 0.0) (* x 0.08333333333333333) (fma x x 2.0)))
   (/ 720.0 (* x (* x (* x (* x (* x x))))))))

C

double code(double x) {
	double tmp;
	if (x <= 6.2) {
		tmp = 2.0 / fma(fma(x, fma(x, x, 0.0), 0.0), (x * 0.08333333333333333), fma(x, x, 2.0));
	} else {
		tmp = 720.0 / (x * (x * (x * (x * (x * x)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 6.2)
		tmp = Float64(2.0 / fma(fma(x, fma(x, x, 0.0), 0.0), Float64(x * 0.08333333333333333), fma(x, x, 2.0)));
	else
		tmp = Float64(720.0 / Float64(x * Float64(x * Float64(x * Float64(x * Float64(x * x))))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 6.2], N[(2.0 / N[(N[(x * N[(x * x + 0.0), $MachinePrecision] + 0.0), $MachinePrecision] * N[(x * 0.08333333333333333), $MachinePrecision] + N[(x * x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(720.0 / N[(x * N[(x * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.20000000000000018

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. +-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \left(x \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right) + \color{blue}{0}\right), 2\right)\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(1 + \frac{1}{12} \cdot {x}^{2}\right)}, 0\right), 2\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(\frac{1}{12} \cdot {x}^{2} + \color{blue}{1}\right), 0\right), 2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \frac{1}{12} + 1\right), 0\right), 2\right)\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{12}}, 1\right), 0\right), 2\right)\right) \]

      10. +-rgt-identity N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(\left({x}^{2} + 0\right), \frac{1}{12}, 1\right), 0\right), 2\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(\left(x \cdot x + 0\right), \frac{1}{12}, 1\right), 0\right), 2\right)\right) \]

      12. accelerator-lowering-fma.f64 93.1%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(x, x, 0\right), \frac{1}{12}, 1\right), 0\right), 2\right)\right) \]

   5. Simplified 93.1%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.08333333333333333, 1\right), 0\right), 2\right)}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{12} + 1\right)\right) + 2\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(x \cdot \left(\left(\left(x \cdot x + 0\right) \cdot \frac{1}{12}\right) \cdot x + 1 \cdot x\right) + 2\right)\right) \]

      3. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(x \cdot \left(\left(\left(x \cdot x + 0\right) \cdot \frac{1}{12}\right) \cdot x + x\right) + 2\right)\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(x \cdot \left(\left(\left(x \cdot x + 0\right) \cdot \frac{1}{12}\right) \cdot x\right) + x \cdot x\right) + 2\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(x \cdot \left(\left(\left(x \cdot x + 0\right) \cdot \frac{1}{12}\right) \cdot x\right) + \color{blue}{\left(x \cdot x + 2\right)}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(x \cdot \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{12}\right)\right) + \left(x \cdot \color{blue}{x} + 2\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{12}\right) + \left(\color{blue}{x \cdot x} + 2\right)\right)\right) \]

      8. +-rgt-identity N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. pow3 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \left({x}^{3} \cdot \left(x \cdot \frac{1}{12}\right) + \left(\color{blue}{x} \cdot x + 2\right)\right)\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(\left({x}^{3}\right), \color{blue}{\left(x \cdot \frac{1}{12}\right)}, \left(x \cdot x + 2\right)\right)\right) \]

   7. Applied egg-rr 93.1%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, x, 0\right), 0\right), x \cdot 0.08333333333333333, \mathsf{fma}\left(x, x, 2\right)\right)}} \]

   if 6.20000000000000018 < x

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. cosh-def N/A

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

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

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

      4. cosh-lowering-cosh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + \color{blue}{1}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}, 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right), 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right), 1\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \color{blue}{\frac{1}{2}}\right), 1\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}\right), 1\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + \frac{1}{2}\right), 1\right)\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}, \frac{1}{2}\right), 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right), \frac{1}{2}\right), 1\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{720} \cdot {x}^{2} + \color{blue}{\frac{1}{24}}\right)\right), \frac{1}{2}\right), 1\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{720} + \frac{1}{24}\right)\right), \frac{1}{2}\right), 1\right)\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}, \frac{1}{24}\right)\right), \frac{1}{2}\right), 1\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left(x \cdot x\right), \frac{1}{720}, \frac{1}{24}\right)\right), \frac{1}{2}\right), 1\right)\right) \]

      14. *-lowering-*.f64 82.3%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}, \frac{1}{24}\right)\right), \frac{1}{2}\right), 1\right)\right) \]

   7. Simplified 82.3%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{720}{{x}^{6}}} \]

   10. Simplified 82.3%

       \[\leadsto \color{blue}{\frac{720}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 92.5% 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. clear-num N/A

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

   2. cosh-def N/A

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

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

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

   4. cosh-lowering-cosh.f64 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + \color{blue}{1}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}, 1\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right), 1\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right), 1\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \color{blue}{\frac{1}{2}}\right), 1\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}\right), 1\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + \frac{1}{2}\right), 1\right)\right) \]

   8. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}, \frac{1}{2}\right), 1\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right), \frac{1}{2}\right), 1\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{720} \cdot {x}^{2} + \color{blue}{\frac{1}{24}}\right)\right), \frac{1}{2}\right), 1\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{720} + \frac{1}{24}\right)\right), \frac{1}{2}\right), 1\right)\right) \]

   12. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}, \frac{1}{24}\right)\right), \frac{1}{2}\right), 1\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left(x \cdot x\right), \frac{1}{720}, \frac{1}{24}\right)\right), \frac{1}{2}\right), 1\right)\right) \]

   14. *-lowering-*.f64 91.4%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}, \frac{1}{24}\right)\right), \frac{1}{2}\right), 1\right)\right) \]

7. Simplified 91.4%

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

Alternative 8: 90.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.2:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{720}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 6.2)
   (/ 1.0 (fma (* x x) (fma (* x x) 0.041666666666666664 0.5) 1.0))
   (/ 720.0 (* x (* x (* x (* x (* x x))))))))

C

double code(double x) {
	double tmp;
	if (x <= 6.2) {
		tmp = 1.0 / fma((x * x), fma((x * x), 0.041666666666666664, 0.5), 1.0);
	} else {
		tmp = 720.0 / (x * (x * (x * (x * (x * x)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 6.2)
		tmp = Float64(1.0 / fma(Float64(x * x), fma(Float64(x * x), 0.041666666666666664, 0.5), 1.0));
	else
		tmp = Float64(720.0 / Float64(x * Float64(x * Float64(x * Float64(x * Float64(x * x))))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 6.2], N[(1.0 / N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(720.0 / N[(x * N[(x * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.20000000000000018

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. cosh-def N/A

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

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

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

      4. cosh-lowering-cosh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 93.1%

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

   7. Simplified 93.1%

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

   if 6.20000000000000018 < x

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. cosh-def N/A

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

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

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

      4. cosh-lowering-cosh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + \color{blue}{1}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}, 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right), 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right), 1\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \color{blue}{\frac{1}{2}}\right), 1\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}\right), 1\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + \frac{1}{2}\right), 1\right)\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}, \frac{1}{2}\right), 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right), \frac{1}{2}\right), 1\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{720} \cdot {x}^{2} + \color{blue}{\frac{1}{24}}\right)\right), \frac{1}{2}\right), 1\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{720} + \frac{1}{24}\right)\right), \frac{1}{2}\right), 1\right)\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}, \frac{1}{24}\right)\right), \frac{1}{2}\right), 1\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\left(x \cdot x\right), \frac{1}{720}, \frac{1}{24}\right)\right), \frac{1}{2}\right), 1\right)\right) \]

      14. *-lowering-*.f64 82.3%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}, \frac{1}{24}\right)\right), \frac{1}{2}\right), 1\right)\right) \]

   7. Simplified 82.3%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{720}{{x}^{6}}} \]

   10. Simplified 82.3%

       \[\leadsto \color{blue}{\frac{720}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 88.4% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

   5. +-rgt-identity N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \left(x \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right) + \color{blue}{0}\right), 2\right)\right) \]

   6. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(1 + \frac{1}{12} \cdot {x}^{2}\right)}, 0\right), 2\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(\frac{1}{12} \cdot {x}^{2} + \color{blue}{1}\right), 0\right), 2\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \frac{1}{12} + 1\right), 0\right), 2\right)\right) \]

   9. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{12}}, 1\right), 0\right), 2\right)\right) \]

   10. +-rgt-identity N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(\left({x}^{2} + 0\right), \frac{1}{12}, 1\right), 0\right), 2\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(\left(x \cdot x + 0\right), \frac{1}{12}, 1\right), 0\right), 2\right)\right) \]

   12. accelerator-lowering-fma.f64 88.4%

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(x, x, 0\right), \frac{1}{12}, 1\right), 0\right), 2\right)\right) \]

5. Simplified 88.4%

   \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.08333333333333333, 1\right), 0\right), 2\right)}} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \left(x \cdot \color{blue}{\left(\left(x \cdot x + 0\right) \cdot \frac{1}{12} + 1\right)}\right), 2\right)\right) \]

   2. distribute-lft-in N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \left(x \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{12}\right) + \color{blue}{x \cdot 1}\right), 2\right)\right) \]

   3. +-rgt-identity N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-rgt-identity N/A

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

   7. associate-*r* N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

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

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

   10. *-lowering-*.f64 88.4%

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

7. Applied egg-rr 88.4%

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

8. Final simplification 88.4%

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

Alternative 10: 69.1% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.20833333333333334, -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (fma x (* x (fma x (* x 0.20833333333333334) -0.5)) 1.0)
   (/ 24.0 (* x (* x (* x x))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = fma(x, (x * fma(x, (x * 0.20833333333333334), -0.5)), 1.0);
	} else {
		tmp = 24.0 / (x * (x * (x * x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = fma(x, Float64(x * fma(x, Float64(x * 0.20833333333333334), -0.5)), 1.0);
	else
		tmp = Float64(24.0 / Float64(x * Float64(x * Float64(x * x))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[(x * N[(x * N[(x * N[(x * 0.20833333333333334), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(24.0 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. cosh-def N/A

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

      3. inv-pow N/A

         \[\leadsto {\cosh x}^{\color{blue}{-1}} \]

      4. sqr-pow N/A

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

      5. pow2 N/A

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

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

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

      7. metadata-eval N/A

         \[\leadsto \mathsf{pow.f64}\left(\left({\cosh x}^{\frac{-1}{2}}\right), 2\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{pow.f64}\left(\left({\cosh x}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), 2\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{pow.f64}\left(\left({\cosh x}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), 2\right) \]

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

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\cosh x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{cosh.f64}\left(x\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), 2\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{cosh.f64}\left(x\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), 2\right) \]

      13. metadata-eval 100.0%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{cosh.f64}\left(x\right), \frac{-1}{2}\right), 2\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left({\cosh x}^{-0.5}\right)}^{2}} \]

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

         \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{5}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)}, 1\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{5}{24} \cdot {x}^{2} - \frac{1}{2}\right)}\right), 1\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{5}{24} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right), 1\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{5}{24} \cdot \left(x \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), 1\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(\frac{5}{24} \cdot x\right) \cdot x + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)\right)\right), 1\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{5}{24} \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)\right)\right), 1\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{5}{24} \cdot x\right) + \frac{-1}{2}\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(\frac{5}{24} \cdot x\right)}, \frac{-1}{2}\right)\right), 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \color{blue}{\frac{5}{24}}\right), \frac{-1}{2}\right)\right), 1\right) \]

      13. *-lowering-*.f64 66.0%

          \[\leadsto \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{5}{24}}\right), \frac{-1}{2}\right)\right), 1\right) \]

   7. Simplified 66.0%

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

   if 1.8500000000000001 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. +-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \left(x \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right) + \color{blue}{0}\right), 2\right)\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(1 + \frac{1}{12} \cdot {x}^{2}\right)}, 0\right), 2\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(\frac{1}{12} \cdot {x}^{2} + \color{blue}{1}\right), 0\right), 2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \frac{1}{12} + 1\right), 0\right), 2\right)\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{12}}, 1\right), 0\right), 2\right)\right) \]

      10. +-rgt-identity N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(\left({x}^{2} + 0\right), \frac{1}{12}, 1\right), 0\right), 2\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(\left(x \cdot x + 0\right), \frac{1}{12}, 1\right), 0\right), 2\right)\right) \]

      12. accelerator-lowering-fma.f64 75.4%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(x, x, 0\right), \frac{1}{12}, 1\right), 0\right), 2\right)\right) \]

   5. Simplified 75.4%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.08333333333333333, 1\right), 0\right), 2\right)}} \]

   6. Taylor expanded in x around inf

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

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

         \[\leadsto \mathsf{/.f64}\left(24, \color{blue}{\left({x}^{4}\right)}\right) \]

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(24, \left(x \cdot {x}^{\color{blue}{3}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(24, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{3}\right)}\right)\right) \]

      9. cube-mult N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 75.4%

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

   8. Simplified 75.4%

      \[\leadsto \color{blue}{\frac{24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 88.0% accurate, 6.6× speedup

Math

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

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(x, Float64(x * Float64(Float64(x * x) * 0.08333333333333333)), 2.0))
end

Wolfram

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

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

   5. +-rgt-identity N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \left(x \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right) + \color{blue}{0}\right), 2\right)\right) \]

   6. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(1 + \frac{1}{12} \cdot {x}^{2}\right)}, 0\right), 2\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(\frac{1}{12} \cdot {x}^{2} + \color{blue}{1}\right), 0\right), 2\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left({x}^{2} \cdot \frac{1}{12} + 1\right), 0\right), 2\right)\right) \]

   9. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{12}}, 1\right), 0\right), 2\right)\right) \]

   10. +-rgt-identity N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(\left({x}^{2} + 0\right), \frac{1}{12}, 1\right), 0\right), 2\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(\left(x \cdot x + 0\right), \frac{1}{12}, 1\right), 0\right), 2\right)\right) \]

   12. accelerator-lowering-fma.f64 88.4%

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(x, x, 0\right), \frac{1}{12}, 1\right), 0\right), 2\right)\right) \]

5. Simplified 88.4%

   \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.08333333333333333, 1\right), 0\right), 2\right)}} \]

6. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(2, \mathsf{fma.f64}\left(x, \color{blue}{\left(\frac{1}{12} \cdot {x}^{3}\right)}, 2\right)\right) \]
7. Step-by-step derivation

   1. unpow3 N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. *-commutative N/A

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 88.3%

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

8. Simplified 88.3%

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

Alternative 12: 88.0% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(1.0 / N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $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. clear-num N/A

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

   2. cosh-def N/A

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

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

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

   4. cosh-lowering-cosh.f64 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

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

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

   3. unpow2 N/A

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

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

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 88.0%

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

7. Simplified 88.0%

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

8. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

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

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 88.0%

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

10. Simplified 88.0%

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

Alternative 13: 63.1% accurate, 9.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. cosh-def N/A

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

      3. inv-pow N/A

         \[\leadsto {\cosh x}^{\color{blue}{-1}} \]

      4. sqr-pow N/A

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

      5. pow2 N/A

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

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

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

      7. metadata-eval N/A

         \[\leadsto \mathsf{pow.f64}\left(\left({\cosh x}^{\frac{-1}{2}}\right), 2\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{pow.f64}\left(\left({\cosh x}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), 2\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{pow.f64}\left(\left({\cosh x}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), 2\right) \]

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

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\cosh x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{cosh.f64}\left(x\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), 2\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{cosh.f64}\left(x\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), 2\right) \]

      13. metadata-eval 100.0%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{cosh.f64}\left(x\right), \frac{-1}{2}\right), 2\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left({\cosh x}^{-0.5}\right)}^{2}} \]

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 66.1%

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

   7. Simplified 66.1%

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

   if 1.25 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. accelerator-lowering-fma.f64 49.3%

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

   5. Simplified 49.3%

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

   6. Taylor expanded in x around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 49.3%

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

   8. Simplified 49.3%

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

Alternative 14: 76.5% 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

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. accelerator-lowering-fma.f64 73.9%

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

5. Simplified 73.9%

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

Alternative 15: 50.9% 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. Simplified 49.4%

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

Reproduce

herbie shell --seed 2024193 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))