Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 9.4s

Alternatives: 19

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. cosh-undef N/A

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

   2. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{2}{2}}{\cosh x}} \]

   3. metadata-eval N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cosh.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 88.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} + e^{-x} \leq 4:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.20833333333333334, -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{360}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (exp x) (exp (- x))) 4.0)
   (fma (* x x) (fma x (* x 0.20833333333333334) -0.5) 1.0)
   (/ 360.0 (* x (* x (* x x))))))

C

double code(double x) {
	double tmp;
	if ((exp(x) + exp(-x)) <= 4.0) {
		tmp = fma((x * x), fma(x, (x * 0.20833333333333334), -0.5), 1.0);
	} else {
		tmp = 360.0 / (x * (x * (x * x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) + exp(Float64(-x))) <= 4.0)
		tmp = fma(Float64(x * x), fma(x, Float64(x * 0.20833333333333334), -0.5), 1.0);
	else
		tmp = Float64(360.0 / Float64(x * Float64(x * Float64(x * x))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 4.0], N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.20833333333333334), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(360.0 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. cosh-undef N/A

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

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{2}}{\cosh x}} \]

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cosh.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   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

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 99.7

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

   7. Simplified 99.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 82.7

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

   5. Simplified 82.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 62.4

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

   8. Simplified 62.4%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(x, \color{blue}{{x}^{3} \cdot \frac{1}{360}}, 2\right)} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 71.2

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

   11. Simplified 71.2%

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

   12. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{360}{{x}^{4}}} \]
   13. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{360}{{x}^{4}}} \]

       2. metadata-eval N/A

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

       3. pow-sqr N/A

          \[\leadsto \frac{360}{\color{blue}{{x}^{2} \cdot {x}^{2}}} \]

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. unpow2 N/A

          \[\leadsto \frac{360}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)} \]

       7. cube-mult N/A

          \[\leadsto \frac{360}{x \cdot \color{blue}{{x}^{3}}} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{360}{\color{blue}{x \cdot {x}^{3}}} \]

       9. cube-mult N/A

          \[\leadsto \frac{360}{x \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}} \]

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

           \[\leadsto \frac{360}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)} \]

       13. lower-*.f64 71.2

           \[\leadsto \frac{360}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)} \]

   14. Simplified 71.2%

       \[\leadsto \color{blue}{\frac{360}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 83.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} + e^{-x} \leq 2 \cdot 10^{+50}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(x, x, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{360}{x \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (exp x) (exp (- x))) 2e+50)
   (/ 2.0 (fma x x 2.0))
   (/ 360.0 (* x (* x x)))))

C

double code(double x) {
	double tmp;
	if ((exp(x) + exp(-x)) <= 2e+50) {
		tmp = 2.0 / fma(x, x, 2.0);
	} else {
		tmp = 360.0 / (x * (x * x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) + exp(Float64(-x))) <= 2e+50)
		tmp = Float64(2.0 / fma(x, x, 2.0));
	else
		tmp = Float64(360.0 / Float64(x * Float64(x * x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 2e+50], N[(2.0 / N[(x * x + 2.0), $MachinePrecision]), $MachinePrecision], N[(360.0 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 2.0000000000000002e50

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 98.2

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

   5. Simplified 98.2%

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

   if 2.0000000000000002e50 < (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 83.8

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

   5. Simplified 83.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 63.3

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

   8. Simplified 63.3%

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

   9. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{\frac{360}{{x}^{3}}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{360}{{x}^{3}}} \]

       2. cube-mult N/A

          \[\leadsto \frac{360}{\color{blue}{x \cdot \left(x \cdot x\right)}} \]

       3. unpow2 N/A

          \[\leadsto \frac{360}{x \cdot \color{blue}{{x}^{2}}} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{360}{\color{blue}{x \cdot {x}^{2}}} \]

       5. unpow2 N/A

          \[\leadsto \frac{360}{x \cdot \color{blue}{\left(x \cdot x\right)}} \]

       6. lower-*.f64 63.3

          \[\leadsto \frac{360}{x \cdot \color{blue}{\left(x \cdot x\right)}} \]

   11. Simplified 63.3%

       \[\leadsto \color{blue}{\frac{360}{x \cdot \left(x \cdot x\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 95.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.002777777777777778, t\_0 \cdot t\_0, 0.08333333333333333\right), 1\right), x \cdot x, 2\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (/
    2.0
    (fma
     (fma
      (* x x)
      (fma 0.002777777777777778 (* t_0 t_0) 0.08333333333333333)
      1.0)
     (* x x)
     2.0))))

C

double code(double x) {
	double t_0 = x * (x * x);
	return 2.0 / fma(fma((x * x), fma(0.002777777777777778, (t_0 * t_0), 0.08333333333333333), 1.0), (x * x), 2.0);
}

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	return Float64(2.0 / fma(fma(Float64(x * x), fma(0.002777777777777778, Float64(t_0 * t_0), 0.08333333333333333), 1.0), Float64(x * x), 2.0))
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 91.1

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

5. Simplified 91.1%

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

6. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. pow-sqr N/A

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

   3. distribute-lft-in N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. pow-sqr N/A

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

   7. metadata-eval N/A

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

   8. lft-mult-inverse N/A

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

   9. metadata-eval N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. unpow2 N/A

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

   13. cube-mult N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\left(x \cdot \color{blue}{{x}^{3}}\right) \cdot \frac{1}{360} + \frac{1}{12}\right), 1\right), 2\right)} \]

   14. associate-*r* N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left({x}^{3} \cdot \frac{1}{360}\right)} + \frac{1}{12}\right), 1\right), 2\right)} \]

   15. *-commutative N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(x \cdot \color{blue}{\left(\frac{1}{360} \cdot {x}^{3}\right)} + \frac{1}{12}\right), 1\right), 2\right)} \]

   16. lower-fma.f64 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{1}{360} \cdot {x}^{3}, \frac{1}{12}\right)}, 1\right), 2\right)} \]

8. Simplified 92.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 92.1

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

10. Applied egg-rr 92.1%

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

11. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 N/A

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

    3. unpow2 N/A

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

    4. lower-*.f64 N/A

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

    5. +-commutative N/A

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

    6. lower-fma.f64 N/A

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

    7. metadata-eval N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{360}, {x}^{\color{blue}{\left(2 \cdot 3\right)}}, \frac{1}{12}\right), 1\right), x \cdot x, 2\right)} \]

    8. pow-sqr N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{360}, \color{blue}{{x}^{3} \cdot {x}^{3}}, \frac{1}{12}\right), 1\right), x \cdot x, 2\right)} \]

    9. lower-*.f64 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{360}, \color{blue}{{x}^{3} \cdot {x}^{3}}, \frac{1}{12}\right), 1\right), x \cdot x, 2\right)} \]

    10. cube-mult N/A

        \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{360}, \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot {x}^{3}, \frac{1}{12}\right), 1\right), x \cdot x, 2\right)} \]

    11. unpow2 N/A

        \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{360}, \left(x \cdot \color{blue}{{x}^{2}}\right) \cdot {x}^{3}, \frac{1}{12}\right), 1\right), x \cdot x, 2\right)} \]

    12. lower-*.f64 N/A

        \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{360}, \color{blue}{\left(x \cdot {x}^{2}\right)} \cdot {x}^{3}, \frac{1}{12}\right), 1\right), x \cdot x, 2\right)} \]

    13. unpow2 N/A

        \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{360}, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot {x}^{3}, \frac{1}{12}\right), 1\right), x \cdot x, 2\right)} \]

    14. lower-*.f64 N/A

        \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{360}, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot {x}^{3}, \frac{1}{12}\right), 1\right), x \cdot x, 2\right)} \]

    15. cube-mult N/A

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

    16. unpow2 N/A

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

    17. lower-*.f64 N/A

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

    18. unpow2 N/A

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

    19. lower-*.f64 93.2

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

13. Simplified 93.2%

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

Alternative 5: 95.0% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 91.1

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

5. Simplified 91.1%

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

6. Taylor expanded in x around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 90.9

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

8. Simplified 90.9%

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

9. Taylor expanded in x around inf

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

    1. mul-1-neg N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{neg}\left({x}^{8}\right)}, 2\right)} \]

    2. metadata-eval N/A

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

    3. pow-sqr N/A

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

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

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

    5. metadata-eval N/A

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

    6. pow-sqr N/A

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

    7. distribute-rgt-neg-out N/A

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

    8. mul-1-neg N/A

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

    9. lower-*.f64 N/A

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

    10. metadata-eval N/A

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

    11. pow-sqr N/A

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

    12. unpow2 N/A

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

    13. associate-*l* N/A

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

    14. unpow2 N/A

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

    15. cube-mult N/A

        \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \left(x \cdot \color{blue}{{x}^{3}}\right) \cdot \left({x}^{2} \cdot \left(-1 \cdot {x}^{2}\right)\right), 2\right)} \]

    16. lower-*.f64 N/A

        \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot {x}^{3}\right)} \cdot \left({x}^{2} \cdot \left(-1 \cdot {x}^{2}\right)\right), 2\right)} \]

    17. cube-mult N/A

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

    18. unpow2 N/A

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

    19. lower-*.f64 N/A

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

    20. unpow2 N/A

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

    21. lower-*.f64 N/A

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

    22. unpow2 N/A

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

    23. associate-*l* N/A

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

    24. *-commutative N/A

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

    25. associate-*r* N/A

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

11. Simplified 92.9%

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

Alternative 6: 94.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \frac{2}{\mathsf{fma}\left(0.08333333333333333 \cdot \left(x \cdot \left(t\_0 \cdot t\_0\right)\right), x \cdot x, 2\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (/ 2.0 (fma (* 0.08333333333333333 (* x (* t_0 t_0))) (* x x) 2.0))))

C

double code(double x) {
	double t_0 = x * (x * x);
	return 2.0 / fma((0.08333333333333333 * (x * (t_0 * t_0))), (x * x), 2.0);
}

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	return Float64(2.0 / fma(Float64(0.08333333333333333 * Float64(x * Float64(t_0 * t_0))), Float64(x * x), 2.0))
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 91.1

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

5. Simplified 91.1%

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

6. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. pow-sqr N/A

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

   3. distribute-lft-in N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. pow-sqr N/A

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

   7. metadata-eval N/A

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

   8. lft-mult-inverse N/A

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

   9. metadata-eval N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. unpow2 N/A

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

   13. cube-mult N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\left(x \cdot \color{blue}{{x}^{3}}\right) \cdot \frac{1}{360} + \frac{1}{12}\right), 1\right), 2\right)} \]

   14. associate-*r* N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left({x}^{3} \cdot \frac{1}{360}\right)} + \frac{1}{12}\right), 1\right), 2\right)} \]

   15. *-commutative N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(x \cdot \color{blue}{\left(\frac{1}{360} \cdot {x}^{3}\right)} + \frac{1}{12}\right), 1\right), 2\right)} \]

   16. lower-fma.f64 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{1}{360} \cdot {x}^{3}, \frac{1}{12}\right)}, 1\right), 2\right)} \]

8. Simplified 92.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 92.1

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

10. Applied egg-rr 92.1%

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

11. Taylor expanded in x around inf

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

    1. *-commutative N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{{x}^{7} \cdot \frac{1}{12}}, x \cdot x, 2\right)} \]

    2. lower-*.f64 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{{x}^{7} \cdot \frac{1}{12}}, x \cdot x, 2\right)} \]

    3. metadata-eval N/A

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

    4. pow-plus N/A

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

    5. *-commutative N/A

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

    6. lower-*.f64 N/A

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

    7. metadata-eval N/A

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

    8. pow-sqr N/A

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

    9. lower-*.f64 N/A

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

    10. cube-mult N/A

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

    11. unpow2 N/A

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

    12. lower-*.f64 N/A

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

    13. unpow2 N/A

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

    14. lower-*.f64 N/A

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

    15. cube-mult N/A

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

    16. unpow2 N/A

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

    17. lower-*.f64 N/A

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

    18. unpow2 N/A

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

    19. lower-*.f64 92.2

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

13. Simplified 92.2%

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

14. Final simplification 92.2%

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

Alternative 7: 94.0% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 9.259259259259259 \cdot 10^{-5}, 0.002777777777777778\right), 0.08333333333333333\right), 1\right), 2\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  2.0
  (fma
   (* x x)
   (fma
    (* x x)
    (fma
     (* x x)
     (fma (* x x) 9.259259259259259e-5 0.002777777777777778)
     0.08333333333333333)
    1.0)
   2.0)))

C

double code(double x) {
	return 2.0 / fma((x * x), fma((x * x), fma((x * x), fma((x * x), 9.259259259259259e-5, 0.002777777777777778), 0.08333333333333333), 1.0), 2.0);
}

Julia

function code(x)
	return Float64(2.0 / fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), 9.259259259259259e-5, 0.002777777777777778), 0.08333333333333333), 1.0), 2.0))
end

Wolfram

code[x_] := N[(2.0 / N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 9.259259259259259e-5 + 0.002777777777777778), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 91.1

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

5. Simplified 91.1%

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

   1. lift-*.f64 N/A

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

   2. flip-+ N/A

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

   3. lower-/.f64 N/A

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

7. Applied egg-rr 66.1%

   \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\frac{\mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), 7.71604938271605 \cdot 10^{-6}, -0.006944444444444444\right)}{\mathsf{fma}\left(x, x \cdot 0.002777777777777778, -0.08333333333333333\right)}}, 1\right), 2\right)} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{10800} \cdot {x}^{2}\right)\right)}, 2\right)} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{10800} \cdot {x}^{2}\right)\right) + 1}, 2\right)} \]

   2. lower-fma.f64 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{10800} \cdot {x}^{2}\right), 1\right)}, 2\right)} \]

   3. unpow2 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{10800} \cdot {x}^{2}\right), 1\right), 2\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{10800} \cdot {x}^{2}\right), 1\right), 2\right)} \]

   5. +-commutative N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{10800} \cdot {x}^{2}\right) + \frac{1}{12}}, 1\right), 2\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{360} + \frac{1}{10800} \cdot {x}^{2}, \frac{1}{12}\right)}, 1\right), 2\right)} \]

   7. unpow2 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{360} + \frac{1}{10800} \cdot {x}^{2}, \frac{1}{12}\right), 1\right), 2\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{360} + \frac{1}{10800} \cdot {x}^{2}, \frac{1}{12}\right), 1\right), 2\right)} \]

   9. +-commutative N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{10800} \cdot {x}^{2} + \frac{1}{360}}, \frac{1}{12}\right), 1\right), 2\right)} \]

   10. *-commutative N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{10800}} + \frac{1}{360}, \frac{1}{12}\right), 1\right), 2\right)} \]

   11. lower-fma.f64 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{10800}, \frac{1}{360}\right)}, \frac{1}{12}\right), 1\right), 2\right)} \]

   12. unpow2 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{10800}, \frac{1}{360}\right), \frac{1}{12}\right), 1\right), 2\right)} \]

   13. lower-*.f64 92.2

       \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 9.259259259259259 \cdot 10^{-5}, 0.002777777777777778\right), 0.08333333333333333\right), 1\right), 2\right)} \]

10. Simplified 92.2%

    \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 9.259259259259259 \cdot 10^{-5}, 0.002777777777777778\right), 0.08333333333333333\right), 1\right)}, 2\right)} \]
11. Add Preprocessing

Alternative 8: 94.0% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 91.1

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

5. Simplified 91.1%

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

6. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. pow-sqr N/A

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

   3. distribute-lft-in N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. pow-sqr N/A

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

   7. metadata-eval N/A

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

   8. lft-mult-inverse N/A

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

   9. metadata-eval N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. unpow2 N/A

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

   13. cube-mult N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\left(x \cdot \color{blue}{{x}^{3}}\right) \cdot \frac{1}{360} + \frac{1}{12}\right), 1\right), 2\right)} \]

   14. associate-*r* N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left({x}^{3} \cdot \frac{1}{360}\right)} + \frac{1}{12}\right), 1\right), 2\right)} \]

   15. *-commutative N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(x \cdot \color{blue}{\left(\frac{1}{360} \cdot {x}^{3}\right)} + \frac{1}{12}\right), 1\right), 2\right)} \]

   16. lower-fma.f64 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{1}{360} \cdot {x}^{3}, \frac{1}{12}\right)}, 1\right), 2\right)} \]

8. Simplified 92.1%

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

9. Final simplification 92.1%

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

Alternative 9: 94.0% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 91.1

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

5. Simplified 91.1%

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

6. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. pow-sqr N/A

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

   3. distribute-lft-in N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. pow-sqr N/A

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

   7. metadata-eval N/A

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

   8. lft-mult-inverse N/A

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

   9. metadata-eval N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. unpow2 N/A

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

   13. cube-mult N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\left(x \cdot \color{blue}{{x}^{3}}\right) \cdot \frac{1}{360} + \frac{1}{12}\right), 1\right), 2\right)} \]

   14. associate-*r* N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left({x}^{3} \cdot \frac{1}{360}\right)} + \frac{1}{12}\right), 1\right), 2\right)} \]

   15. *-commutative N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(x \cdot \color{blue}{\left(\frac{1}{360} \cdot {x}^{3}\right)} + \frac{1}{12}\right), 1\right), 2\right)} \]

   16. lower-fma.f64 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{1}{360} \cdot {x}^{3}, \frac{1}{12}\right)}, 1\right), 2\right)} \]

8. Simplified 92.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 92.1

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

10. Applied egg-rr 92.1%

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

11. Taylor expanded in x around inf

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

    1. unpow2 N/A

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

    2. lower-*.f64 92.1

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

13. Simplified 92.1%

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

Alternative 10: 93.1% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 91.1

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

5. Simplified 91.1%

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

6. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{3}\right)}, 1\right), 2\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{360} \cdot {x}^{3} + \frac{1}{12}\right)}, 1\right), 2\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{3} \cdot \frac{1}{360}} + \frac{1}{12}\right), 1\right), 2\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{1}{360}, \frac{1}{12}\right)}, 1\right), 2\right)} \]

   4. cube-mult N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 92.0

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

8. Simplified 92.0%

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

Alternative 11: 93.1% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 91.1

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

5. Simplified 91.1%

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

6. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. pow-sqr N/A

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

   3. distribute-lft-in N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. pow-sqr N/A

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

   7. metadata-eval N/A

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

   8. lft-mult-inverse N/A

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

   9. metadata-eval N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. unpow2 N/A

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

   13. cube-mult N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\left(x \cdot \color{blue}{{x}^{3}}\right) \cdot \frac{1}{360} + \frac{1}{12}\right), 1\right), 2\right)} \]

   14. associate-*r* N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left({x}^{3} \cdot \frac{1}{360}\right)} + \frac{1}{12}\right), 1\right), 2\right)} \]

   15. *-commutative N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(x \cdot \color{blue}{\left(\frac{1}{360} \cdot {x}^{3}\right)} + \frac{1}{12}\right), 1\right), 2\right)} \]

   16. lower-fma.f64 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{1}{360} \cdot {x}^{3}, \frac{1}{12}\right)}, 1\right), 2\right)} \]

8. Simplified 92.1%

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

9. Taylor expanded in x around inf

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

    1. unpow2 N/A

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

    2. lower-*.f64 92.0

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

11. Simplified 92.0%

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

Alternative 12: 93.0% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 91.1

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

5. Simplified 91.1%

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

6. Taylor expanded in x around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 90.9

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

8. Simplified 90.9%

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

9. Taylor expanded in x around -inf

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

    1. cube-mult N/A

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

    2. unpow2 N/A

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

    3. lower-*.f64 N/A

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

    4. unpow2 N/A

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

    5. lower-*.f64 91.9

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

11. Simplified 91.9%

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

Alternative 13: 92.1% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 91.1

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

5. Simplified 91.1%

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

6. Taylor expanded in x around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 90.9

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

8. Simplified 90.9%

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

Alternative 14: 87.9% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{2}{2}}{\cosh x}} \]

   3. metadata-eval N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cosh.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 85.1

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

7. Simplified 85.1%

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

Alternative 15: 87.6% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 91.1

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

5. Simplified 91.1%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 80.0

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

8. Simplified 80.0%

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

9. Taylor expanded in x around inf

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

    1. *-commutative N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \color{blue}{{x}^{3} \cdot \frac{1}{360}}, 2\right)} \]

    2. cube-mult N/A

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

    3. unpow2 N/A

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

    4. associate-*l* N/A

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

    5. *-commutative N/A

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

    6. lower-*.f64 N/A

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

    7. *-commutative N/A

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

    8. lower-*.f64 N/A

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

    9. unpow2 N/A

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

    10. lower-*.f64 84.9

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

11. Simplified 84.9%

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

12. Taylor expanded in x around inf

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

    1. unpow2 N/A

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

    2. lower-*.f64 84.9

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

14. Simplified 84.9%

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

Alternative 16: 83.6% accurate, 9.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 91.1

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

5. Simplified 91.1%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 80.0

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

8. Simplified 80.0%

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

9. Taylor expanded in x around inf

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

    1. *-commutative N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \color{blue}{{x}^{3} \cdot \frac{1}{360}}, 2\right)} \]

    2. cube-mult N/A

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

    3. unpow2 N/A

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

    4. associate-*l* N/A

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

    5. *-commutative N/A

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

    6. lower-*.f64 N/A

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

    7. *-commutative N/A

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

    8. lower-*.f64 N/A

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

    9. unpow2 N/A

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

    10. lower-*.f64 84.9

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

11. Simplified 84.9%

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

12. Taylor expanded in x around inf

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

    1. unpow2 N/A

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

    2. lower-*.f64 80.4

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

14. Simplified 80.4%

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

Alternative 17: 76.6% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. lower-fma.f64 74.9

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

5. Simplified 74.9%

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

Alternative 18: 52.1% 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. Step-by-step derivation

   1. cosh-undef N/A

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

   2. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{2}{2}}{\cosh x}} \]

   3. metadata-eval N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cosh.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

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

7. Simplified 50.1%

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

Alternative 19: 11.1% accurate, 217.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.5)

C

double code(double x) {
	return 0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0
end function

Java

public static double code(double x) {
	return 0.5;
}

Python

def code(x):
	return 0.5

Julia

function code(x)
	return 0.5
end

MATLAB

function tmp = code(x)
	tmp = 0.5;
end

Wolfram

code[x_] := 0.5

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

5. Simplified 10.8%

   \[\leadsto \color{blue}{0.5} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))