Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 9.3s

Alternatives: 13

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: 91.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} + e^{-x} \leq 4:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{8}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (exp x) (exp (- x))) 4.0)
   (/ 1.0 (fma (* x x) (fma (* x x) 0.041666666666666664 0.5) 1.0))
   (/ 8.0 (* x (* x (* x (* x (* x x))))))))

C

double code(double x) {
	double tmp;
	if ((exp(x) + exp(-x)) <= 4.0) {
		tmp = 1.0 / fma((x * x), fma((x * x), 0.041666666666666664, 0.5), 1.0);
	} else {
		tmp = 8.0 / (x * (x * (x * (x * (x * x)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) + exp(Float64(-x))) <= 4.0)
		tmp = Float64(1.0 / fma(Float64(x * x), fma(Float64(x * x), 0.041666666666666664, 0.5), 1.0));
	else
		tmp = Float64(8.0 / Float64(x * Float64(x * 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[(1.0 / N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(8.0 / N[(x * N[(x * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.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

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 99.6

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 99.5%

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

   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

      \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2}}} \]
   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 49.4

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

   5. Simplified 49.4%

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

      1. lift-*.f64 N/A

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

      2. flip3-+ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{\left(x \cdot x\right)}^{3} + {2}^{3}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 \cdot 2 - \left(x \cdot x\right) \cdot 2\right)}}} \]

      3. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{2}{{\left(x \cdot x\right)}^{3} + {2}^{3}} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 \cdot 2 - \left(x \cdot x\right) \cdot 2\right)\right)} \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 \cdot 2 - \left(x \cdot x\right) \cdot 2\right)\right)}{{\left(x \cdot x\right)}^{3} + {2}^{3}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 \cdot 2 - \left(x \cdot x\right) \cdot 2\right)\right)}{{\left(x \cdot x\right)}^{3} + {2}^{3}}} \]

   7. Applied egg-rr 9.2%

      \[\leadsto \color{blue}{\frac{2 \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x, -2\right), 4\right)}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), x \cdot \left(x \cdot x\right), 8\right)}} \]

   8. Taylor expanded in x around 0

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

   10. Simplified 82.4%

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

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{8}{{x}^{6}}} \]
   12. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{8}{{x}^{6}}} \]

       2. metadata-eval N/A

          \[\leadsto \frac{8}{{x}^{\color{blue}{\left(5 + 1\right)}}} \]

       3. metadata-eval N/A

          \[\leadsto \frac{8}{{x}^{\left(\color{blue}{\left(4 + 1\right)} + 1\right)}} \]

       4. pow-plus N/A

          \[\leadsto \frac{8}{\color{blue}{{x}^{\left(4 + 1\right)} \cdot x}} \]

       5. pow-plus N/A

          \[\leadsto \frac{8}{\color{blue}{\left({x}^{4} \cdot x\right)} \cdot x} \]

       6. *-commutative N/A

          \[\leadsto \frac{8}{\color{blue}{x \cdot \left({x}^{4} \cdot x\right)}} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{8}{\color{blue}{x \cdot \left({x}^{4} \cdot x\right)}} \]

       8. *-commutative N/A

          \[\leadsto \frac{8}{x \cdot \color{blue}{\left(x \cdot {x}^{4}\right)}} \]

       9. lower-*.f64 N/A

          \[\leadsto \frac{8}{x \cdot \color{blue}{\left(x \cdot {x}^{4}\right)}} \]

       10. metadata-eval N/A

           \[\leadsto \frac{8}{x \cdot \left(x \cdot {x}^{\color{blue}{\left(3 + 1\right)}}\right)} \]

       11. pow-plus N/A

           \[\leadsto \frac{8}{x \cdot \left(x \cdot \color{blue}{\left({x}^{3} \cdot x\right)}\right)} \]

       12. *-commutative N/A

           \[\leadsto \frac{8}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot {x}^{3}\right)}\right)} \]

       13. lower-*.f64 N/A

           \[\leadsto \frac{8}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot {x}^{3}\right)}\right)} \]

       14. cube-mult N/A

           \[\leadsto \frac{8}{x \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}\right)\right)} \]

       15. unpow2 N/A

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

       16. lower-*.f64 N/A

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

       17. unpow2 N/A

           \[\leadsto \frac{8}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)} \]

       18. lower-*.f64 82.4

           \[\leadsto \frac{8}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)} \]

   13. Simplified 82.4%

       \[\leadsto \color{blue}{\frac{8}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 88.0% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (exp x) (exp (- x))) 4.0)
   (fma x (* x (fma (* x x) 0.20833333333333334 -0.5)) 1.0)
   (/ 2.0 (* x (fma 0.08333333333333333 (* 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 = 2.0 / (x * fma(0.08333333333333333, (x * (x * x)), x));
	}
	return tmp;
}

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 99.4

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

   5. Simplified 99.4%

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

   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

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 74.3

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

   5. Simplified 74.3%

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

   6. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

         \[\leadsto \frac{2}{\color{blue}{{x}^{4} \cdot \frac{1}{12} + {x}^{4} \cdot \frac{1}{{x}^{2}}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{1}{12} \cdot {x}^{4}} + {x}^{4} \cdot \frac{1}{{x}^{2}}} \]

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

      10. associate-/l* N/A

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

      11. *-rgt-identity N/A

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

      12. associate-*r/ N/A

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

      13. rgt-mult-inverse N/A

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

      14. *-rgt-identity N/A

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

      15. distribute-lft1-in N/A

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

      16. +-commutative N/A

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

      17. *-commutative N/A

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

      18. unpow2 N/A

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

      19. associate-*l* N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

   8. Simplified 74.3%

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

Alternative 4: 88.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} + e^{-x} \leq 4:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.20833333333333334, -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (exp x) (exp (- x))) 4.0)
   (fma x (* x (fma (* x x) 0.20833333333333334 -0.5)) 1.0)
   (/ 24.0 (* x (* x (* x x))))))

C

double code(double x) {
	double tmp;
	if ((exp(x) + exp(-x)) <= 4.0) {
		tmp = fma(x, (x * fma((x * x), 0.20833333333333334, -0.5)), 1.0);
	} else {
		tmp = 24.0 / (x * (x * (x * x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) + exp(Float64(-x))) <= 4.0)
		tmp = fma(x, Float64(x * fma(Float64(x * x), 0.20833333333333334, -0.5)), 1.0);
	else
		tmp = Float64(24.0 / Float64(x * Float64(x * Float64(x * x))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 4.0], N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.20833333333333334 + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(24.0 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 99.4

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

   5. Simplified 99.4%

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

   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

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 74.3

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

   5. Simplified 74.3%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{24}{{x}^{4}}} \]

      2. metadata-eval N/A

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

      3. pow-sqr N/A

         \[\leadsto \frac{24}{\color{blue}{{x}^{2} \cdot {x}^{2}}} \]

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. cube-mult N/A

         \[\leadsto \frac{24}{x \cdot \color{blue}{{x}^{3}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{24}{\color{blue}{x \cdot {x}^{3}}} \]

      9. cube-mult N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 74.3

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

   8. Simplified 74.3%

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

Alternative 5: 73.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ t_1 := \left(x \cdot x\right) \cdot t\_0\\ \mathbf{if}\;x \leq 2.4 \cdot 10^{+51}:\\ \;\;\;\;\frac{8}{\mathsf{fma}\left(t\_1, t\_1, -64\right)} \cdot \mathsf{fma}\left(x \cdot x, t\_0, -8\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{8}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) (* x x))) (t_1 (* (* x x) t_0)))
   (if (<= x 2.4e+51)
     (* (/ 8.0 (fma t_1 t_1 -64.0)) (fma (* x x) t_0 -8.0))
     (/ 8.0 (* x (* x (* x (* x (* x x)))))))))

C

double code(double x) {
	double t_0 = (x * x) * (x * x);
	double t_1 = (x * x) * t_0;
	double tmp;
	if (x <= 2.4e+51) {
		tmp = (8.0 / fma(t_1, t_1, -64.0)) * fma((x * x), t_0, -8.0);
	} else {
		tmp = 8.0 / (x * (x * (x * (x * (x * x)))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * Float64(x * x))
	t_1 = Float64(Float64(x * x) * t_0)
	tmp = 0.0
	if (x <= 2.4e+51)
		tmp = Float64(Float64(8.0 / fma(t_1, t_1, -64.0)) * fma(Float64(x * x), t_0, -8.0));
	else
		tmp = Float64(8.0 / Float64(x * Float64(x * Float64(x * Float64(x * Float64(x * x))))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[x, 2.4e+51], N[(N[(8.0 / N[(t$95$1 * t$95$1 + -64.0), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * t$95$0 + -8.0), $MachinePrecision]), $MachinePrecision], N[(8.0 / N[(x * N[(x * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.3999999999999999e51

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 77.1

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

   5. Simplified 77.1%

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

      1. lift-*.f64 N/A

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

      2. flip3-+ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{\left(x \cdot x\right)}^{3} + {2}^{3}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 \cdot 2 - \left(x \cdot x\right) \cdot 2\right)}}} \]

      3. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{2}{{\left(x \cdot x\right)}^{3} + {2}^{3}} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 \cdot 2 - \left(x \cdot x\right) \cdot 2\right)\right)} \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 \cdot 2 - \left(x \cdot x\right) \cdot 2\right)\right)}{{\left(x \cdot x\right)}^{3} + {2}^{3}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 \cdot 2 - \left(x \cdot x\right) \cdot 2\right)\right)}{{\left(x \cdot x\right)}^{3} + {2}^{3}}} \]

   7. Applied egg-rr 65.1%

      \[\leadsto \color{blue}{\frac{2 \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x, -2\right), 4\right)}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), x \cdot \left(x \cdot x\right), 8\right)}} \]

   8. Taylor expanded in x around 0

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

   10. Simplified 87.7%

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

       1. lift-*.f64 N/A

          \[\leadsto \frac{8}{\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) + 8} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{8}{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(x \cdot \left(x \cdot x\right)\right) + 8} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{8}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) + 8} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{8}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} + 8} \]

       5. flip-+ N/A

          \[\leadsto \frac{8}{\color{blue}{\frac{\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) - 8 \cdot 8}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) - 8}}} \]

       6. associate-/r/ N/A

          \[\leadsto \color{blue}{\frac{8}{\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) - 8 \cdot 8} \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) - 8\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{8}{\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) - 8 \cdot 8} \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) - 8\right)} \]

   12. Applied egg-rr 67.7%

       \[\leadsto \color{blue}{\frac{8}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), -64\right)} \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \left(x \cdot x\right), -8\right)} \]

   if 2.3999999999999999e51 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2}}} \]
   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 62.2

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

   5. Simplified 62.2%

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

      1. lift-*.f64 N/A

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

      2. flip3-+ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{\left(x \cdot x\right)}^{3} + {2}^{3}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 \cdot 2 - \left(x \cdot x\right) \cdot 2\right)}}} \]

      3. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{2}{{\left(x \cdot x\right)}^{3} + {2}^{3}} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 \cdot 2 - \left(x \cdot x\right) \cdot 2\right)\right)} \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 \cdot 2 - \left(x \cdot x\right) \cdot 2\right)\right)}{{\left(x \cdot x\right)}^{3} + {2}^{3}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 \cdot 2 - \left(x \cdot x\right) \cdot 2\right)\right)}{{\left(x \cdot x\right)}^{3} + {2}^{3}}} \]

   7. Applied egg-rr 9.6%

      \[\leadsto \color{blue}{\frac{2 \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x, -2\right), 4\right)}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), x \cdot \left(x \cdot x\right), 8\right)}} \]

   8. Taylor expanded in x around 0

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

   10. Simplified 100.0%

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

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{8}{{x}^{6}}} \]
   12. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{8}{{x}^{6}}} \]

       2. metadata-eval N/A

          \[\leadsto \frac{8}{{x}^{\color{blue}{\left(5 + 1\right)}}} \]

       3. metadata-eval N/A

          \[\leadsto \frac{8}{{x}^{\left(\color{blue}{\left(4 + 1\right)} + 1\right)}} \]

       4. pow-plus N/A

          \[\leadsto \frac{8}{\color{blue}{{x}^{\left(4 + 1\right)} \cdot x}} \]

       5. pow-plus N/A

          \[\leadsto \frac{8}{\color{blue}{\left({x}^{4} \cdot x\right)} \cdot x} \]

       6. *-commutative N/A

          \[\leadsto \frac{8}{\color{blue}{x \cdot \left({x}^{4} \cdot x\right)}} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{8}{\color{blue}{x \cdot \left({x}^{4} \cdot x\right)}} \]

       8. *-commutative N/A

          \[\leadsto \frac{8}{x \cdot \color{blue}{\left(x \cdot {x}^{4}\right)}} \]

       9. lower-*.f64 N/A

          \[\leadsto \frac{8}{x \cdot \color{blue}{\left(x \cdot {x}^{4}\right)}} \]

       10. metadata-eval N/A

           \[\leadsto \frac{8}{x \cdot \left(x \cdot {x}^{\color{blue}{\left(3 + 1\right)}}\right)} \]

       11. pow-plus N/A

           \[\leadsto \frac{8}{x \cdot \left(x \cdot \color{blue}{\left({x}^{3} \cdot x\right)}\right)} \]

       12. *-commutative N/A

           \[\leadsto \frac{8}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot {x}^{3}\right)}\right)} \]

       13. lower-*.f64 N/A

           \[\leadsto \frac{8}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot {x}^{3}\right)}\right)} \]

       14. cube-mult N/A

           \[\leadsto \frac{8}{x \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}\right)\right)} \]

       15. unpow2 N/A

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

       16. lower-*.f64 N/A

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

       17. unpow2 N/A

           \[\leadsto \frac{8}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)} \]

       18. lower-*.f64 100.0

           \[\leadsto \frac{8}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)} \]

   13. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{8}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 73.5% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (fma x (* x 0.041666666666666664) 0.5))))
   (if (<= x 2e+77)
     (/ (fma x t_0 -1.0) (fma x (* t_0 (* x t_0)) -1.0))
     (/ 24.0 (* x (* x (* x x)))))))

C

double code(double x) {
	double t_0 = x * fma(x, (x * 0.041666666666666664), 0.5);
	double tmp;
	if (x <= 2e+77) {
		tmp = fma(x, t_0, -1.0) / fma(x, (t_0 * (x * t_0)), -1.0);
	} else {
		tmp = 24.0 / (x * (x * (x * x)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(x * fma(x, Float64(x * 0.041666666666666664), 0.5))
	tmp = 0.0
	if (x <= 2e+77)
		tmp = Float64(fma(x, t_0, -1.0) / fma(x, Float64(t_0 * Float64(x * t_0)), -1.0));
	else
		tmp = Float64(24.0 / Float64(x * Float64(x * Float64(x * x))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2e+77], N[(N[(x * t$95$0 + -1.0), $MachinePrecision] / N[(x * N[(t$95$0 * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(24.0 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999999999997e77

   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

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 88.5

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

   7. Simplified 88.5%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 83.9%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-fma.f64 N/A

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

       4. flip-+ N/A

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

       5. clear-num N/A

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

       6. lower-/.f64 N/A

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

   12. Applied egg-rr 69.4%

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

   if 1.99999999999999997e77 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{24}{{x}^{4}}} \]

      2. metadata-eval N/A

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

      3. pow-sqr N/A

         \[\leadsto \frac{24}{\color{blue}{{x}^{2} \cdot {x}^{2}}} \]

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. cube-mult N/A

         \[\leadsto \frac{24}{x \cdot \color{blue}{{x}^{3}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{24}{\color{blue}{x \cdot {x}^{3}}} \]

      9. cube-mult N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 100.0

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

   8. Simplified 100.0%

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

Alternative 7: 91.7% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(1.0 / N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $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

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 90.6

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

7. Simplified 90.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower-*.f64 90.6

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

   8. lift-fma.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l* N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-*.f64 90.6

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

9. Applied egg-rr 90.6%

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

10. Final simplification 90.6%

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

Alternative 8: 91.7% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(1.0 / N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. 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

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 90.6

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

7. Simplified 90.6%

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

Alternative 9: 91.3% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \frac{8}{\mathsf{fma}\left(t\_0, t\_0, 8\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x)))) (/ 8.0 (fma t_0 t_0 8.0))))

C

double code(double x) {
	double t_0 = x * (x * x);
	return 8.0 / fma(t_0, t_0, 8.0);
}

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	return Float64(8.0 / fma(t_0, t_0, 8.0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, N[(8.0 / N[(t$95$0 * t$95$0 + 8.0), $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 74.1

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

5. Simplified 74.1%

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

   1. lift-*.f64 N/A

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

   2. flip3-+ N/A

      \[\leadsto \frac{2}{\color{blue}{\frac{{\left(x \cdot x\right)}^{3} + {2}^{3}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 \cdot 2 - \left(x \cdot x\right) \cdot 2\right)}}} \]

   3. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{2}{{\left(x \cdot x\right)}^{3} + {2}^{3}} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 \cdot 2 - \left(x \cdot x\right) \cdot 2\right)\right)} \]

   4. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 \cdot 2 - \left(x \cdot x\right) \cdot 2\right)\right)}{{\left(x \cdot x\right)}^{3} + {2}^{3}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 \cdot 2 - \left(x \cdot x\right) \cdot 2\right)\right)}{{\left(x \cdot x\right)}^{3} + {2}^{3}}} \]

7. Applied egg-rr 53.9%

   \[\leadsto \color{blue}{\frac{2 \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x, -2\right), 4\right)}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), x \cdot \left(x \cdot x\right), 8\right)}} \]

8. Taylor expanded in x around 0

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

10. Simplified 90.2%

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

Alternative 10: 88.0% 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

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 90.6

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

7. Simplified 90.6%

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

8. Taylor expanded in x around 0

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

10. Simplified 86.8%

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

Alternative 11: 62.8% accurate, 9.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 66.3

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

   5. Simplified 66.3%

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

   if 1.25 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 51.2

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

   5. Simplified 51.2%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 51.2

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

   8. Simplified 51.2%

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

Alternative 12: 76.4% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. lower-fma.f64 74.1

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

5. Simplified 74.1%

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

Alternative 13: 50.3% accurate, 217.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

5. Simplified 50.3%

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

Reproduce

herbie shell --seed 2024211 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))