Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 9.0s

Alternatives: 12

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 rewrites 100.0%

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

Alternative 2: 92.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{e^{x} + e^{-x}} \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{720}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.08472222222222223, 0.20833333333333334\right), -0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ 2.0 (+ (exp x) (exp (- x)))) 2e-9)
   (/ 720.0 (* (* x x) (* x (* x (* x x)))))
   (fma
    (* x x)
    (fma (* x x) (fma x (* x -0.08472222222222223) 0.20833333333333334) -0.5)
    1.0)))

C

double code(double x) {
	double tmp;
	if ((2.0 / (exp(x) + exp(-x))) <= 2e-9) {
		tmp = 720.0 / ((x * x) * (x * (x * (x * x))));
	} else {
		tmp = fma((x * x), fma((x * x), fma(x, (x * -0.08472222222222223), 0.20833333333333334), -0.5), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(2.0 / Float64(exp(x) + exp(Float64(-x)))) <= 2e-9)
		tmp = Float64(720.0 / Float64(Float64(x * x) * Float64(x * Float64(x * Float64(x * x)))));
	else
		tmp = fma(Float64(x * x), fma(Float64(x * x), fma(x, Float64(x * -0.08472222222222223), 0.20833333333333334), -0.5), 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(2.0 / N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-9], N[(720.0 / N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.08472222222222223), $MachinePrecision] + 0.20833333333333334), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))) < 2.00000000000000012e-9

   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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 91.1%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{720}{{x}^{\color{blue}{\left(5 + 1\right)}}} \]

      3. pow-plus N/A

         \[\leadsto \frac{720}{\color{blue}{{x}^{5} \cdot x}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{720}{{x}^{\color{blue}{\left(4 + 1\right)}} \cdot x} \]

      5. pow-plus N/A

         \[\leadsto \frac{720}{\color{blue}{\left({x}^{4} \cdot x\right)} \cdot x} \]

      6. associate-*r* N/A

         \[\leadsto \frac{720}{\color{blue}{{x}^{4} \cdot \left(x \cdot x\right)}} \]

      7. unpow2 N/A

         \[\leadsto \frac{720}{{x}^{4} \cdot \color{blue}{{x}^{2}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{720}{\color{blue}{{x}^{2} \cdot {x}^{4}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{720}{\color{blue}{{x}^{2} \cdot {x}^{4}}} \]

      10. unpow2 N/A

          \[\leadsto \frac{720}{\color{blue}{\left(x \cdot x\right)} \cdot {x}^{4}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{720}{\color{blue}{\left(x \cdot x\right)} \cdot {x}^{4}} \]

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

      16. unpow2 N/A

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

      17. cube-mult N/A

          \[\leadsto \frac{720}{\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{{x}^{3}}\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{720}{\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot {x}^{3}\right)}} \]

      19. cube-mult N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 N/A

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

      22. unpow2 N/A

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

      23. lower-*.f64 91.1

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

   8. Applied rewrites 91.1%

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

   if 2.00000000000000012e-9 < (/.f64 #s(literal 2 binary64) (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x))))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 99.4

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

   5. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.08472222222222223, 0.20833333333333334\right), -0.5\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 92.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 99.4

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

   5. Applied rewrites 99.4%

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

   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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 91.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. distribute-rgt-in N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+l+ N/A

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

   7. Applied rewrites 91.1%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 91.1%

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

Alternative 4: 87.9% accurate, 0.9× speedup

Math

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

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 x (* x (* x 0.08333333333333333)) 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(x, (x * (x * 0.08333333333333333)), 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(x, Float64(x * Float64(x * 0.08333333333333333)), x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 4.0], N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.20833333333333334 + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(2.0 / N[(x * N[(x * N[(x * N[(x * 0.08333333333333333), $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 99.2

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

   5. Applied rewrites 99.2%

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

   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 83.7

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

   5. Applied rewrites 83.7%

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

   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. associate-/l* N/A

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

      8. *-rgt-identity N/A

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

      9. associate-*r/ N/A

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

      10. rgt-mult-inverse N/A

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

      11. *-rgt-identity N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. pow-sqr N/A

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

      15. associate-*l* N/A

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

      16. *-lft-identity N/A

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

      17. distribute-rgt-in 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

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

   8. Applied rewrites 83.7%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 83.7

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

   10. Applied rewrites 83.7%

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

4. Final simplification 92.3%

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

Alternative 5: 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 99.2

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

   5. Applied rewrites 99.2%

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

   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 83.7

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

   5. Applied rewrites 83.7%

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

   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 83.7

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

   8. Applied rewrites 83.7%

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

Alternative 6: 76.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 99.1

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

   5. Applied rewrites 99.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 57.5

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

   5. Applied rewrites 57.5%

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

   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 57.5

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

   8. Applied rewrites 57.5%

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

Alternative 7: 92.1% accurate, 4.8× speedup

Math

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

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(x, Float64(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[(x * N[(x * 0.001388888888888889), $MachinePrecision] + 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 rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. 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. unpow2 N/A

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

   12. associate-*l* N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-*.f64 95.8

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

7. Applied rewrites 95.8%

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

Alternative 8: 91.9% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(2.0 / N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * 0.002777777777777778), $MachinePrecision]), $MachinePrecision] + 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. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-fma.f64 N/A

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

5. Applied rewrites 95.8%

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

6. Taylor expanded in x around inf

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

   1. unpow3 N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 95.5

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

8. Applied rewrites 95.5%

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

Alternative 9: 91.6% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(2.0 / N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * 0.002777777777777778), $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. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-fma.f64 N/A

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

5. Applied rewrites 95.8%

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

6. Taylor expanded in x around inf

   \[\leadsto \frac{2}{\mathsf{fma}\left(x, \color{blue}{\frac{1}{360} \cdot {x}^{5}}, 2\right)} \]
7. Step-by-step derivation

   1. metadata-eval N/A

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

   2. pow-plus N/A

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

   3. associate-*l* N/A

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

   4. metadata-eval N/A

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

   5. pow-sqr N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. associate-*r* N/A

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

   10. unpow2 N/A

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

   11. unpow3 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow3 N/A

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

   16. unpow2 N/A

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

   17. associate-*r* N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 N/A

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

   20. *-commutative N/A

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

   21. lower-*.f64 N/A

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

   22. unpow2 N/A

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

   23. lower-*.f64 95.1

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

8. Applied rewrites 95.1%

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

Alternative 10: 87.9% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right)}} \]
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 92.3

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

5. Applied rewrites 92.3%

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

Alternative 11: 76.0% 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 80.4

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

5. Applied rewrites 80.4%

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

Alternative 12: 50.1% 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. Taylor expanded in x around 0

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

5. Applied rewrites 55.7%

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

Reproduce

herbie shell --seed 2024216 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))