Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 10.3s

Alternatives: 16

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \frac{2}{t\_0 + \frac{1}{t\_0}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x)))) (/ 2.0 (+ t_0 (/ 1.0 t_0)))))

C

double code(double x) {
	double t_0 = exp(-x);
	return 2.0 / (t_0 + (1.0 / t_0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = exp(-x)
    code = 2.0d0 / (t_0 + (1.0d0 / t_0))
end function

Java

public static double code(double x) {
	double t_0 = Math.exp(-x);
	return 2.0 / (t_0 + (1.0 / t_0));
}

Python

def code(x):
	t_0 = math.exp(-x)
	return 2.0 / (t_0 + (1.0 / t_0))

Julia

function code(x)
	t_0 = exp(Float64(-x))
	return Float64(2.0 / Float64(t_0 + Float64(1.0 / t_0)))
end

MATLAB

function tmp = code(x)
	t_0 = exp(-x);
	tmp = 2.0 / (t_0 + (1.0 / t_0));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. /-rgt-identity N/A

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

   2. clear-num N/A

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

   3. exp-neg N/A

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

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

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

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

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

   6. neg-lowering-neg.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

   \[\leadsto \frac{2}{e^{-x} + \frac{1}{e^{-x}}} \]
6. Add Preprocessing

Alternative 2: 91.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{e^{-x} + e^{x}} \leq 10^{-270}:\\ \;\;\;\;\frac{720}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.08333333333333333, 0.20833333333333334\right), -0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ 2.0 (+ (exp (- x)) (exp x))) 1e-270)
   (/ 720.0 (* x (* x (* x (* x (* x x))))))
   (fma
    (* x x)
    (fma (* x x) (fma x (* x -0.08333333333333333) 0.20833333333333334) -0.5)
    1.0)))

C

double code(double x) {
	double tmp;
	if ((2.0 / (exp(-x) + exp(x))) <= 1e-270) {
		tmp = 720.0 / (x * (x * (x * (x * (x * x)))));
	} else {
		tmp = fma((x * x), fma((x * x), fma(x, (x * -0.08333333333333333), 0.20833333333333334), -0.5), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(2.0 / Float64(exp(Float64(-x)) + exp(x))) <= 1e-270)
		tmp = Float64(720.0 / Float64(x * Float64(x * Float64(x * Float64(x * Float64(x * x))))));
	else
		tmp = fma(Float64(x * x), fma(Float64(x * x), fma(x, Float64(x * -0.08333333333333333), 0.20833333333333334), -0.5), 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(2.0 / N[(N[Exp[(-x)], $MachinePrecision] + N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-270], N[(720.0 / N[(x * N[(x * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.08333333333333333), $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)))) < 1e-270

   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. accelerator-lowering-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. Simplified 82.9%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \color{blue}{{x}^{5} \cdot \left(\frac{1}{360} + \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right)}, 2\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(x, \color{blue}{\left(\frac{1}{360} + \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{5}}, 2\right)} \]

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

         \[\leadsto \frac{2}{\mathsf{fma}\left(x, \color{blue}{\left(\frac{1}{360} + \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{5}}, 2\right)} \]

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

         \[\leadsto \frac{2}{\mathsf{fma}\left(x, \color{blue}{\left(\frac{1}{360} + \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right)} \cdot {x}^{5}, 2\right)} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(x, \left(\frac{1}{360} + \color{blue}{\frac{\frac{1}{12} \cdot 1}{{x}^{2}}}\right) \cdot {x}^{5}, 2\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(x, \left(\frac{1}{360} + \frac{\color{blue}{\frac{1}{12}}}{{x}^{2}}\right) \cdot {x}^{5}, 2\right)} \]

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

         \[\leadsto \frac{2}{\mathsf{fma}\left(x, \left(\frac{1}{360} + \color{blue}{\frac{\frac{1}{12}}{{x}^{2}}}\right) \cdot {x}^{5}, 2\right)} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(x, \left(\frac{1}{360} + \frac{\frac{1}{12}}{\color{blue}{x \cdot x}}\right) \cdot {x}^{5}, 2\right)} \]

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

         \[\leadsto \frac{2}{\mathsf{fma}\left(x, \left(\frac{1}{360} + \frac{\frac{1}{12}}{\color{blue}{x \cdot x}}\right) \cdot {x}^{5}, 2\right)} \]

      9. metadata-eval N/A

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

      10. pow-plus N/A

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

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

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

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

      16. unpow2 N/A

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

      17. cube-mult N/A

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

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

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

      19. cube-mult N/A

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

      20. unpow2 N/A

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

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

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

      22. unpow2 N/A

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

      23. *-lowering-*.f64 82.9

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

   8. Simplified 82.9%

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

   9. Taylor expanded in x around inf

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

       1. /-lowering-/.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. *-commutative N/A

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

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

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

       6. metadata-eval N/A

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

       7. pow-plus N/A

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

       8. *-commutative N/A

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

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

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

       10. metadata-eval N/A

           \[\leadsto \frac{720}{x \cdot \left(x \cdot {x}^{\color{blue}{\left(3 + 1\right)}}\right)} \]

       11. pow-plus N/A

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

       12. *-commutative N/A

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

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

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

       14. cube-mult N/A

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

       15. unpow2 N/A

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

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

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

       17. unpow2 N/A

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

       18. *-lowering-*.f64 83.6

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

   11. Simplified 83.6%

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

   if 1e-270 < (/.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

      \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right)}} \]
   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. accelerator-lowering-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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.f64 98.9

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

   5. Simplified 98.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{5}{24} + \frac{-1}{12} \cdot {x}^{2}\right) - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{5}{24} + \frac{-1}{12} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{5}{24} + \frac{-1}{12} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{5}{24} + \frac{-1}{12} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{5}{24} + \frac{-1}{12} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{5}{24} + \frac{-1}{12} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \left(\frac{5}{24} + \frac{-1}{12} \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{2}}, 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{5}{24} + \frac{-1}{12} \cdot {x}^{2}, \frac{-1}{2}\right)}, 1\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{5}{24} + \frac{-1}{12} \cdot {x}^{2}, \frac{-1}{2}\right), 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{5}{24} + \frac{-1}{12} \cdot {x}^{2}, \frac{-1}{2}\right), 1\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{12} \cdot {x}^{2} + \frac{5}{24}}, \frac{-1}{2}\right), 1\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{12} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{5}{24}, \frac{-1}{2}\right), 1\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{-1}{12} \cdot x\right) \cdot x} + \frac{5}{24}, \frac{-1}{2}\right), 1\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{-1}{12} \cdot x\right)} + \frac{5}{24}, \frac{-1}{2}\right), 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \frac{-1}{12} \cdot x, \frac{5}{24}\right)}, \frac{-1}{2}\right), 1\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{12}}, \frac{5}{24}\right), \frac{-1}{2}\right), 1\right) \]

      16. *-lowering-*.f64 99.0

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot -0.08333333333333333}, 0.20833333333333334\right), -0.5\right), 1\right) \]

   8. Simplified 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.08333333333333333, 0.20833333333333334\right), -0.5\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{e^{-x} + e^{x}} \leq 10^{-270}:\\ \;\;\;\;\frac{720}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.08333333333333333, 0.20833333333333334\right), -0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := \left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq 4 \cdot 10^{+51}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(t\_1, \left(t\_1 \cdot t\_1\right) \cdot 7.71604938271605 \cdot 10^{-6}, -4\right)} \cdot \mathsf{fma}\left(x, x \cdot \left(x \cdot \left(0.002777777777777778 \cdot t\_0\right)\right), -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{720}{x \cdot \left(x \cdot \left(x \cdot t\_0\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))) (t_1 (* (* x x) (* x x))))
   (if (<= x 4e+51)
     (*
      (/ 2.0 (fma t_1 (* (* t_1 t_1) 7.71604938271605e-6) -4.0))
      (fma x (* x (* x (* 0.002777777777777778 t_0))) -2.0))
     (/ 720.0 (* x (* x (* x t_0)))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = (x * x) * (x * x);
	double tmp;
	if (x <= 4e+51) {
		tmp = (2.0 / fma(t_1, ((t_1 * t_1) * 7.71604938271605e-6), -4.0)) * fma(x, (x * (x * (0.002777777777777778 * t_0))), -2.0);
	} else {
		tmp = 720.0 / (x * (x * (x * t_0)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	t_1 = Float64(Float64(x * x) * Float64(x * x))
	tmp = 0.0
	if (x <= 4e+51)
		tmp = Float64(Float64(2.0 / fma(t_1, Float64(Float64(t_1 * t_1) * 7.71604938271605e-6), -4.0)) * fma(x, Float64(x * Float64(x * Float64(0.002777777777777778 * t_0))), -2.0));
	else
		tmp = Float64(720.0 / Float64(x * Float64(x * Float64(x * t_0))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 4e+51], N[(N[(2.0 / N[(t$95$1 * N[(N[(t$95$1 * t$95$1), $MachinePrecision] * 7.71604938271605e-6), $MachinePrecision] + -4.0), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * N[(x * N[(0.002777777777777778 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision], N[(720.0 / N[(x * N[(x * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 4e51

   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. accelerator-lowering-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. Simplified 90.2%

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

   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

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

      5. pow-plus N/A

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

      6. associate-*l* N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. unpow2 N/A

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

      12. *-commutative N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. associate-*r* N/A

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

      19. unpow2 N/A

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

      20. unpow3 N/A

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

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

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

      22. unpow3 N/A

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

      23. unpow2 N/A

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

   8. Simplified 89.1%

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

      1. flip-+ N/A

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

      2. associate-/r/ N/A

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

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

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

   10. Applied egg-rr 70.1%

       \[\leadsto \color{blue}{\frac{2}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right), \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 7.71604938271605 \cdot 10^{-6}, -4\right)} \cdot \mathsf{fma}\left(x, x \cdot \left(x \cdot \left(0.002777777777777778 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right), -2\right)} \]

   if 4e51 < 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. accelerator-lowering-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. Simplified 98.2%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \color{blue}{{x}^{5} \cdot \left(\frac{1}{360} + \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right)}, 2\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(x, \color{blue}{\left(\frac{1}{360} + \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{5}}, 2\right)} \]

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

         \[\leadsto \frac{2}{\mathsf{fma}\left(x, \color{blue}{\left(\frac{1}{360} + \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{5}}, 2\right)} \]

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

         \[\leadsto \frac{2}{\mathsf{fma}\left(x, \color{blue}{\left(\frac{1}{360} + \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right)} \cdot {x}^{5}, 2\right)} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(x, \left(\frac{1}{360} + \color{blue}{\frac{\frac{1}{12} \cdot 1}{{x}^{2}}}\right) \cdot {x}^{5}, 2\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(x, \left(\frac{1}{360} + \frac{\color{blue}{\frac{1}{12}}}{{x}^{2}}\right) \cdot {x}^{5}, 2\right)} \]

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

         \[\leadsto \frac{2}{\mathsf{fma}\left(x, \left(\frac{1}{360} + \color{blue}{\frac{\frac{1}{12}}{{x}^{2}}}\right) \cdot {x}^{5}, 2\right)} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(x, \left(\frac{1}{360} + \frac{\frac{1}{12}}{\color{blue}{x \cdot x}}\right) \cdot {x}^{5}, 2\right)} \]

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

         \[\leadsto \frac{2}{\mathsf{fma}\left(x, \left(\frac{1}{360} + \frac{\frac{1}{12}}{\color{blue}{x \cdot x}}\right) \cdot {x}^{5}, 2\right)} \]

      9. metadata-eval N/A

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

      10. pow-plus N/A

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

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

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

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

      16. unpow2 N/A

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

      17. cube-mult N/A

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

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

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

      19. cube-mult N/A

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

      20. unpow2 N/A

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

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

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

      22. unpow2 N/A

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

      23. *-lowering-*.f64 98.2

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

   8. Simplified 98.2%

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

   9. Taylor expanded in x around inf

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

       1. /-lowering-/.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. *-commutative N/A

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

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

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

       6. metadata-eval N/A

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

       7. pow-plus N/A

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

       8. *-commutative N/A

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

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

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

       10. metadata-eval N/A

           \[\leadsto \frac{720}{x \cdot \left(x \cdot {x}^{\color{blue}{\left(3 + 1\right)}}\right)} \]

       11. pow-plus N/A

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

       12. *-commutative N/A

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

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

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

       14. cube-mult N/A

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

       15. unpow2 N/A

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

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

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

       17. unpow2 N/A

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

       18. *-lowering-*.f64 100.0

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

   11. Simplified 100.0%

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

Alternative 4: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. clear-num N/A

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

   2. cosh-def N/A

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

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

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

   4. cosh-lowering-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 5: 73.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.08333333333333333, x\right)\\ t_1 := x \cdot t\_0\\ \mathbf{if}\;x \leq 2 \cdot 10^{+77}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(t\_1, t\_1, -4\right)} \cdot \mathsf{fma}\left(x, t\_0, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma x (* (* x x) 0.08333333333333333) x)) (t_1 (* x t_0)))
   (if (<= x 2e+77)
     (* (/ 2.0 (fma t_1 t_1 -4.0)) (fma x t_0 -2.0))
     (/ 24.0 (* x (* x (* x x)))))))

C

double code(double x) {
	double t_0 = fma(x, ((x * x) * 0.08333333333333333), x);
	double t_1 = x * t_0;
	double tmp;
	if (x <= 2e+77) {
		tmp = (2.0 / fma(t_1, t_1, -4.0)) * fma(x, t_0, -2.0);
	} else {
		tmp = 24.0 / (x * (x * (x * x)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(x, Float64(Float64(x * x) * 0.08333333333333333), x)
	t_1 = Float64(x * t_0)
	tmp = 0.0
	if (x <= 2e+77)
		tmp = Float64(Float64(2.0 / fma(t_1, t_1, -4.0)) * fma(x, t_0, -2.0));
	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[(N[(x * x), $MachinePrecision] * 0.08333333333333333), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, If[LessEqual[x, 2e+77], N[(N[(2.0 / N[(t$95$1 * t$95$1 + -4.0), $MachinePrecision]), $MachinePrecision] * N[(x * t$95$0 + -2.0), $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. 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. accelerator-lowering-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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.f64 84.4

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

   5. Simplified 84.4%

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

      1. flip-+ N/A

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

      2. associate-/r/ N/A

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

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

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

   7. Applied egg-rr 72.1%

      \[\leadsto \color{blue}{\frac{2}{\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.08333333333333333, x\right), x \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.08333333333333333, x\right), -4\right)} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.08333333333333333, x\right), -2\right)} \]

   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. accelerator-lowering-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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.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. /-lowering-/.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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 6: 91.5% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(2.0 / N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * 0.002777777777777778), $MachinePrecision] + 0.08333333333333333), $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. accelerator-lowering-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. Simplified 91.6%

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

Alternative 7: 91.4% 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. accelerator-lowering-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. Simplified 91.6%

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

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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 91.1

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

8. Simplified 91.1%

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

Alternative 8: 91.0% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. 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. accelerator-lowering-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. Simplified 91.6%

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

6. Taylor expanded in x around inf

   \[\leadsto \frac{2}{\mathsf{fma}\left(x, \color{blue}{{x}^{5} \cdot \left(\frac{1}{360} + \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right)}, 2\right)} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \color{blue}{\left(\frac{1}{360} + \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{5}}, 2\right)} \]

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

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \color{blue}{\left(\frac{1}{360} + \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{5}}, 2\right)} \]

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

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \color{blue}{\left(\frac{1}{360} + \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right)} \cdot {x}^{5}, 2\right)} \]

   4. associate-*r/ N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \left(\frac{1}{360} + \color{blue}{\frac{\frac{1}{12} \cdot 1}{{x}^{2}}}\right) \cdot {x}^{5}, 2\right)} \]

   5. metadata-eval N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \left(\frac{1}{360} + \frac{\color{blue}{\frac{1}{12}}}{{x}^{2}}\right) \cdot {x}^{5}, 2\right)} \]

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

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \left(\frac{1}{360} + \color{blue}{\frac{\frac{1}{12}}{{x}^{2}}}\right) \cdot {x}^{5}, 2\right)} \]

   7. unpow2 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \left(\frac{1}{360} + \frac{\frac{1}{12}}{\color{blue}{x \cdot x}}\right) \cdot {x}^{5}, 2\right)} \]

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

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \left(\frac{1}{360} + \frac{\frac{1}{12}}{\color{blue}{x \cdot x}}\right) \cdot {x}^{5}, 2\right)} \]

   9. metadata-eval N/A

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

   10. pow-plus N/A

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

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

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

   12. metadata-eval N/A

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

   13. pow-sqr N/A

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

   14. unpow2 N/A

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

   15. associate-*l* N/A

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

   16. unpow2 N/A

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

   17. cube-mult N/A

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

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

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

   19. cube-mult N/A

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

   20. unpow2 N/A

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

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

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

   22. unpow2 N/A

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

   23. *-lowering-*.f64 62.6

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

8. Simplified 62.6%

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

9. Taylor expanded in x around inf

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

11. Simplified 90.7%

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

12. Final simplification 90.7%

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

Alternative 9: 67.9% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.20833333333333334, -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot \mathsf{fma}\left(x, x \cdot \left(x \cdot 0.08333333333333333\right), x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.4)
   (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 (x <= 1.4) {
		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 (x <= 1.4)
		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[x, 1.4], 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 x < 1.3999999999999999

   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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f64 66.7

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

   5. Simplified 66.7%

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

   if 1.3999999999999999 < 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. accelerator-lowering-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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.f64 78.5

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

   5. Simplified 78.5%

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

      1. distribute-rgt-in N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      9. accelerator-lowering-fma.f64 78.5

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

   7. Applied egg-rr 78.5%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{2}{\color{blue}{{x}^{4} \cdot \left(\frac{1}{12} + \frac{1}{{x}^{2}}\right)}} \]
   9. 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. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. associate-/l* N/A

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

      8. *-rgt-identity N/A

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

      9. associate-*r/ N/A

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

      10. rgt-mult-inverse N/A

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

      11. *-rgt-identity N/A

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

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. associate-*l* N/A

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

      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. unpow2 N/A

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

      18. associate-*r* N/A

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

      19. *-commutative N/A

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

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

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

      21. *-commutative N/A

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

   10. Simplified 78.5%

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

Alternative 10: 87.7% 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. accelerator-lowering-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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.f64 86.9

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

5. Simplified 86.9%

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

Alternative 11: 68.0% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.20833333333333334, -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.9)
   (fma x (* x (fma (* x x) 0.20833333333333334 -0.5)) 1.0)
   (/ 24.0 (* x (* x (* x x))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = fma(x, (x * fma((x * x), 0.20833333333333334, -0.5)), 1.0);
	} else {
		tmp = 24.0 / (x * (x * (x * x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.9)
		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[x, 1.9], 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 x < 1.8999999999999999

   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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f64 66.7

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

   5. Simplified 66.7%

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

   if 1.8999999999999999 < 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. accelerator-lowering-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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.f64 78.5

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

   5. Simplified 78.5%

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

   6. Taylor expanded in x around inf

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

      1. /-lowering-/.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 78.5

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

   8. Simplified 78.5%

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

Alternative 12: 87.3% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right)}} \]
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. accelerator-lowering-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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.f64 86.9

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

5. Simplified 86.9%

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

6. Taylor expanded in x around inf

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

   1. unpow3 N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. *-commutative N/A

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 86.2

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

8. Simplified 86.2%

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

Alternative 13: 61.7% 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. accelerator-lowering-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. *-lowering-*.f64 66.5

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

   5. Simplified 66.5%

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

   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. accelerator-lowering-fma.f64 57.3

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

   5. Simplified 57.3%

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

   6. Taylor expanded in x around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 57.3

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

   8. Simplified 57.3%

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

Alternative 14: 75.3% 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. accelerator-lowering-fma.f64 76.6

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

5. Simplified 76.6%

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

Alternative 15: 49.7% accurate, 14.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return 2.0 / (2.0 + x);
}

Python

def code(x):
	return 2.0 / (2.0 + x)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. /-rgt-identity N/A

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

   2. clear-num N/A

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

   3. exp-neg N/A

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

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

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

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

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

   6. neg-lowering-neg.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

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

7. Simplified 73.1%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. +-lowering-+.f64 54.1

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

10. Simplified 54.1%

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

11. Final simplification 54.1%

    \[\leadsto \frac{2}{2 + x} \]
12. Add Preprocessing

Alternative 16: 49.0% 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 53.3%

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

Reproduce

herbie shell --seed 2024199 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))