Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 10.7s

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

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (exp (- x_m)))) (/ 2.0 (+ t_0 (/ 1.0 t_0)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[Exp[(-x$95$m)], $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. lift-exp.f64 100.0

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

   2. /-rgt-identity N/A

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

   3. clear-num N/A

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

   4. lift-exp.f64 N/A

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

   5. exp-neg N/A

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

   6. lift-neg.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lower-/.f64 100.0

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

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: 91.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2}{e^{-x\_m} + e^{x\_m}} \leq 2 \cdot 10^{-33}:\\ \;\;\;\;\frac{720}{\left(x\_m \cdot x\_m\right) \cdot \left(x\_m \cdot \left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot 0.041666666666666664, 0.5\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (/ 2.0 (+ (exp (- x_m)) (exp x_m))) 2e-33)
   (/ 720.0 (* (* x_m x_m) (* x_m (* x_m (* x_m x_m)))))
   (/ 1.0 (fma x_m (* x_m (fma x_m (* x_m 0.041666666666666664) 0.5)) 1.0))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if ((2.0 / (exp(-x_m) + exp(x_m))) <= 2e-33) {
		tmp = 720.0 / ((x_m * x_m) * (x_m * (x_m * (x_m * x_m))));
	} else {
		tmp = 1.0 / fma(x_m, (x_m * fma(x_m, (x_m * 0.041666666666666664), 0.5)), 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (Float64(2.0 / Float64(exp(Float64(-x_m)) + exp(x_m))) <= 2e-33)
		tmp = Float64(720.0 / Float64(Float64(x_m * x_m) * Float64(x_m * Float64(x_m * Float64(x_m * x_m)))));
	else
		tmp = Float64(1.0 / fma(x_m, Float64(x_m * fma(x_m, Float64(x_m * 0.041666666666666664), 0.5)), 1.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[(2.0 / N[(N[Exp[(-x$95$m)], $MachinePrecision] + N[Exp[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-33], N[(720.0 / N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))) < 2.0000000000000001e-33

   1. Initial program 100.0%

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

      1. cosh-undef N/A

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

      2. associate-/r* N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cosh.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 80.1

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

   7. Simplified 80.1%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{720}{{x}^{\color{blue}{\left(2 \cdot 3\right)}}} \]

      3. pow-sqr N/A

         \[\leadsto \frac{720}{\color{blue}{{x}^{3} \cdot {x}^{3}}} \]

      4. cube-mult N/A

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

      5. unpow2 N/A

         \[\leadsto \frac{720}{\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot {x}^{3}} \]

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. swap-sqr N/A

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

      9. unpow2 N/A

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

      10. pow-sqr N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. pow-plus N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. cube-mult N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 N/A

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

      22. unpow2 N/A

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

      23. lower-*.f64 80.1

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

   10. Simplified 80.1%

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

   if 2.0000000000000001e-33 < (/.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. Step-by-step derivation

      1. cosh-undef N/A

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

      2. associate-/r* N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cosh.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 99.1

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

   7. Simplified 99.1%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{e^{-x} + e^{x}} \leq 2 \cdot 10^{-33}:\\ \;\;\;\;\frac{720}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2}{e^{-x\_m} + e^{x\_m}} \leq 0.5:\\ \;\;\;\;\frac{12}{x\_m \cdot \left(x\_m \cdot x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot 0.20833333333333334, -0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (/ 2.0 (+ (exp (- x_m)) (exp x_m))) 0.5)
   (/ 12.0 (* x_m (* x_m x_m)))
   (fma x_m (* x_m (fma x_m (* x_m 0.20833333333333334) -0.5)) 1.0)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if ((2.0 / (exp(-x_m) + exp(x_m))) <= 0.5) {
		tmp = 12.0 / (x_m * (x_m * x_m));
	} else {
		tmp = fma(x_m, (x_m * fma(x_m, (x_m * 0.20833333333333334), -0.5)), 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (Float64(2.0 / Float64(exp(Float64(-x_m)) + exp(x_m))) <= 0.5)
		tmp = Float64(12.0 / Float64(x_m * Float64(x_m * x_m)));
	else
		tmp = fma(x_m, Float64(x_m * fma(x_m, Float64(x_m * 0.20833333333333334), -0.5)), 1.0);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[(2.0 / N[(N[Exp[(-x$95$m)], $MachinePrecision] + N[Exp[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], N[(12.0 / N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.20833333333333334), $MachinePrecision] + -0.5), $MachinePrecision]), $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)))) < 0.5

   1. Initial program 100.0%

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

      1. lift-exp.f64 100.0

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

      2. /-rgt-identity N/A

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

      3. clear-num N/A

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

      4. lift-exp.f64 N/A

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

      5. exp-neg N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lower-/.f64 100.0

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

   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 54.7%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 66.8

         \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)}, 1\right), 2\right)} \]

   10. Simplified 66.8%

       \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 2\right)}} \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{12}{{x}^{3}}} \]
   12. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{12}{{x}^{3}}} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 67.3

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

   13. Simplified 67.3%

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

   if 0.5 < (/.f64 #s(literal 2 binary64) (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x))))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.6

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

   5. Simplified 99.6%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{e^{-x} + e^{x}} \leq 0.5:\\ \;\;\;\;\frac{12}{x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.20833333333333334, -0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2}{e^{-x\_m} + e^{x\_m}} \leq 2 \cdot 10^{-33}:\\ \;\;\;\;\frac{12}{x\_m \cdot \left(x\_m \cdot x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x\_m, x\_m \cdot 0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (/ 2.0 (+ (exp (- x_m)) (exp x_m))) 2e-33)
   (/ 12.0 (* x_m (* x_m x_m)))
   (/ 1.0 (fma x_m (* x_m 0.5) 1.0))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if ((2.0 / (exp(-x_m) + exp(x_m))) <= 2e-33) {
		tmp = 12.0 / (x_m * (x_m * x_m));
	} else {
		tmp = 1.0 / fma(x_m, (x_m * 0.5), 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (Float64(2.0 / Float64(exp(Float64(-x_m)) + exp(x_m))) <= 2e-33)
		tmp = Float64(12.0 / Float64(x_m * Float64(x_m * x_m)));
	else
		tmp = Float64(1.0 / fma(x_m, Float64(x_m * 0.5), 1.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[(2.0 / N[(N[Exp[(-x$95$m)], $MachinePrecision] + N[Exp[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-33], N[(12.0 / N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x$95$m * N[(x$95$m * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))) < 2.0000000000000001e-33

   1. Initial program 100.0%

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

      1. lift-exp.f64 100.0

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

      2. /-rgt-identity N/A

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

      3. clear-num N/A

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

      4. lift-exp.f64 N/A

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

      5. exp-neg N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lower-/.f64 100.0

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

   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 55.0%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 67.0

         \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)}, 1\right), 2\right)} \]

   10. Simplified 67.0%

       \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 2\right)}} \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{12}{{x}^{3}}} \]
   12. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{12}{{x}^{3}}} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 67.7

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

   13. Simplified 67.7%

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

   if 2.0000000000000001e-33 < (/.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. Step-by-step derivation

      1. cosh-undef N/A

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

      2. associate-/r* N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cosh.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 98.9

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

   7. Simplified 98.9%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{e^{-x} + e^{x}} \leq 2 \cdot 10^{-33}:\\ \;\;\;\;\frac{12}{x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, x \cdot 0.5, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;e^{-x\_m} + e^{x\_m} \leq 4:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m \cdot \mathsf{fma}\left(x\_m, x\_m \cdot 0.20833333333333334, -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x\_m \cdot x\_m\right) \cdot \mathsf{fma}\left(x\_m, x\_m \cdot 0.041666666666666664, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (+ (exp (- x_m)) (exp x_m)) 4.0)
   (fma x_m (* x_m (fma x_m (* x_m 0.20833333333333334) -0.5)) 1.0)
   (/ 1.0 (* (* x_m x_m) (fma x_m (* x_m 0.041666666666666664) 0.5)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if ((exp(-x_m) + exp(x_m)) <= 4.0) {
		tmp = fma(x_m, (x_m * fma(x_m, (x_m * 0.20833333333333334), -0.5)), 1.0);
	} else {
		tmp = 1.0 / ((x_m * x_m) * fma(x_m, (x_m * 0.041666666666666664), 0.5));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (Float64(exp(Float64(-x_m)) + exp(x_m)) <= 4.0)
		tmp = fma(x_m, Float64(x_m * fma(x_m, Float64(x_m * 0.20833333333333334), -0.5)), 1.0);
	else
		tmp = Float64(1.0 / Float64(Float64(x_m * x_m) * fma(x_m, Float64(x_m * 0.041666666666666664), 0.5)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[(N[Exp[(-x$95$m)], $MachinePrecision] + N[Exp[x$95$m], $MachinePrecision]), $MachinePrecision], 4.0], N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.20833333333333334), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(1.0 / N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.6

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

   5. Simplified 99.6%

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

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

   1. Initial program 100.0%

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

      1. cosh-undef N/A

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

      2. associate-/r* N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cosh.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 74.6

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

   7. Simplified 74.6%

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

   8. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-*l* N/A

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

      7. lft-mult-inverse N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 74.5

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

   10. Simplified 74.5%

       \[\leadsto \frac{1}{\color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} + e^{x} \leq 4:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.20833333333333334, -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2}{e^{-x\_m} + e^{x\_m}} \leq 0.5:\\ \;\;\;\;\frac{12}{x\_m \cdot \left(x\_m \cdot x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, x\_m \cdot x\_m, 1\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (/ 2.0 (+ (exp (- x_m)) (exp x_m))) 0.5)
   (/ 12.0 (* x_m (* x_m x_m)))
   (fma -0.5 (* x_m x_m) 1.0)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if ((2.0 / (exp(-x_m) + exp(x_m))) <= 0.5) {
		tmp = 12.0 / (x_m * (x_m * x_m));
	} else {
		tmp = fma(-0.5, (x_m * x_m), 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (Float64(2.0 / Float64(exp(Float64(-x_m)) + exp(x_m))) <= 0.5)
		tmp = Float64(12.0 / Float64(x_m * Float64(x_m * x_m)));
	else
		tmp = fma(-0.5, Float64(x_m * x_m), 1.0);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[(2.0 / N[(N[Exp[(-x$95$m)], $MachinePrecision] + N[Exp[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], N[(12.0 / N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(x$95$m * x$95$m), $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)))) < 0.5

   1. Initial program 100.0%

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

      1. lift-exp.f64 100.0

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

      2. /-rgt-identity N/A

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

      3. clear-num N/A

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

      4. lift-exp.f64 N/A

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

      5. exp-neg N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lower-/.f64 100.0

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

   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 54.7%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 66.8

         \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)}, 1\right), 2\right)} \]

   10. Simplified 66.8%

       \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 2\right)}} \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{12}{{x}^{3}}} \]
   12. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{12}{{x}^{3}}} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 67.3

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

   13. Simplified 67.3%

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

   if 0.5 < (/.f64 #s(literal 2 binary64) (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x))))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 99.5

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

   5. Simplified 99.5%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{e^{-x} + e^{x}} \leq 0.5:\\ \;\;\;\;\frac{12}{x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.8% accurate, 0.9× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (+ (exp (- x_m)) (exp x_m)) 4.0)
   (fma x_m (* x_m (fma x_m (* x_m 0.20833333333333334) -0.5)) 1.0)
   (/ 24.0 (* x_m (* x_m (* x_m x_m))))))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[(N[Exp[(-x$95$m)], $MachinePrecision] + N[Exp[x$95$m], $MachinePrecision]), $MachinePrecision], 4.0], N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.20833333333333334), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(24.0 / N[(x$95$m * N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.6

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

   5. Simplified 99.6%

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

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

   1. Initial program 100.0%

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

      1. cosh-undef N/A

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

      2. associate-/r* N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cosh.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 74.6

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

   7. Simplified 74.6%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 74.5

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

   10. Simplified 74.5%

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

   11. Taylor expanded in x around inf

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

       1. lower-/.f64 N/A

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

       2. metadata-eval N/A

          \[\leadsto \frac{24}{{x}^{\color{blue}{\left(3 + 1\right)}}} \]

       3. pow-plus N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. cube-mult N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 74.5

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

   13. Simplified 74.5%

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

4. Final simplification 86.9%

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

Alternative 8: 76.1% accurate, 1.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (+ (exp (- x_m)) (exp x_m)) 4.0)
   (fma -0.5 (* x_m x_m) 1.0)
   (/ 2.0 (* x_m x_m))))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[(N[Exp[(-x$95$m)], $MachinePrecision] + N[Exp[x$95$m], $MachinePrecision]), $MachinePrecision], 4.0], N[(-0.5 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision], N[(2.0 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 99.5

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

   5. Simplified 99.5%

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

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

   1. Initial program 100.0%

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

      1. cosh-undef N/A

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

      2. associate-/r* N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cosh.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 47.1

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

   7. Simplified 47.1%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 47.1

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

   10. Simplified 47.1%

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

4. Final simplification 72.9%

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

Alternative 9: 100.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{1}{\cosh x\_m} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (/ 1.0 (cosh x_m)))

C

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

Fortran

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

Java

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

Python

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

Julia

x_m = abs(x)
function code(x_m)
	return Float64(1.0 / cosh(x_m))
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(1.0 / N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. cosh-undef N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cosh.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 10: 91.7% accurate, 4.8× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (/
  1.0
  (fma
   (* x_m x_m)
   (fma
    x_m
    (* x_m (fma (* x_m x_m) 0.001388888888888889 0.041666666666666664))
    0.5)
   1.0)))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(1.0 / N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. cosh-undef N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cosh.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. +-commutative N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 89.7

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

7. Simplified 89.7%

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

Alternative 11: 91.6% accurate, 4.9× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (/
  1.0
  (fma
   (* x_m x_m)
   (fma x_m (* 0.001388888888888889 (* x_m (* x_m x_m))) 0.5)
   1.0)))

C

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

Julia

x_m = abs(x)
function code(x_m)
	return Float64(1.0 / fma(Float64(x_m * x_m), fma(x_m, Float64(0.001388888888888889 * Float64(x_m * Float64(x_m * x_m))), 0.5), 1.0))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(1.0 / N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(0.001388888888888889 * N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. cosh-undef N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cosh.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. +-commutative N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 89.7

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

7. Simplified 89.7%

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

8. Taylor expanded in x around inf

   \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{720} \cdot {x}^{3}}, \frac{1}{2}\right), 1\right)} \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{720} \cdot {x}^{3}}, \frac{1}{2}\right), 1\right)} \]

   2. cube-mult N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 89.5

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

10. Simplified 89.5%

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

Alternative 12: 91.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{1}{\mathsf{fma}\left(x\_m \cdot x\_m, x\_m \cdot \left(0.001388888888888889 \cdot \left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right)\right), 1\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (/
  1.0
  (fma (* x_m x_m) (* x_m (* 0.001388888888888889 (* x_m (* x_m x_m)))) 1.0)))

C

x_m = fabs(x);
double code(double x_m) {
	return 1.0 / fma((x_m * x_m), (x_m * (0.001388888888888889 * (x_m * (x_m * x_m)))), 1.0);
}

Julia

x_m = abs(x)
function code(x_m)
	return Float64(1.0 / fma(Float64(x_m * x_m), Float64(x_m * Float64(0.001388888888888889 * Float64(x_m * Float64(x_m * x_m)))), 1.0))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(1.0 / N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(0.001388888888888889 * N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. cosh-undef N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cosh.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. +-commutative N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 89.7

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

7. Simplified 89.7%

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

8. Taylor expanded in x around inf

   \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{720} \cdot {x}^{4}}, 1\right)} \]
9. Step-by-step derivation

   1. metadata-eval N/A

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

   2. pow-plus N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. cube-mult N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 89.2

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

10. Simplified 89.2%

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

Alternative 13: 87.8% accurate, 6.4× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (/ 1.0 (fma x_m (* x_m (fma x_m (* x_m 0.041666666666666664) 0.5)) 1.0)))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(1.0 / N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. cosh-undef N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cosh.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-*.f64 86.9

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

7. Simplified 86.9%

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

Alternative 14: 87.4% accurate, 6.6× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (/ 1.0 (fma x_m (* x_m (* (* x_m x_m) 0.041666666666666664)) 1.0)))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(1.0 / N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. cosh-undef N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cosh.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-*.f64 86.9

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

7. Simplified 86.9%

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

8. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 86.6

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

10. Simplified 86.6%

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

11. Final simplification 86.6%

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

Alternative 15: 51.1% accurate, 14.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{2}{2 + x\_m} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (/ 2.0 (+ 2.0 x_m)))

C

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

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 2.0 / (2.0 + x_m);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 2.0 / (2.0 + x_m)

Julia

x_m = abs(x)
function code(x_m)
	return Float64(2.0 / Float64(2.0 + x_m))
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(2.0 / N[(2.0 + x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. lift-exp.f64 100.0

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

   2. /-rgt-identity N/A

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

   3. clear-num N/A

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

   4. lift-exp.f64 N/A

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

   5. exp-neg N/A

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

   6. lift-neg.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lower-/.f64 100.0

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

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 75.8%

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

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. lower-+.f64 50.7

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

10. Simplified 50.7%

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

11. Final simplification 50.7%

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

Alternative 16: 50.4% accurate, 217.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 1 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 1.0)

C

x_m = fabs(x);
double code(double x_m) {
	return 1.0;
}

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 1.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 1.0

Julia

x_m = abs(x)
function code(x_m)
	return 1.0
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

5. Simplified 50.4%

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

Reproduce

herbie shell --seed 2024215 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))