Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 11.1s

Alternatives: 10

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. clear-num N/A

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

   2. cosh-def N/A

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

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

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

   4. cosh-lowering-cosh.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 70.9% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.08472222222222223, 0.20833333333333334\right), -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, 0.002777777777777778, 0.08333333333333333\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.45)
   (fma
    (* x x)
    (fma (* x x) (fma (* x x) -0.08472222222222223 0.20833333333333334) -0.5)
    1.0)
   (/
    2.0
    (*
     (* x x)
     (* (* x x) (fma (* x x) 0.002777777777777778 0.08333333333333333))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.45) {
		tmp = fma((x * x), fma((x * x), fma((x * x), -0.08472222222222223, 0.20833333333333334), -0.5), 1.0);
	} else {
		tmp = 2.0 / ((x * x) * ((x * x) * fma((x * x), 0.002777777777777778, 0.08333333333333333)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.45)
		tmp = fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), -0.08472222222222223, 0.20833333333333334), -0.5), 1.0);
	else
		tmp = Float64(2.0 / Float64(Float64(x * x) * Float64(Float64(x * x) * fma(Float64(x * x), 0.002777777777777778, 0.08333333333333333))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.45], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08472222222222223 + 0.20833333333333334), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(2.0 / N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.002777777777777778 + 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.44999999999999996

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. cosh-def N/A

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

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

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

      4. cosh-lowering-cosh.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 66.8

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

   7. Simplified 66.8%

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

   if 1.44999999999999996 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. +-rgt-identity N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{{x}^{2} + 0}, 1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right), 2\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{x \cdot x} + 0, 1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right), 2\right)} \]

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

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x, 0\right)}, 1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right), 2\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{{x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) + 1}, 2\right)} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) + 1, 2\right)} \]

      8. associate-*l* N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)} + 1, 2\right)} \]

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

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right), 1\right)}, 2\right)} \]

      10. +-rgt-identity N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) + 0}, 1\right), 2\right)} \]

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

          \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{12} + \frac{1}{360} \cdot {x}^{2}, 0\right)}, 1\right), 2\right)} \]

      12. +-commutative N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{360} \cdot {x}^{2} + \frac{1}{12}}, 0\right), 1\right), 2\right)} \]

      13. *-commutative N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \frac{1}{360}} + \frac{1}{12}, 0\right), 1\right), 2\right)} \]

      14. unpow2 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{360} + \frac{1}{12}, 0\right), 1\right), 2\right)} \]

      15. associate-*l* N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(x \cdot \frac{1}{360}\right)} + \frac{1}{12}, 0\right), 1\right), 2\right)} \]

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

          \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{360}, \frac{1}{12}\right)}, 0\right), 1\right), 2\right)} \]

      17. +-rgt-identity N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{360} + 0}, \frac{1}{12}\right), 0\right), 1\right), 2\right)} \]

      18. accelerator-lowering-fma.f64 86.3

          \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.002777777777777778, 0\right)}, 0.08333333333333333\right), 0\right), 1\right), 2\right)} \]

   5. Simplified 86.3%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.002777777777777778, 0\right), 0.08333333333333333\right), 0\right), 1\right), 2\right)}} \]

   6. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. cube-prod N/A

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

      4. unpow2 N/A

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

      5. unpow3 N/A

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

      6. pow-sqr N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. associate-*l* N/A

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

      12. lft-mult-inverse N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. pow-sqr N/A

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

      16. associate-*r* N/A

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

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

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

      18. unpow2 N/A

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

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

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

   8. Simplified 86.3%

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

Alternative 3: 91.9% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(1.0 / N[(x * N[(x * N[(x * N[(x * N[(x * N[(x * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $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. clear-num N/A

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

   2. cosh-def N/A

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

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

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

   4. cosh-lowering-cosh.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. 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. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

7. Simplified 94.2%

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

Alternative 4: 70.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.5:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.08472222222222223, 0.20833333333333334\right), -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{720}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.5)
   (fma
    (* x x)
    (fma (* x x) (fma (* x x) -0.08472222222222223 0.20833333333333334) -0.5)
    1.0)
   (/ 720.0 (* (* x x) (* x (* x (* x x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.5) {
		tmp = fma((x * x), fma((x * x), fma((x * x), -0.08472222222222223, 0.20833333333333334), -0.5), 1.0);
	} else {
		tmp = 720.0 / ((x * x) * (x * (x * (x * x))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.5)
		tmp = fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), -0.08472222222222223, 0.20833333333333334), -0.5), 1.0);
	else
		tmp = Float64(720.0 / Float64(Float64(x * x) * Float64(x * Float64(x * Float64(x * x)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.5], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08472222222222223 + 0.20833333333333334), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(720.0 / N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. cosh-def N/A

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

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

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

      4. cosh-lowering-cosh.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 66.8

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

   7. Simplified 66.8%

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

   if 1.5 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. +-rgt-identity N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{{x}^{2} + 0}, 1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right), 2\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{x \cdot x} + 0, 1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right), 2\right)} \]

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

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x, 0\right)}, 1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right), 2\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{{x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) + 1}, 2\right)} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) + 1, 2\right)} \]

      8. associate-*l* N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)} + 1, 2\right)} \]

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

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right), 1\right)}, 2\right)} \]

      10. +-rgt-identity N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) + 0}, 1\right), 2\right)} \]

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

          \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{12} + \frac{1}{360} \cdot {x}^{2}, 0\right)}, 1\right), 2\right)} \]

      12. +-commutative N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{360} \cdot {x}^{2} + \frac{1}{12}}, 0\right), 1\right), 2\right)} \]

      13. *-commutative N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \frac{1}{360}} + \frac{1}{12}, 0\right), 1\right), 2\right)} \]

      14. unpow2 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{360} + \frac{1}{12}, 0\right), 1\right), 2\right)} \]

      15. associate-*l* N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(x \cdot \frac{1}{360}\right)} + \frac{1}{12}, 0\right), 1\right), 2\right)} \]

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

          \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{360}, \frac{1}{12}\right)}, 0\right), 1\right), 2\right)} \]

      17. +-rgt-identity N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{360} + 0}, \frac{1}{12}\right), 0\right), 1\right), 2\right)} \]

      18. accelerator-lowering-fma.f64 86.3

          \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.002777777777777778, 0\right)}, 0.08333333333333333\right), 0\right), 1\right), 2\right)} \]

   5. Simplified 86.3%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.002777777777777778, 0\right), 0.08333333333333333\right), 0\right), 1\right), 2\right)}} \]

   6. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. cube-prod N/A

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

      5. unpow2 N/A

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

      6. cube-unmult N/A

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

      7. pow-sqr N/A

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

      8. metadata-eval N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

      16. unpow2 N/A

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

      17. cube-mult N/A

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

      18. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 86.3

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

   8. Simplified 86.3%

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

Alternative 5: 69.1% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.20833333333333334, -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.4)
   (fma (* x x) (fma x (* x 0.20833333333333334) -0.5) 1.0)
   (/ 1.0 (* (* x x) (fma x (* x 0.041666666666666664) 0.5)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.4) {
		tmp = fma((x * x), fma(x, (x * 0.20833333333333334), -0.5), 1.0);
	} else {
		tmp = 1.0 / ((x * x) * fma(x, (x * 0.041666666666666664), 0.5));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.4)
		tmp = fma(Float64(x * x), fma(x, Float64(x * 0.20833333333333334), -0.5), 1.0);
	else
		tmp = Float64(1.0 / Float64(Float64(x * x) * fma(x, Float64(x * 0.041666666666666664), 0.5)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.4], N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.20833333333333334), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(1.0 / N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. cosh-def N/A

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

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

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

      4. cosh-lowering-cosh.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. sub-neg N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 66.7

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

   7. Simplified 66.7%

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

   if 1.3999999999999999 < x

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. cosh-def N/A

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

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

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

      4. cosh-lowering-cosh.f64 100.0

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

   4. Applied egg-rr 100.0%

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

      10. *-lowering-*.f64 77.7

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

   7. Simplified 77.7%

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

   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. +-commutative N/A

         \[\leadsto \frac{1}{{x}^{4} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}} + \frac{1}{24}\right)}} \]

      2. distribute-rgt-in N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{4} + \frac{1}{24} \cdot {x}^{4}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{2} \cdot 1}{{x}^{2}}} \cdot {x}^{4} + \frac{1}{24} \cdot {x}^{4}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{1}{2}}}{{x}^{2}} \cdot {x}^{4} + \frac{1}{24} \cdot {x}^{4}} \]

      5. associate-*l/ N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{2} \cdot {x}^{4}}{{x}^{2}}} + \frac{1}{24} \cdot {x}^{4}} \]

      6. associate-/l* N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{2} \cdot \frac{{x}^{4}}{{x}^{2}}} + \frac{1}{24} \cdot {x}^{4}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{1}{2} \cdot \frac{{x}^{\color{blue}{\left(2 \cdot 2\right)}}}{{x}^{2}} + \frac{1}{24} \cdot {x}^{4}} \]

      8. pow-sqr N/A

         \[\leadsto \frac{1}{\frac{1}{2} \cdot \frac{\color{blue}{{x}^{2} \cdot {x}^{2}}}{{x}^{2}} + \frac{1}{24} \cdot {x}^{4}} \]

      9. associate-/l* N/A

         \[\leadsto \frac{1}{\frac{1}{2} \cdot \color{blue}{\left({x}^{2} \cdot \frac{{x}^{2}}{{x}^{2}}\right)} + \frac{1}{24} \cdot {x}^{4}} \]

      10. *-rgt-identity N/A

          \[\leadsto \frac{1}{\frac{1}{2} \cdot \left({x}^{2} \cdot \frac{\color{blue}{{x}^{2} \cdot 1}}{{x}^{2}}\right) + \frac{1}{24} \cdot {x}^{4}} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{1}{\frac{1}{2} \cdot \left({x}^{2} \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{{x}^{2}}\right)}\right) + \frac{1}{24} \cdot {x}^{4}} \]

      12. rgt-mult-inverse N/A

          \[\leadsto \frac{1}{\frac{1}{2} \cdot \left({x}^{2} \cdot \color{blue}{1}\right) + \frac{1}{24} \cdot {x}^{4}} \]

      13. *-rgt-identity N/A

          \[\leadsto \frac{1}{\frac{1}{2} \cdot \color{blue}{{x}^{2}} + \frac{1}{24} \cdot {x}^{4}} \]

      14. metadata-eval N/A

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

      15. pow-sqr N/A

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

      16. associate-*l* N/A

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

      17. distribute-rgt-in N/A

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

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

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

   10. Simplified 77.7%

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

Alternative 6: 88.1% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. clear-num N/A

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

   2. cosh-def N/A

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

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

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

   4. cosh-lowering-cosh.f64 100.0

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

4. Applied egg-rr 100.0%

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

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

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

   3. unpow2 N/A

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

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

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

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

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

   10. *-lowering-*.f64 89.0

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

7. Simplified 89.0%

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

Alternative 7: 69.1% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.86:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.20833333333333334, -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.86)
   (fma (* x x) (fma x (* x 0.20833333333333334) -0.5) 1.0)
   (/ 24.0 (* x (* x (* x x))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.86) {
		tmp = fma((x * x), fma(x, (x * 0.20833333333333334), -0.5), 1.0);
	} else {
		tmp = 24.0 / (x * (x * (x * x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.86)
		tmp = fma(Float64(x * x), fma(x, Float64(x * 0.20833333333333334), -0.5), 1.0);
	else
		tmp = Float64(24.0 / Float64(x * Float64(x * Float64(x * x))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.86], N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.20833333333333334), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(24.0 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8600000000000001

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. cosh-def N/A

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

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

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

      4. cosh-lowering-cosh.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. sub-neg N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 66.7

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

   7. Simplified 66.7%

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

   if 1.8600000000000001 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. +-rgt-identity N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right) + 0}, 2\right)} \]

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

         \[\leadsto \frac{2}{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 1 + \frac{1}{12} \cdot {x}^{2}, 0\right)}, 2\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{12} \cdot {x}^{2} + 1}, 0\right), 2\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \frac{1}{12}} + 1, 0\right), 2\right)} \]

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

         \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{12}, 1\right)}, 0\right), 2\right)} \]

      10. +-rgt-identity N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\color{blue}{{x}^{2} + 0}, \frac{1}{12}, 1\right), 0\right), 2\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\color{blue}{x \cdot x} + 0, \frac{1}{12}, 1\right), 0\right), 2\right)} \]

      12. accelerator-lowering-fma.f64 77.7

          \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x, 0\right)}, 0.08333333333333333, 1\right), 0\right), 2\right)} \]

   5. Simplified 77.7%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.08333333333333333, 1\right), 0\right), 2\right)}} \]

   6. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. cube-mult N/A

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

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

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 77.7

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

   8. Simplified 77.7%

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

Alternative 8: 63.3% accurate, 9.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. cosh-def N/A

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

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

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

      4. cosh-lowering-cosh.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 66.6

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

   7. Simplified 66.6%

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

   if 1.25 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. accelerator-lowering-fma.f64 65.5

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

   5. Simplified 65.5%

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

   6. Taylor expanded in x around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 65.5

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

   8. Simplified 65.5%

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

Alternative 9: 76.3% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. accelerator-lowering-fma.f64 79.9

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

5. Simplified 79.9%

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

Alternative 10: 51.1% accurate, 217.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

5. Simplified 53.0%

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

Reproduce

herbie shell --seed 2024197 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))