Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 11.0s

Alternatives: 18

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: 87.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{e^{x} + e^{-x}} \leq 0.004:\\ \;\;\;\;\frac{2}{x \cdot \mathsf{fma}\left(x, x \cdot \left(x \cdot 0.08333333333333333\right), x\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.20833333333333334, -0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ 2.0 (+ (exp x) (exp (- x)))) 0.004)
   (/ 2.0 (* x (fma x (* x (* x 0.08333333333333333)) x)))
   (fma x (* x (fma (* x x) 0.20833333333333334 -0.5)) 1.0)))

C

double code(double x) {
	double tmp;
	if ((2.0 / (exp(x) + exp(-x))) <= 0.004) {
		tmp = 2.0 / (x * fma(x, (x * (x * 0.08333333333333333)), x));
	} else {
		tmp = fma(x, (x * fma((x * x), 0.20833333333333334, -0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(2.0 / Float64(exp(x) + exp(Float64(-x)))) <= 0.004)
		tmp = Float64(2.0 / Float64(x * fma(x, Float64(x * Float64(x * 0.08333333333333333)), x)));
	else
		tmp = fma(x, Float64(x * fma(Float64(x * x), 0.20833333333333334, -0.5)), 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(2.0 / N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.004], N[(2.0 / N[(x * N[(x * N[(x * N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.20833333333333334 + -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)))) < 0.0040000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 83.9

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

   5. Simplified 83.9%

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

   6. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

      10. associate-/l* N/A

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

      11. *-rgt-identity N/A

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

      12. associate-*r/ N/A

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

      13. rgt-mult-inverse N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow2 N/A

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

      17. +-commutative N/A

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

      18. associate-*l* N/A

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

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

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

      20. +-commutative N/A

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

      21. distribute-lft-in N/A

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

      22. *-rgt-identity N/A

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

   8. Simplified 83.9%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-lowering-*.f64 83.9

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

   10. Applied egg-rr 83.9%

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

   if 0.0040000000000000001 < (/.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. accelerator-lowering-fma.f64 N/A

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

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

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 99.6

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

   5. Simplified 99.6%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{e^{x} + e^{-x}} \leq 0.004:\\ \;\;\;\;\frac{2}{x \cdot \mathsf{fma}\left(x, x \cdot \left(x \cdot 0.08333333333333333\right), x\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.20833333333333334, -0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{e^{x} + e^{-x}} \leq 0.004:\\ \;\;\;\;\frac{24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.20833333333333334, -0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ 2.0 (+ (exp x) (exp (- x)))) 0.004)
   (/ 24.0 (* x (* x (* x x))))
   (fma x (* x (fma (* x x) 0.20833333333333334 -0.5)) 1.0)))

C

double code(double x) {
	double tmp;
	if ((2.0 / (exp(x) + exp(-x))) <= 0.004) {
		tmp = 24.0 / (x * (x * (x * x)));
	} else {
		tmp = fma(x, (x * fma((x * x), 0.20833333333333334, -0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(2.0 / Float64(exp(x) + exp(Float64(-x)))) <= 0.004)
		tmp = Float64(24.0 / Float64(x * Float64(x * Float64(x * x))));
	else
		tmp = fma(x, Float64(x * fma(Float64(x * x), 0.20833333333333334, -0.5)), 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(2.0 / N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.004], N[(24.0 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.20833333333333334 + -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)))) < 0.0040000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 83.9

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

   5. Simplified 83.9%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{24}{{x}^{4}}} \]
   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 83.9

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

   8. Simplified 83.9%

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

   if 0.0040000000000000001 < (/.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. accelerator-lowering-fma.f64 N/A

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

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

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 99.6

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

   5. Simplified 99.6%

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

Alternative 4: 92.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} + e^{-x} \leq 4:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.08472222222222223, 0.20833333333333334\right), -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{720}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (exp x) (exp (- x))) 4.0)
   (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 ((exp(x) + exp(-x)) <= 4.0) {
		tmp = fma((x * x), fma((x * x), fma(x, (x * -0.08472222222222223), 0.20833333333333334), -0.5), 1.0);
	} else {
		tmp = 720.0 / ((x * x) * (x * (x * (x * x))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) + exp(Float64(-x))) <= 4.0)
		tmp = fma(Float64(x * x), fma(Float64(x * x), fma(x, Float64(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[N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 4.0], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.08472222222222223), $MachinePrecision] + 0.20833333333333334), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(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 (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 4

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. 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. unpow2 N/A

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

      13. associate-*l* N/A

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

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

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

      15. *-lowering-*.f64 99.9

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

   5. Simplified 99.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

   5. Simplified 88.6%

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

   6. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. metadata-eval N/A

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

      5. pow-plus N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. *-commutative N/A

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

      9. *-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 88.6

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

   8. Simplified 88.6%

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

Alternative 5: 96.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 7.71604938271605 \cdot 10^{-6}, -0.006944444444444444\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.00044444444444444447, -0.013333333333333334\right), -0.4\right), -12\right), x\right), 2\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  2.0
  (fma
   x
   (fma
    (*
     (* x (* x x))
     (fma (* x x) (* (* x x) 7.71604938271605e-6) -0.006944444444444444))
    (fma
     (* x x)
     (fma
      (* x x)
      (fma (* x x) -0.00044444444444444447 -0.013333333333333334)
      -0.4)
     -12.0)
    x)
   2.0)))

C

double code(double x) {
	return 2.0 / fma(x, fma(((x * (x * x)) * fma((x * x), ((x * x) * 7.71604938271605e-6), -0.006944444444444444)), fma((x * x), fma((x * x), fma((x * x), -0.00044444444444444447, -0.013333333333333334), -0.4), -12.0), x), 2.0);
}

Julia

function code(x)
	return Float64(2.0 / fma(x, fma(Float64(Float64(x * Float64(x * x)) * fma(Float64(x * x), Float64(Float64(x * x) * 7.71604938271605e-6), -0.006944444444444444)), fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), -0.00044444444444444447, -0.013333333333333334), -0.4), -12.0), x), 2.0))
end

Wolfram

code[x_] := N[(2.0 / N[(x * N[(N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 7.71604938271605e-6), $MachinePrecision] + -0.006944444444444444), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.00044444444444444447 + -0.013333333333333334), $MachinePrecision] + -0.4), $MachinePrecision] + -12.0), $MachinePrecision] + x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

5. Simplified 94.5%

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

   1. associate-*r* N/A

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

   2. pow3 N/A

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

   3. flip-+ N/A

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

   4. associate-*r/ N/A

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

   5. div-inv N/A

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

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

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

7. Applied egg-rr 69.5%

   \[\leadsto \frac{2}{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 7.71604938271605 \cdot 10^{-6}, -0.006944444444444444\right), \frac{1}{\mathsf{fma}\left(x, x \cdot 0.002777777777777778, -0.08333333333333333\right)}, x\right)}, 2\right)} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{2250} \cdot {x}^{2} - \frac{1}{75}\right) - \frac{2}{5}\right) - 12}, x\right), 2\right)} \]
9. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{2250} \cdot {x}^{2} - \frac{1}{75}\right) - \frac{2}{5}\right) + \left(\mathsf{neg}\left(12\right)\right)}, x\right), 2\right)} \]

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

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{-1}{2250} \cdot {x}^{2} - \frac{1}{75}\right) - \frac{2}{5}, \mathsf{neg}\left(12\right)\right)}, x\right), 2\right)} \]

   3. unpow2 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{-1}{2250} \cdot {x}^{2} - \frac{1}{75}\right) - \frac{2}{5}, \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

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

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{-1}{2250} \cdot {x}^{2} - \frac{1}{75}\right) - \frac{2}{5}, \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

   5. sub-neg N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{-1}{2250} \cdot {x}^{2} - \frac{1}{75}\right) + \left(\mathsf{neg}\left(\frac{2}{5}\right)\right)}, \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

   6. metadata-eval N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \left(\frac{-1}{2250} \cdot {x}^{2} - \frac{1}{75}\right) + \color{blue}{\frac{-2}{5}}, \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

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

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{2250} \cdot {x}^{2} - \frac{1}{75}, \frac{-2}{5}\right)}, \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

   8. unpow2 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2250} \cdot {x}^{2} - \frac{1}{75}, \frac{-2}{5}\right), \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

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

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2250} \cdot {x}^{2} - \frac{1}{75}, \frac{-2}{5}\right), \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

   10. sub-neg N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{2250} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{75}\right)\right)}, \frac{-2}{5}\right), \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

   11. *-commutative N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{-1}{2250}} + \left(\mathsf{neg}\left(\frac{1}{75}\right)\right), \frac{-2}{5}\right), \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

   12. metadata-eval N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \frac{-1}{2250} + \color{blue}{\frac{-1}{75}}, \frac{-2}{5}\right), \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

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

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{2250}, \frac{-1}{75}\right)}, \frac{-2}{5}\right), \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

   14. unpow2 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2250}, \frac{-1}{75}\right), \frac{-2}{5}\right), \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

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

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2250}, \frac{-1}{75}\right), \frac{-2}{5}\right), \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

   16. metadata-eval 97.8

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 7.71604938271605 \cdot 10^{-6}, -0.006944444444444444\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.00044444444444444447, -0.013333333333333334\right), -0.4\right), \color{blue}{-12}\right), x\right), 2\right)} \]

10. Simplified 97.8%

    \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 7.71604938271605 \cdot 10^{-6}, -0.006944444444444444\right), \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.00044444444444444447, -0.013333333333333334\right), -0.4\right), -12\right)}, x\right), 2\right)} \]
11. Add Preprocessing

Alternative 6: 96.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 7.71604938271605 \cdot 10^{-6}, -0.006944444444444444\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot -0.00044444444444444447, -0.4\right), -12\right), x\right), 2\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  2.0
  (fma
   x
   (fma
    (*
     (* x (* x x))
     (fma (* x x) (* (* x x) 7.71604938271605e-6) -0.006944444444444444))
    (fma (* x x) (fma (* x x) (* (* x x) -0.00044444444444444447) -0.4) -12.0)
    x)
   2.0)))

C

double code(double x) {
	return 2.0 / fma(x, fma(((x * (x * x)) * fma((x * x), ((x * x) * 7.71604938271605e-6), -0.006944444444444444)), fma((x * x), fma((x * x), ((x * x) * -0.00044444444444444447), -0.4), -12.0), x), 2.0);
}

Julia

function code(x)
	return Float64(2.0 / fma(x, fma(Float64(Float64(x * Float64(x * x)) * fma(Float64(x * x), Float64(Float64(x * x) * 7.71604938271605e-6), -0.006944444444444444)), fma(Float64(x * x), fma(Float64(x * x), Float64(Float64(x * x) * -0.00044444444444444447), -0.4), -12.0), x), 2.0))
end

Wolfram

code[x_] := N[(2.0 / N[(x * N[(N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 7.71604938271605e-6), $MachinePrecision] + -0.006944444444444444), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.00044444444444444447), $MachinePrecision] + -0.4), $MachinePrecision] + -12.0), $MachinePrecision] + x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

5. Simplified 94.5%

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

   1. associate-*r* N/A

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

   2. pow3 N/A

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

   3. flip-+ N/A

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

   4. associate-*r/ N/A

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

   5. div-inv N/A

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

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

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

7. Applied egg-rr 69.5%

   \[\leadsto \frac{2}{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 7.71604938271605 \cdot 10^{-6}, -0.006944444444444444\right), \frac{1}{\mathsf{fma}\left(x, x \cdot 0.002777777777777778, -0.08333333333333333\right)}, x\right)}, 2\right)} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{2250} \cdot {x}^{2} - \frac{1}{75}\right) - \frac{2}{5}\right) - 12}, x\right), 2\right)} \]
9. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{2250} \cdot {x}^{2} - \frac{1}{75}\right) - \frac{2}{5}\right) + \left(\mathsf{neg}\left(12\right)\right)}, x\right), 2\right)} \]

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

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{-1}{2250} \cdot {x}^{2} - \frac{1}{75}\right) - \frac{2}{5}, \mathsf{neg}\left(12\right)\right)}, x\right), 2\right)} \]

   3. unpow2 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{-1}{2250} \cdot {x}^{2} - \frac{1}{75}\right) - \frac{2}{5}, \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

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

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{-1}{2250} \cdot {x}^{2} - \frac{1}{75}\right) - \frac{2}{5}, \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

   5. sub-neg N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{-1}{2250} \cdot {x}^{2} - \frac{1}{75}\right) + \left(\mathsf{neg}\left(\frac{2}{5}\right)\right)}, \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

   6. metadata-eval N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \left(\frac{-1}{2250} \cdot {x}^{2} - \frac{1}{75}\right) + \color{blue}{\frac{-2}{5}}, \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

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

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{2250} \cdot {x}^{2} - \frac{1}{75}, \frac{-2}{5}\right)}, \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

   8. unpow2 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2250} \cdot {x}^{2} - \frac{1}{75}, \frac{-2}{5}\right), \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

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

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2250} \cdot {x}^{2} - \frac{1}{75}, \frac{-2}{5}\right), \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

   10. sub-neg N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{2250} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{75}\right)\right)}, \frac{-2}{5}\right), \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

   11. *-commutative N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{-1}{2250}} + \left(\mathsf{neg}\left(\frac{1}{75}\right)\right), \frac{-2}{5}\right), \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

   12. metadata-eval N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \frac{-1}{2250} + \color{blue}{\frac{-1}{75}}, \frac{-2}{5}\right), \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

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

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{2250}, \frac{-1}{75}\right)}, \frac{-2}{5}\right), \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

   14. unpow2 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2250}, \frac{-1}{75}\right), \frac{-2}{5}\right), \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

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

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2250}, \frac{-1}{75}\right), \frac{-2}{5}\right), \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

   16. metadata-eval 97.8

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 7.71604938271605 \cdot 10^{-6}, -0.006944444444444444\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.00044444444444444447, -0.013333333333333334\right), -0.4\right), \color{blue}{-12}\right), x\right), 2\right)} \]

10. Simplified 97.8%

    \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 7.71604938271605 \cdot 10^{-6}, -0.006944444444444444\right), \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.00044444444444444447, -0.013333333333333334\right), -0.4\right), -12\right)}, x\right), 2\right)} \]

11. Taylor expanded in x around inf

    \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{2250} \cdot {x}^{2}}, \frac{-2}{5}\right), -12\right), x\right), 2\right)} \]
12. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{-1}{2250}}, \frac{-2}{5}\right), -12\right), x\right), 2\right)} \]

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

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{-1}{2250}}, \frac{-2}{5}\right), -12\right), x\right), 2\right)} \]

    3. unpow2 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{2250}, \frac{-2}{5}\right), -12\right), x\right), 2\right)} \]

    4. *-lowering-*.f64 97.8

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 7.71604938271605 \cdot 10^{-6}, -0.006944444444444444\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot -0.00044444444444444447, -0.4\right), -12\right), x\right), 2\right)} \]

13. Simplified 97.8%

    \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 7.71604938271605 \cdot 10^{-6}, -0.006944444444444444\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right) \cdot -0.00044444444444444447}, -0.4\right), -12\right), x\right), 2\right)} \]
14. Add Preprocessing

Alternative 7: 96.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 7.71604938271605 \cdot 10^{-6}, -0.006944444444444444\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.013333333333333334, -0.4\right), -12\right), x\right), 2\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  2.0
  (fma
   x
   (fma
    (*
     (* x (* x x))
     (fma (* x x) (* (* x x) 7.71604938271605e-6) -0.006944444444444444))
    (fma (* x x) (fma x (* x -0.013333333333333334) -0.4) -12.0)
    x)
   2.0)))

C

double code(double x) {
	return 2.0 / fma(x, fma(((x * (x * x)) * fma((x * x), ((x * x) * 7.71604938271605e-6), -0.006944444444444444)), fma((x * x), fma(x, (x * -0.013333333333333334), -0.4), -12.0), x), 2.0);
}

Julia

function code(x)
	return Float64(2.0 / fma(x, fma(Float64(Float64(x * Float64(x * x)) * fma(Float64(x * x), Float64(Float64(x * x) * 7.71604938271605e-6), -0.006944444444444444)), fma(Float64(x * x), fma(x, Float64(x * -0.013333333333333334), -0.4), -12.0), x), 2.0))
end

Wolfram

code[x_] := N[(2.0 / N[(x * N[(N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 7.71604938271605e-6), $MachinePrecision] + -0.006944444444444444), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.013333333333333334), $MachinePrecision] + -0.4), $MachinePrecision] + -12.0), $MachinePrecision] + x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

5. Simplified 94.5%

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

   1. associate-*r* N/A

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

   2. pow3 N/A

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

   3. flip-+ N/A

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

   4. associate-*r/ N/A

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

   5. div-inv N/A

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

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

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

7. Applied egg-rr 69.5%

   \[\leadsto \frac{2}{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 7.71604938271605 \cdot 10^{-6}, -0.006944444444444444\right), \frac{1}{\mathsf{fma}\left(x, x \cdot 0.002777777777777778, -0.08333333333333333\right)}, x\right)}, 2\right)} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \color{blue}{{x}^{2} \cdot \left(\frac{-1}{75} \cdot {x}^{2} - \frac{2}{5}\right) - 12}, x\right), 2\right)} \]
9. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \color{blue}{{x}^{2} \cdot \left(\frac{-1}{75} \cdot {x}^{2} - \frac{2}{5}\right) + \left(\mathsf{neg}\left(12\right)\right)}, x\right), 2\right)} \]

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

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{75} \cdot {x}^{2} - \frac{2}{5}, \mathsf{neg}\left(12\right)\right)}, x\right), 2\right)} \]

   3. unpow2 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{75} \cdot {x}^{2} - \frac{2}{5}, \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

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

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{75} \cdot {x}^{2} - \frac{2}{5}, \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

   5. sub-neg N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{75} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{2}{5}\right)\right)}, \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{-1}{75}} + \left(\mathsf{neg}\left(\frac{2}{5}\right)\right), \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

   7. unpow2 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{75} + \left(\mathsf{neg}\left(\frac{2}{5}\right)\right), \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

   8. associate-*l* N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{-1}{75}\right)} + \left(\mathsf{neg}\left(\frac{2}{5}\right)\right), \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

   9. metadata-eval N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \frac{-1}{75}\right) + \color{blue}{\frac{-2}{5}}, \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

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

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{75}, \frac{-2}{5}\right)}, \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

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

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{75}}, \frac{-2}{5}\right), \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

   12. metadata-eval 97.4

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 7.71604938271605 \cdot 10^{-6}, -0.006944444444444444\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.013333333333333334, -0.4\right), \color{blue}{-12}\right), x\right), 2\right)} \]

10. Simplified 97.4%

    \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 7.71604938271605 \cdot 10^{-6}, -0.006944444444444444\right), \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.013333333333333334, -0.4\right), -12\right)}, x\right), 2\right)} \]
11. Add Preprocessing

Alternative 8: 95.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot -0.006944444444444444, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.00044444444444444447, -0.013333333333333334\right), -0.4\right), -12\right), x\right), 2\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  2.0
  (fma
   x
   (fma
    (* (* x (* x x)) -0.006944444444444444)
    (fma
     (* x x)
     (fma
      (* x x)
      (fma (* x x) -0.00044444444444444447 -0.013333333333333334)
      -0.4)
     -12.0)
    x)
   2.0)))

C

double code(double x) {
	return 2.0 / fma(x, fma(((x * (x * x)) * -0.006944444444444444), fma((x * x), fma((x * x), fma((x * x), -0.00044444444444444447, -0.013333333333333334), -0.4), -12.0), x), 2.0);
}

Julia

function code(x)
	return Float64(2.0 / fma(x, fma(Float64(Float64(x * Float64(x * x)) * -0.006944444444444444), fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), -0.00044444444444444447, -0.013333333333333334), -0.4), -12.0), x), 2.0))
end

Wolfram

code[x_] := N[(2.0 / N[(x * N[(N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * -0.006944444444444444), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.00044444444444444447 + -0.013333333333333334), $MachinePrecision] + -0.4), $MachinePrecision] + -12.0), $MachinePrecision] + x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

5. Simplified 94.5%

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

   1. associate-*r* N/A

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

   2. pow3 N/A

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

   3. flip-+ N/A

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

   4. associate-*r/ N/A

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

   5. div-inv N/A

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

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

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

7. Applied egg-rr 69.5%

   \[\leadsto \frac{2}{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 7.71604938271605 \cdot 10^{-6}, -0.006944444444444444\right), \frac{1}{\mathsf{fma}\left(x, x \cdot 0.002777777777777778, -0.08333333333333333\right)}, x\right)}, 2\right)} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{2250} \cdot {x}^{2} - \frac{1}{75}\right) - \frac{2}{5}\right) - 12}, x\right), 2\right)} \]
9. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{2250} \cdot {x}^{2} - \frac{1}{75}\right) - \frac{2}{5}\right) + \left(\mathsf{neg}\left(12\right)\right)}, x\right), 2\right)} \]

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

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{-1}{2250} \cdot {x}^{2} - \frac{1}{75}\right) - \frac{2}{5}, \mathsf{neg}\left(12\right)\right)}, x\right), 2\right)} \]

   3. unpow2 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{-1}{2250} \cdot {x}^{2} - \frac{1}{75}\right) - \frac{2}{5}, \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

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

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{-1}{2250} \cdot {x}^{2} - \frac{1}{75}\right) - \frac{2}{5}, \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

   5. sub-neg N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{-1}{2250} \cdot {x}^{2} - \frac{1}{75}\right) + \left(\mathsf{neg}\left(\frac{2}{5}\right)\right)}, \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

   6. metadata-eval N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \left(\frac{-1}{2250} \cdot {x}^{2} - \frac{1}{75}\right) + \color{blue}{\frac{-2}{5}}, \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

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

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{2250} \cdot {x}^{2} - \frac{1}{75}, \frac{-2}{5}\right)}, \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

   8. unpow2 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2250} \cdot {x}^{2} - \frac{1}{75}, \frac{-2}{5}\right), \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

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

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2250} \cdot {x}^{2} - \frac{1}{75}, \frac{-2}{5}\right), \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

   10. sub-neg N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{2250} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{75}\right)\right)}, \frac{-2}{5}\right), \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

   11. *-commutative N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{-1}{2250}} + \left(\mathsf{neg}\left(\frac{1}{75}\right)\right), \frac{-2}{5}\right), \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

   12. metadata-eval N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \frac{-1}{2250} + \color{blue}{\frac{-1}{75}}, \frac{-2}{5}\right), \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

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

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{2250}, \frac{-1}{75}\right)}, \frac{-2}{5}\right), \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

   14. unpow2 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2250}, \frac{-1}{75}\right), \frac{-2}{5}\right), \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

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

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2250}, \frac{-1}{75}\right), \frac{-2}{5}\right), \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

   16. metadata-eval 97.8

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 7.71604938271605 \cdot 10^{-6}, -0.006944444444444444\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.00044444444444444447, -0.013333333333333334\right), -0.4\right), \color{blue}{-12}\right), x\right), 2\right)} \]

10. Simplified 97.8%

    \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 7.71604938271605 \cdot 10^{-6}, -0.006944444444444444\right), \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.00044444444444444447, -0.013333333333333334\right), -0.4\right), -12\right)}, x\right), 2\right)} \]

11. Taylor expanded in x around 0

    \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\color{blue}{\frac{-1}{144} \cdot {x}^{3}}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{2250}, \frac{-1}{75}\right), \frac{-2}{5}\right), -12\right), x\right), 2\right)} \]
12. Step-by-step derivation

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

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\color{blue}{\frac{-1}{144} \cdot {x}^{3}}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{2250}, \frac{-1}{75}\right), \frac{-2}{5}\right), -12\right), x\right), 2\right)} \]

    2. cube-mult N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{-1}{144} \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{2250}, \frac{-1}{75}\right), \frac{-2}{5}\right), -12\right), x\right), 2\right)} \]

    3. unpow2 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{-1}{144} \cdot \left(x \cdot \color{blue}{{x}^{2}}\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{2250}, \frac{-1}{75}\right), \frac{-2}{5}\right), -12\right), x\right), 2\right)} \]

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

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{-1}{144} \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{2250}, \frac{-1}{75}\right), \frac{-2}{5}\right), -12\right), x\right), 2\right)} \]

    5. unpow2 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{-1}{144} \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{2250}, \frac{-1}{75}\right), \frac{-2}{5}\right), -12\right), x\right), 2\right)} \]

    6. *-lowering-*.f64 97.1

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(-0.006944444444444444 \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.00044444444444444447, -0.013333333333333334\right), -0.4\right), -12\right), x\right), 2\right)} \]

13. Simplified 97.1%

    \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\color{blue}{-0.006944444444444444 \cdot \left(x \cdot \left(x \cdot x\right)\right)}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.00044444444444444447, -0.013333333333333334\right), -0.4\right), -12\right), x\right), 2\right)} \]

14. Final simplification 97.1%

    \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot -0.006944444444444444, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.00044444444444444447, -0.013333333333333334\right), -0.4\right), -12\right), x\right), 2\right)} \]
15. Add Preprocessing

Alternative 9: 95.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 7.71604938271605 \cdot 10^{-6}, -0.006944444444444444\right), \mathsf{fma}\left(x, x \cdot -0.4, -12\right), x\right), 2\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  2.0
  (fma
   x
   (fma
    (*
     (* x (* x x))
     (fma (* x x) (* (* x x) 7.71604938271605e-6) -0.006944444444444444))
    (fma x (* x -0.4) -12.0)
    x)
   2.0)))

C

double code(double x) {
	return 2.0 / fma(x, fma(((x * (x * x)) * fma((x * x), ((x * x) * 7.71604938271605e-6), -0.006944444444444444)), fma(x, (x * -0.4), -12.0), x), 2.0);
}

Julia

function code(x)
	return Float64(2.0 / fma(x, fma(Float64(Float64(x * Float64(x * x)) * fma(Float64(x * x), Float64(Float64(x * x) * 7.71604938271605e-6), -0.006944444444444444)), fma(x, Float64(x * -0.4), -12.0), x), 2.0))
end

Wolfram

code[x_] := N[(2.0 / N[(x * N[(N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 7.71604938271605e-6), $MachinePrecision] + -0.006944444444444444), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * -0.4), $MachinePrecision] + -12.0), $MachinePrecision] + x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

5. Simplified 94.5%

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

   1. associate-*r* N/A

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

   2. pow3 N/A

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

   3. flip-+ N/A

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

   4. associate-*r/ N/A

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

   5. div-inv N/A

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

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

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

7. Applied egg-rr 69.5%

   \[\leadsto \frac{2}{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 7.71604938271605 \cdot 10^{-6}, -0.006944444444444444\right), \frac{1}{\mathsf{fma}\left(x, x \cdot 0.002777777777777778, -0.08333333333333333\right)}, x\right)}, 2\right)} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \color{blue}{\frac{-2}{5} \cdot {x}^{2} - 12}, x\right), 2\right)} \]
9. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \color{blue}{\frac{-2}{5} \cdot {x}^{2} + \left(\mathsf{neg}\left(12\right)\right)}, x\right), 2\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \color{blue}{{x}^{2} \cdot \frac{-2}{5}} + \left(\mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

   3. unpow2 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \color{blue}{\left(x \cdot x\right)} \cdot \frac{-2}{5} + \left(\mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

   4. associate-*l* N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \color{blue}{x \cdot \left(x \cdot \frac{-2}{5}\right)} + \left(\mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

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

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-2}{5}, \mathsf{neg}\left(12\right)\right)}, x\right), 2\right)} \]

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

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-2}{5}}, \mathsf{neg}\left(12\right)\right), x\right), 2\right)} \]

   7. metadata-eval 97.0

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 7.71604938271605 \cdot 10^{-6}, -0.006944444444444444\right), \mathsf{fma}\left(x, x \cdot -0.4, \color{blue}{-12}\right), x\right), 2\right)} \]

10. Simplified 97.0%

    \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 7.71604938271605 \cdot 10^{-6}, -0.006944444444444444\right), \color{blue}{\mathsf{fma}\left(x, x \cdot -0.4, -12\right)}, x\right), 2\right)} \]
11. Add Preprocessing

Alternative 10: 94.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 7.71604938271605 \cdot 10^{-6}, -0.006944444444444444\right), -12, x\right), 2\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  2.0
  (fma
   x
   (fma
    (*
     (* x (* x x))
     (fma (* x x) (* (* x x) 7.71604938271605e-6) -0.006944444444444444))
    -12.0
    x)
   2.0)))

C

double code(double x) {
	return 2.0 / fma(x, fma(((x * (x * x)) * fma((x * x), ((x * x) * 7.71604938271605e-6), -0.006944444444444444)), -12.0, x), 2.0);
}

Julia

function code(x)
	return Float64(2.0 / fma(x, fma(Float64(Float64(x * Float64(x * x)) * fma(Float64(x * x), Float64(Float64(x * x) * 7.71604938271605e-6), -0.006944444444444444)), -12.0, x), 2.0))
end

Wolfram

code[x_] := N[(2.0 / N[(x * N[(N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 7.71604938271605e-6), $MachinePrecision] + -0.006944444444444444), $MachinePrecision]), $MachinePrecision] * -12.0 + x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

5. Simplified 94.5%

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

   1. associate-*r* N/A

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

   2. pow3 N/A

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

   3. flip-+ N/A

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

   4. associate-*r/ N/A

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

   5. div-inv N/A

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

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

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

7. Applied egg-rr 69.5%

   \[\leadsto \frac{2}{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 7.71604938271605 \cdot 10^{-6}, -0.006944444444444444\right), \frac{1}{\mathsf{fma}\left(x, x \cdot 0.002777777777777778, -0.08333333333333333\right)}, x\right)}, 2\right)} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{129600}, \frac{-1}{144}\right), \color{blue}{-12}, x\right), 2\right)} \]
9. Step-by-step derivation

10. Simplified 96.5%

    \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 7.71604938271605 \cdot 10^{-6}, -0.006944444444444444\right), \color{blue}{-12}, x\right), 2\right)} \]
11. Add Preprocessing

Alternative 11: 92.4% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(2.0 / N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * 0.002777777777777778), $MachinePrecision] + 0.08333333333333333), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

5. Simplified 94.5%

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

Alternative 12: 92.2% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

5. Simplified 94.5%

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

6. Taylor expanded in x around inf

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

   1. unpow3 N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. *-commutative N/A

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 94.2

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

8. Simplified 94.2%

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

Alternative 13: 91.9% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

5. Simplified 94.5%

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

6. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. pow-plus N/A

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

   3. associate-*l* N/A

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

   4. metadata-eval N/A

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

   5. pow-sqr N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. associate-*r* N/A

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

   10. unpow2 N/A

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

   11. unpow3 N/A

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

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

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

   13. unpow2 N/A

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

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

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

   15. unpow3 N/A

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

   16. unpow2 N/A

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

   17. associate-*r* N/A

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

   18. *-commutative N/A

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

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

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

   20. *-commutative N/A

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

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

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

   22. unpow2 N/A

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

   23. *-lowering-*.f64 94.0

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

8. Simplified 94.0%

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

Alternative 14: 88.0% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

   5. +-commutative N/A

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

   6. distribute-lft-in N/A

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

   7. *-rgt-identity N/A

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

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

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

   9. *-commutative N/A

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

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

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

   11. unpow2 N/A

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

   12. *-lowering-*.f64 92.2

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

5. Simplified 92.2%

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

Alternative 15: 87.6% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

5. Simplified 94.5%

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

6. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}} \]
7. 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. unpow2 N/A

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

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

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

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

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

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

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

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

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

   13. unpow2 N/A

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

   14. *-lowering-*.f64 94.5

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

8. Simplified 94.5%

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

9. Taylor expanded in x around inf

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

    1. distribute-rgt-in N/A

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

    2. metadata-eval N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \frac{1}{360} \cdot {x}^{\color{blue}{\left(3 + 1\right)}} + \left(\frac{1}{12} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{4}, 2\right)} \]

    3. pow-plus N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \frac{1}{360} \cdot \color{blue}{\left({x}^{3} \cdot x\right)} + \left(\frac{1}{12} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{4}, 2\right)} \]

    4. associate-*l* N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{1}{360} \cdot {x}^{3}\right) \cdot x} + \left(\frac{1}{12} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{4}, 2\right)} \]

    5. associate-*r/ N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \left(\frac{1}{360} \cdot {x}^{3}\right) \cdot x + \color{blue}{\frac{\frac{1}{12} \cdot 1}{{x}^{2}}} \cdot {x}^{4}, 2\right)} \]

    6. metadata-eval N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \left(\frac{1}{360} \cdot {x}^{3}\right) \cdot x + \frac{\color{blue}{\frac{1}{12}}}{{x}^{2}} \cdot {x}^{4}, 2\right)} \]

    7. associate-*l/ N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot x, \left(\frac{1}{360} \cdot {x}^{3}\right) \cdot x + \color{blue}{\frac{\frac{1}{12} \cdot {x}^{4}}{{x}^{2}}}, 2\right)} \]

    8. metadata-eval N/A

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

    9. pow-sqr N/A

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

    10. associate-*l* N/A

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

    11. associate-/l* N/A

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

    12. *-rgt-identity N/A

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

    13. associate-*r/ N/A

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

    14. rgt-mult-inverse N/A

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

    15. *-rgt-identity N/A

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

    16. unpow2 N/A

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

    17. associate-*r* N/A

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

    18. distribute-rgt-in N/A

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

11. Simplified 94.0%

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

12. Taylor expanded in x around 0

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

    1. *-commutative N/A

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

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

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

    3. unpow2 N/A

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

    4. *-lowering-*.f64 91.8

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

14. Simplified 91.8%

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

Alternative 16: 63.8% accurate, 9.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. 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 67.7

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

   5. Simplified 67.7%

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

   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 48.9

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

   5. Simplified 48.9%

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

   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 48.9

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

   8. Simplified 48.9%

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

Alternative 17: 76.4% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. accelerator-lowering-fma.f64 78.4

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

5. Simplified 78.4%

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

Alternative 18: 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 2024204 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))