Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 8.0s

Alternatives: 12

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. cosh-undef N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cosh.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 91.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} + e^{-x} \leq 4:\\ \;\;\;\;\mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (exp x) (exp (- x))) 4.0)
   (fma -0.5 (* x x) 1.0)
   (/
    1.0
    (*
     (* x x)
     (* (* x x) (fma x (* x 0.001388888888888889) 0.041666666666666664))))))

C

double code(double x) {
	double tmp;
	if ((exp(x) + exp(-x)) <= 4.0) {
		tmp = fma(-0.5, (x * x), 1.0);
	} else {
		tmp = 1.0 / ((x * x) * ((x * x) * fma(x, (x * 0.001388888888888889), 0.041666666666666664)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) + exp(Float64(-x))) <= 4.0)
		tmp = fma(-0.5, Float64(x * x), 1.0);
	else
		tmp = Float64(1.0 / Float64(Float64(x * x) * Float64(Float64(x * x) * fma(x, Float64(x * 0.001388888888888889), 0.041666666666666664))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 4.0], N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[(1.0 / N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $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 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   5. Simplified 100.0%

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

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

   1. Initial program 100.0%

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

      1. cosh-undef N/A

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

      2. associate-/r* N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cosh.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 83.8

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

   7. Simplified 83.8%

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

   8. Taylor expanded in x around inf

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{{x}^{6} \cdot \color{blue}{\left(\frac{1}{24} \cdot \frac{1}{{x}^{2}} + \frac{1}{720}\right)}} \]

      2. distribute-rgt-in N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{24} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{6} + \frac{1}{720} \cdot {x}^{6}}} \]

   10. Simplified 83.8%

       \[\leadsto \frac{1}{\color{blue}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 91.9% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (exp x) (exp (- x))) 4.0)
   (fma -0.5 (* x x) 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(-0.5, (x * x), 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(-0.5, Float64(x * x), 1.0);
	else
		tmp = Float64(720.0 / Float64(Float64(x * x) * Float64(Float64(x * 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[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[(720.0 / N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   5. Simplified 100.0%

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

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

   1. Initial program 100.0%

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

      1. cosh-undef N/A

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

      2. associate-/r* N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cosh.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 83.8

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

   7. Simplified 83.8%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. flip-+ N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-+ N/A

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

      9. lift-fma.f64 N/A

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

      10. lower-/.f64 83.8

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

   9. Applied egg-rr 83.8%

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

   10. Taylor expanded in x around inf

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

       1. lower-/.f64 N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

       4. pow-plus N/A

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

       5. pow-plus N/A

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

       6. associate-*r* N/A

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

       7. unpow2 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. metadata-eval N/A

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

       13. pow-sqr N/A

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

       14. lower-*.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 N/A

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

       17. unpow2 N/A

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

       18. lower-*.f64 83.8

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

   12. Simplified 83.8%

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

Alternative 4: 76.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   5. Simplified 100.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 52.4

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

   5. Simplified 52.4%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 52.4

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

   8. Simplified 52.4%

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

Alternative 5: 75.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right)\\ t_1 := x \cdot t\_0\\ \mathbf{if}\;x \leq 10^{+75}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, t\_1 \cdot t\_1, -0.25\right) \cdot \frac{1}{\mathsf{fma}\left(x \cdot x, t\_0, -0.5\right)}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (* x x) 0.001388888888888889 0.041666666666666664))
        (t_1 (* x t_0)))
   (if (<= x 1e+75)
     (/
      1.0
      (fma
       (* x x)
       (* (fma (* x x) (* t_1 t_1) -0.25) (/ 1.0 (fma (* x x) t_0 -0.5)))
       1.0))
     (/ 24.0 (* x (* x (* x x)))))))

C

double code(double x) {
	double t_0 = fma((x * x), 0.001388888888888889, 0.041666666666666664);
	double t_1 = x * t_0;
	double tmp;
	if (x <= 1e+75) {
		tmp = 1.0 / fma((x * x), (fma((x * x), (t_1 * t_1), -0.25) * (1.0 / fma((x * x), t_0, -0.5))), 1.0);
	} else {
		tmp = 24.0 / (x * (x * (x * x)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664)
	t_1 = Float64(x * t_0)
	tmp = 0.0
	if (x <= 1e+75)
		tmp = Float64(1.0 / fma(Float64(x * x), Float64(fma(Float64(x * x), Float64(t_1 * t_1), -0.25) * Float64(1.0 / fma(Float64(x * x), t_0, -0.5))), 1.0));
	else
		tmp = Float64(24.0 / Float64(x * Float64(x * Float64(x * x))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, If[LessEqual[x, 1e+75], N[(1.0 / N[(N[(x * x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision] + -0.25), $MachinePrecision] * N[(1.0 / N[(N[(x * x), $MachinePrecision] * t$95$0 + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(24.0 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 9.99999999999999927e74

   1. Initial program 100.0%

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

      1. cosh-undef N/A

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

      2. associate-/r* N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cosh.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 91.2

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

   7. Simplified 91.2%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. flip-+ N/A

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

   9. Applied egg-rr 72.8%

      \[\leadsto \frac{1}{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x \cdot x, \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right)\right), -0.25\right) \cdot \frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), -0.5\right)}}, 1\right)} \]

   if 9.99999999999999927e74 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 100.0

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

   8. Simplified 100.0%

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

Alternative 6: 73.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;x \leq 2.35 \cdot 10^{+51}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(t\_0, t\_0, -16\right)}{\mathsf{fma}\left(x, x, -2\right) \cdot \mathsf{fma}\left(x \cdot x, x \cdot x, 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{720}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x (* x x)))))
   (if (<= x 2.35e+51)
     (/
      2.0
      (/ (fma t_0 t_0 -16.0) (* (fma x x -2.0) (fma (* x x) (* x x) 4.0))))
     (/ 720.0 (* (* x x) (* (* x x) (* x x)))))))

C

double code(double x) {
	double t_0 = x * (x * (x * x));
	double tmp;
	if (x <= 2.35e+51) {
		tmp = 2.0 / (fma(t_0, t_0, -16.0) / (fma(x, x, -2.0) * fma((x * x), (x * x), 4.0)));
	} else {
		tmp = 720.0 / ((x * x) * ((x * x) * (x * x)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(x * Float64(x * Float64(x * x)))
	tmp = 0.0
	if (x <= 2.35e+51)
		tmp = Float64(2.0 / Float64(fma(t_0, t_0, -16.0) / Float64(fma(x, x, -2.0) * fma(Float64(x * x), Float64(x * x), 4.0))));
	else
		tmp = Float64(720.0 / Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(x * x))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.35e+51], N[(2.0 / N[(N[(t$95$0 * t$95$0 + -16.0), $MachinePrecision] / N[(N[(x * x + -2.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(720.0 / N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.3500000000000001e51

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 81.0

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

   5. Simplified 81.0%

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

      1. lift-*.f64 N/A

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

      2. flip-+ N/A

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

      3. flip-- N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   7. Applied egg-rr 69.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), x \cdot \left(x \cdot \left(x \cdot x\right)\right), -16\right)}{\mathsf{fma}\left(x, x, -2\right) \cdot \mathsf{fma}\left(x \cdot x, x \cdot x, 4\right)}}} \]

   if 2.3500000000000001e51 < x

   1. Initial program 100.0%

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

      1. cosh-undef N/A

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

      2. associate-/r* N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cosh.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 100.0

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

   7. Simplified 100.0%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. flip-+ N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-+ N/A

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

      9. lift-fma.f64 N/A

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

      10. lower-/.f64 100.0

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around inf

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

       1. lower-/.f64 N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

       4. pow-plus N/A

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

       5. pow-plus N/A

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

       6. associate-*r* N/A

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

       7. unpow2 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. metadata-eval N/A

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

       13. pow-sqr N/A

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

       14. lower-*.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 N/A

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

       17. unpow2 N/A

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

       18. lower-*.f64 100.0

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

   12. Simplified 100.0%

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

Alternative 7: 92.0% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(1.0 / N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. cosh-undef N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cosh.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-*.f64 92.6

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

7. Simplified 92.6%

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

Alternative 8: 69.7% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.2:\\ \;\;\;\;\mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.08333333333333333, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.2)
   (fma -0.5 (* x x) 1.0)
   (/ 2.0 (* x (fma x (* (* x x) 0.08333333333333333) x)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.2) {
		tmp = fma(-0.5, (x * x), 1.0);
	} else {
		tmp = 2.0 / (x * fma(x, ((x * x) * 0.08333333333333333), x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.2)
		tmp = fma(-0.5, Float64(x * x), 1.0);
	else
		tmp = Float64(2.0 / Float64(x * fma(x, Float64(Float64(x * x) * 0.08333333333333333), x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.2], N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[(2.0 / N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.08333333333333333), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.19999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 71.8

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

   5. Simplified 71.8%

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

   if 1.19999999999999996 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 68.9

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

   5. Simplified 68.9%

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

   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-rgt-in N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. associate-*l* N/A

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

      5. associate-*l/ N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. associate-/l* N/A

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

      10. *-rgt-identity N/A

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

      11. associate-*r/ N/A

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

      12. rgt-mult-inverse N/A

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

      13. *-rgt-identity N/A

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

      14. unpow2 N/A

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

      15. distribute-rgt-in N/A

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

      16. unpow3 N/A

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

      17. unpow2 N/A

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

      18. associate-*r* N/A

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

      19. *-lft-identity N/A

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

      20. distribute-rgt-in N/A

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

      21. +-commutative N/A

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

   8. Simplified 68.9%

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

Alternative 9: 87.7% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. distribute-lft-in N/A

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

   7. *-rgt-identity N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 88.5

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

5. Simplified 88.5%

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

Alternative 10: 69.7% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.4) (fma -0.5 (* x x) 1.0) (/ 24.0 (* x (* x (* x x))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.4) {
		tmp = fma(-0.5, (x * x), 1.0);
	} else {
		tmp = 24.0 / (x * (x * (x * x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.4)
		tmp = fma(-0.5, Float64(x * x), 1.0);
	else
		tmp = Float64(24.0 / Float64(x * Float64(x * Float64(x * x))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.4], N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[(24.0 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 71.8

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

   5. Simplified 71.8%

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

   if 1.3999999999999999 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 68.9

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

   5. Simplified 68.9%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 68.9

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

   8. Simplified 68.9%

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

Alternative 11: 76.0% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. lower-fma.f64 78.2

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

5. Simplified 78.2%

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

Alternative 12: 51.8% 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 55.2%

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

Reproduce

herbie shell --seed 2024207 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))