Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 9.5s

Alternatives: 14

Speedup: 1.0×

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, 1.0× 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]

Derivation

1. Initial program 100.0%

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

Alternative 2: 83.8% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ 2.0 (+ (exp x) (exp (- x)))) 0.5)
   (/ 12.0 (* 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))) <= 0.5) {
		tmp = 12.0 / (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)))) <= 0.5)
		tmp = Float64(12.0 / Float64(x * Float64(x * x)));
	else
		tmp = fma(x, Float64(x * fma(x, Float64(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], 0.5], N[(12.0 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(x * N[(x * 0.20833333333333334), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 76.2

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

   5. Simplified 76.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 66.7

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

   8. Simplified 66.7%

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

   9. Taylor expanded in x around inf

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

       1. lower-/.f64 N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 67.3

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

   11. Simplified 67.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.6

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

   5. Simplified 99.6%

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

Alternative 3: 87.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} + e^{-x} \leq 4:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.20833333333333334, -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (exp x) (exp (- x))) 4.0)
   (fma x (* x (fma x (* x 0.20833333333333334) -0.5)) 1.0)
   (/ 2.0 (* x (* x (* x (fma x 0.041666666666666664 0.16666666666666666)))))))

C

double code(double x) {
	double tmp;
	if ((exp(x) + exp(-x)) <= 4.0) {
		tmp = fma(x, (x * fma(x, (x * 0.20833333333333334), -0.5)), 1.0);
	} else {
		tmp = 2.0 / (x * (x * (x * fma(x, 0.041666666666666664, 0.16666666666666666))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) + exp(Float64(-x))) <= 4.0)
		tmp = fma(x, Float64(x * fma(x, Float64(x * 0.20833333333333334), -0.5)), 1.0);
	else
		tmp = Float64(2.0 / Float64(x * Float64(x * Float64(x * fma(x, 0.041666666666666664, 0.16666666666666666)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 4.0], N[(x * N[(x * N[(x * N[(x * 0.20833333333333334), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(2.0 / N[(x * N[(x * N[(x * N[(x * 0.041666666666666664 + 0.16666666666666666), $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

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.6

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

   5. Simplified 99.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 76.2

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

   5. Simplified 76.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. associate-*r* N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 74.5

          \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right)}, x\right), 2\right)} \]

   8. Simplified 74.5%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), x\right), 2\right)}} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{2}{\mathsf{fma}\left(x, \color{blue}{{x}^{3} \cdot \left(\frac{1}{24} + \frac{1}{6} \cdot \frac{1}{x}\right)}, 2\right)} \]
   10. Step-by-step derivation

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. distribute-rgt-in N/A

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

       8. *-commutative N/A

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

       9. associate-*l* N/A

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

       10. lft-mult-inverse N/A

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

       11. metadata-eval N/A

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

       12. lower-fma.f64 74.5

           \[\leadsto \frac{2}{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right)}, 2\right)} \]

   11. Simplified 74.5%

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right)}, 2\right)} \]

   12. Taylor expanded in x around inf

       \[\leadsto \frac{2}{\color{blue}{{x}^{4} \cdot \left(\frac{1}{24} + \frac{1}{6} \cdot \frac{1}{x}\right)}} \]
   13. Step-by-step derivation

       1. distribute-rgt-in N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{1}{24} \cdot {x}^{4} + \left(\frac{1}{6} \cdot \frac{1}{x}\right) \cdot {x}^{4}}} \]

       2. metadata-eval N/A

          \[\leadsto \frac{2}{\frac{1}{24} \cdot {x}^{\color{blue}{\left(3 + 1\right)}} + \left(\frac{1}{6} \cdot \frac{1}{x}\right) \cdot {x}^{4}} \]

       3. pow-plus N/A

          \[\leadsto \frac{2}{\frac{1}{24} \cdot \color{blue}{\left({x}^{3} \cdot x\right)} + \left(\frac{1}{6} \cdot \frac{1}{x}\right) \cdot {x}^{4}} \]

       4. associate-*l* N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right) \cdot x} + \left(\frac{1}{6} \cdot \frac{1}{x}\right) \cdot {x}^{4}} \]

       5. associate-*r/ N/A

          \[\leadsto \frac{2}{\left(\frac{1}{24} \cdot {x}^{3}\right) \cdot x + \color{blue}{\frac{\frac{1}{6} \cdot 1}{x}} \cdot {x}^{4}} \]

       6. metadata-eval N/A

          \[\leadsto \frac{2}{\left(\frac{1}{24} \cdot {x}^{3}\right) \cdot x + \frac{\color{blue}{\frac{1}{6}}}{x} \cdot {x}^{4}} \]

       7. associate-*l/ N/A

          \[\leadsto \frac{2}{\left(\frac{1}{24} \cdot {x}^{3}\right) \cdot x + \color{blue}{\frac{\frac{1}{6} \cdot {x}^{4}}{x}}} \]

       8. associate-/l* N/A

          \[\leadsto \frac{2}{\left(\frac{1}{24} \cdot {x}^{3}\right) \cdot x + \color{blue}{\frac{1}{6} \cdot \frac{{x}^{4}}{x}}} \]

       9. metadata-eval N/A

          \[\leadsto \frac{2}{\left(\frac{1}{24} \cdot {x}^{3}\right) \cdot x + \frac{1}{6} \cdot \frac{{x}^{\color{blue}{\left(3 + 1\right)}}}{x}} \]

       10. pow-plus N/A

           \[\leadsto \frac{2}{\left(\frac{1}{24} \cdot {x}^{3}\right) \cdot x + \frac{1}{6} \cdot \frac{\color{blue}{{x}^{3} \cdot x}}{x}} \]

       11. associate-*l/ N/A

           \[\leadsto \frac{2}{\left(\frac{1}{24} \cdot {x}^{3}\right) \cdot x + \frac{1}{6} \cdot \color{blue}{\left(\frac{{x}^{3}}{x} \cdot x\right)}} \]

       12. *-lft-identity N/A

           \[\leadsto \frac{2}{\left(\frac{1}{24} \cdot {x}^{3}\right) \cdot x + \frac{1}{6} \cdot \left(\frac{\color{blue}{1 \cdot {x}^{3}}}{x} \cdot x\right)} \]

       13. associate-*l/ N/A

           \[\leadsto \frac{2}{\left(\frac{1}{24} \cdot {x}^{3}\right) \cdot x + \frac{1}{6} \cdot \left(\color{blue}{\left(\frac{1}{x} \cdot {x}^{3}\right)} \cdot x\right)} \]

       14. cube-mult N/A

           \[\leadsto \frac{2}{\left(\frac{1}{24} \cdot {x}^{3}\right) \cdot x + \frac{1}{6} \cdot \left(\left(\frac{1}{x} \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}\right) \cdot x\right)} \]

       15. unpow2 N/A

           \[\leadsto \frac{2}{\left(\frac{1}{24} \cdot {x}^{3}\right) \cdot x + \frac{1}{6} \cdot \left(\left(\frac{1}{x} \cdot \left(x \cdot \color{blue}{{x}^{2}}\right)\right) \cdot x\right)} \]

       16. associate-*r* N/A

           \[\leadsto \frac{2}{\left(\frac{1}{24} \cdot {x}^{3}\right) \cdot x + \frac{1}{6} \cdot \left(\color{blue}{\left(\left(\frac{1}{x} \cdot x\right) \cdot {x}^{2}\right)} \cdot x\right)} \]

       17. lft-mult-inverse N/A

           \[\leadsto \frac{2}{\left(\frac{1}{24} \cdot {x}^{3}\right) \cdot x + \frac{1}{6} \cdot \left(\left(\color{blue}{1} \cdot {x}^{2}\right) \cdot x\right)} \]

       18. *-lft-identity N/A

           \[\leadsto \frac{2}{\left(\frac{1}{24} \cdot {x}^{3}\right) \cdot x + \frac{1}{6} \cdot \left(\color{blue}{{x}^{2}} \cdot x\right)} \]

       19. unpow2 N/A

           \[\leadsto \frac{2}{\left(\frac{1}{24} \cdot {x}^{3}\right) \cdot x + \frac{1}{6} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right)} \]

       20. unpow3 N/A

           \[\leadsto \frac{2}{\left(\frac{1}{24} \cdot {x}^{3}\right) \cdot x + \frac{1}{6} \cdot \color{blue}{{x}^{3}}} \]

   14. Simplified 74.5%

       \[\leadsto \frac{2}{\color{blue}{x \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 83.7% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ 2.0 (+ (exp x) (exp (- x)))) 2e-33)
   (/ 12.0 (* x (* x x)))
   (/ 2.0 (fma x x 2.0))))

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[N[(2.0 / N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-33], N[(12.0 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(x * x + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 76.6

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

   5. Simplified 76.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 67.0

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

   8. Simplified 67.0%

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

   9. Taylor expanded in x around inf

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

       1. lower-/.f64 N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 67.7

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

   11. Simplified 67.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 98.8

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

   5. Simplified 98.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2}}} \]
   7. 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 98.9

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

   8. Simplified 98.9%

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

Alternative 5: 87.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} + e^{-x} \leq 10:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.20833333333333334, -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{48}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (exp x) (exp (- x))) 10.0)
   (fma x (* x (fma x (* x 0.20833333333333334) -0.5)) 1.0)
   (/ 48.0 (* x (* x (* x x))))))

C

double code(double x) {
	double tmp;
	if ((exp(x) + exp(-x)) <= 10.0) {
		tmp = fma(x, (x * fma(x, (x * 0.20833333333333334), -0.5)), 1.0);
	} else {
		tmp = 48.0 / (x * (x * (x * x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) + exp(Float64(-x))) <= 10.0)
		tmp = fma(x, Float64(x * fma(x, Float64(x * 0.20833333333333334), -0.5)), 1.0);
	else
		tmp = Float64(48.0 / 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], 10.0], N[(x * N[(x * N[(x * N[(x * 0.20833333333333334), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(48.0 / N[(x * N[(x * 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))) < 10

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.0

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

   5. Simplified 99.0%

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

   if 10 < (+.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

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 76.6

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

   5. Simplified 76.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. associate-*r* N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 74.9

          \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right)}, x\right), 2\right)} \]

   8. Simplified 74.9%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), x\right), 2\right)}} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{48}{{x}^{4}}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{48}{{x}^{4}}} \]

       2. metadata-eval N/A

          \[\leadsto \frac{48}{{x}^{\color{blue}{\left(3 + 1\right)}}} \]

       3. pow-plus N/A

          \[\leadsto \frac{48}{\color{blue}{{x}^{3} \cdot x}} \]

       4. *-commutative N/A

          \[\leadsto \frac{48}{\color{blue}{x \cdot {x}^{3}}} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{48}{\color{blue}{x \cdot {x}^{3}}} \]

       6. cube-mult N/A

          \[\leadsto \frac{48}{x \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}} \]

       7. unpow2 N/A

          \[\leadsto \frac{48}{x \cdot \left(x \cdot \color{blue}{{x}^{2}}\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{48}{x \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}} \]

       9. unpow2 N/A

          \[\leadsto \frac{48}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)} \]

       10. lower-*.f64 74.9

           \[\leadsto \frac{48}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)} \]

   11. Simplified 74.9%

       \[\leadsto \color{blue}{\frac{48}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 76.1% 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 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 99.5

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

   5. Simplified 99.5%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 76.2

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

   5. Simplified 76.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2}}} \]
   7. 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 47.1

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

   8. Simplified 47.1%

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

   9. Taylor expanded in x around inf

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

       1. lower-/.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 47.1

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

   11. Simplified 47.1%

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

Alternative 7: 87.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{2}{e^{x} + \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 2.0 (+ (exp x) (fma x (fma x 0.5 -1.0) 1.0))))

C

double code(double x) {
	return 2.0 / (exp(x) + fma(x, fma(x, 0.5, -1.0), 1.0));
}

Julia

function code(x)
	return Float64(2.0 / Float64(exp(x) + fma(x, fma(x, 0.5, -1.0), 1.0)))
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 87.6

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

5. Simplified 87.6%

   \[\leadsto \frac{2}{e^{x} + \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right)}} \]
6. Add Preprocessing

Alternative 8: 74.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{2}{e^{x} + 1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (+ (exp x) 1.0)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(2.0 / N[(N[Exp[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

5. Simplified 75.8%

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

Alternative 9: 87.6% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), x\right), 2\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  2.0
  (fma
   x
   (fma (* x x) (fma x 0.041666666666666664 0.16666666666666666) x)
   2.0)))

C

double code(double x) {
	return 2.0 / fma(x, fma((x * x), fma(x, 0.041666666666666664, 0.16666666666666666), x), 2.0);
}

Julia

function code(x)
	return Float64(2.0 / fma(x, fma(Float64(x * x), fma(x, 0.041666666666666664, 0.16666666666666666), x), 2.0))
end

Wolfram

code[x_] := N[(2.0 / N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * 0.041666666666666664 + 0.16666666666666666), $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

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 87.6

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

5. Simplified 87.6%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. +-commutative N/A

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

   8. distribute-lft-in N/A

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

   9. *-rgt-identity N/A

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

   10. associate-*r* N/A

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

   11. unpow2 N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. +-commutative N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f64 86.8

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right)}, x\right), 2\right)} \]

8. Simplified 86.8%

   \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), x\right), 2\right)}} \]
9. Add Preprocessing

Alternative 10: 87.4% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 2\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  2.0
  (fma x (* (* x x) (fma x 0.041666666666666664 0.16666666666666666)) 2.0)))

C

double code(double x) {
	return 2.0 / fma(x, ((x * x) * fma(x, 0.041666666666666664, 0.16666666666666666)), 2.0);
}

Julia

function code(x)
	return Float64(2.0 / fma(x, Float64(Float64(x * x) * fma(x, 0.041666666666666664, 0.16666666666666666)), 2.0))
end

Wolfram

code[x_] := N[(2.0 / N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * 0.041666666666666664 + 0.16666666666666666), $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

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 87.6

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

5. Simplified 87.6%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. +-commutative N/A

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

   8. distribute-lft-in N/A

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

   9. *-rgt-identity N/A

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

   10. associate-*r* N/A

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

   11. unpow2 N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. +-commutative N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f64 86.8

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right)}, x\right), 2\right)} \]

8. Simplified 86.8%

   \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), x\right), 2\right)}} \]

9. Taylor expanded in x around inf

   \[\leadsto \frac{2}{\mathsf{fma}\left(x, \color{blue}{{x}^{3} \cdot \left(\frac{1}{24} + \frac{1}{6} \cdot \frac{1}{x}\right)}, 2\right)} \]
10. Step-by-step derivation

    1. unpow3 N/A

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

    2. unpow2 N/A

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

    3. associate-*l* N/A

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

    4. lower-*.f64 N/A

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

    5. unpow2 N/A

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

    6. lower-*.f64 N/A

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

    7. distribute-rgt-in N/A

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

    8. *-commutative N/A

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

    9. associate-*l* N/A

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

    10. lft-mult-inverse N/A

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

    11. metadata-eval N/A

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

    12. lower-fma.f64 86.6

        \[\leadsto \frac{2}{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right)}, 2\right)} \]

11. Simplified 86.6%

    \[\leadsto \frac{2}{\mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right)}, 2\right)} \]
12. Add Preprocessing

Alternative 11: 87.4% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(x, 0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right), 2\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 2.0 (fma x (* 0.041666666666666664 (* x (* x x))) 2.0)))

C

double code(double x) {
	return 2.0 / fma(x, (0.041666666666666664 * (x * (x * x))), 2.0);
}

Julia

function code(x)
	return Float64(2.0 / fma(x, Float64(0.041666666666666664 * Float64(x * Float64(x * x))), 2.0))
end

Wolfram

code[x_] := N[(2.0 / N[(x * N[(0.041666666666666664 * N[(x * N[(x * x), $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

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 87.6

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

5. Simplified 87.6%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. +-commutative N/A

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

   8. distribute-lft-in N/A

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

   9. *-rgt-identity N/A

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

   10. associate-*r* N/A

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

   11. unpow2 N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. +-commutative N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f64 86.8

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right)}, x\right), 2\right)} \]

8. Simplified 86.8%

   \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), x\right), 2\right)}} \]

9. Taylor expanded in x around inf

   \[\leadsto \frac{2}{\mathsf{fma}\left(x, \color{blue}{\frac{1}{24} \cdot {x}^{3}}, 2\right)} \]
10. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, \color{blue}{\frac{1}{24} \cdot {x}^{3}}, 2\right)} \]

    2. cube-mult N/A

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

    3. unpow2 N/A

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

    4. lower-*.f64 N/A

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

    5. unpow2 N/A

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

    6. lower-*.f64 86.5

       \[\leadsto \frac{2}{\mathsf{fma}\left(x, 0.041666666666666664 \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right), 2\right)} \]

11. Simplified 86.5%

    \[\leadsto \frac{2}{\mathsf{fma}\left(x, \color{blue}{0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)}, 2\right)} \]
12. Add Preprocessing

Alternative 12: 76.1% 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

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 87.6

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

5. Simplified 87.6%

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

6. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2}}} \]
7. 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 72.9

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

8. Simplified 72.9%

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

Alternative 13: 51.1% accurate, 14.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

5. Simplified 75.8%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 50.7

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

8. Simplified 50.7%

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

9. Final simplification 50.7%

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

Alternative 14: 50.4% 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 50.4%

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

Reproduce

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