Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 7.3s

Alternatives: 13

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. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-+.f64 N/A

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

   4. lift-exp.f64 N/A

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

   5. lift-exp.f64 N/A

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

   6. lift-neg.f64 N/A

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

   7. cosh-def N/A

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

   8. lower-/.f64 N/A

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

   9. lower-cosh.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 92.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{e^{-x} + e^{x}} \leq 0:\\ \;\;\;\;\frac{2}{\left(\left(0.002777777777777778 \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.08472222222222223, x \cdot x, 0.20833333333333334\right), x \cdot x, -0.5\right), x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ 2.0 (+ (exp (- x)) (exp x))) 0.0)
   (/ 2.0 (* (* (* 0.002777777777777778 (* x x)) x) (* (* x x) x)))
   (fma
    (fma (fma -0.08472222222222223 (* x x) 0.20833333333333334) (* x x) -0.5)
    (* x x)
    1.0)))

C

double code(double x) {
	double tmp;
	if ((2.0 / (exp(-x) + exp(x))) <= 0.0) {
		tmp = 2.0 / (((0.002777777777777778 * (x * x)) * x) * ((x * x) * x));
	} else {
		tmp = fma(fma(fma(-0.08472222222222223, (x * x), 0.20833333333333334), (x * x), -0.5), (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(2.0 / Float64(exp(Float64(-x)) + exp(x))) <= 0.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(0.002777777777777778 * Float64(x * x)) * x) * Float64(Float64(x * x) * x)));
	else
		tmp = fma(fma(fma(-0.08472222222222223, Float64(x * x), 0.20833333333333334), Float64(x * x), -0.5), Float64(x * x), 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], 0.0], N[(2.0 / N[(N[(N[(0.002777777777777778 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.08472222222222223 * N[(x * x), $MachinePrecision] + 0.20833333333333334), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.5), $MachinePrecision] * N[(x * x), $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)))) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 79.5

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

   5. Applied rewrites 79.5%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{2}{\frac{1}{360} \cdot \color{blue}{{x}^{6}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 79.5%

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

   if 0.0 < (/.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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.08472222222222223, x \cdot x, 0.20833333333333334\right), x \cdot x, -0.5\right), x \cdot x, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{e^{-x} + e^{x}} \leq 0:\\ \;\;\;\;\frac{2}{\left(\left(0.002777777777777778 \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.08472222222222223, x \cdot x, 0.20833333333333334\right), x \cdot x, -0.5\right), x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 7.71604938271605 \cdot 10^{-6}, x \cdot x, -0.006944444444444444\right) \cdot \left(x \cdot x\right), \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.00044444444444444447, x \cdot x, -0.013333333333333334\right), x \cdot x, -0.4\right), x \cdot x, -12\right), 1\right), x \cdot x, 2\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  2.0
  (fma
   (fma
    (*
     (fma (* (* x x) 7.71604938271605e-6) (* x x) -0.006944444444444444)
     (* x x))
    (fma
     (fma
      (fma -0.00044444444444444447 (* x x) -0.013333333333333334)
      (* x x)
      -0.4)
     (* x x)
     -12.0)
    1.0)
   (* x x)
   2.0)))

C

double code(double x) {
	return 2.0 / fma(fma((fma(((x * x) * 7.71604938271605e-6), (x * x), -0.006944444444444444) * (x * x)), fma(fma(fma(-0.00044444444444444447, (x * x), -0.013333333333333334), (x * x), -0.4), (x * x), -12.0), 1.0), (x * x), 2.0);
}

Julia

function code(x)
	return Float64(2.0 / fma(fma(Float64(fma(Float64(Float64(x * x) * 7.71604938271605e-6), Float64(x * x), -0.006944444444444444) * Float64(x * x)), fma(fma(fma(-0.00044444444444444447, Float64(x * x), -0.013333333333333334), Float64(x * x), -0.4), Float64(x * x), -12.0), 1.0), Float64(x * x), 2.0))
end

Wolfram

code[x_] := N[(2.0 / N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 7.71604938271605e-6), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.006944444444444444), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.00044444444444444447 * N[(x * x), $MachinePrecision] + -0.013333333333333334), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.4), $MachinePrecision] * N[(x * x), $MachinePrecision] + -12.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 90.0

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

5. Applied rewrites 90.0%

   \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot x, 2\right)}} \]
6. Step-by-step derivation

7. Applied rewrites 90.0%

   \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right)}}, \color{blue}{x} \cdot x, 2\right)} \]
8. Step-by-step derivation

9. Applied rewrites 69.3%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 96.3%

    \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(7.71604938271605 \cdot 10^{-6} \cdot \left(x \cdot x\right), x \cdot x, -0.006944444444444444\right) \cdot \left(x \cdot x\right), \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.00044444444444444447, x \cdot x, -0.013333333333333334\right), x \cdot x, -0.4\right), x \cdot x, -12\right), 1\right), x \cdot x, 2\right)} \]

13. Final simplification 96.3%

    \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 7.71604938271605 \cdot 10^{-6}, x \cdot x, -0.006944444444444444\right) \cdot \left(x \cdot x\right), \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.00044444444444444447, x \cdot x, -0.013333333333333334\right), x \cdot x, -0.4\right), x \cdot x, -12\right), 1\right), x \cdot x, 2\right)} \]
14. Add Preprocessing

Alternative 4: 96.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 7.71604938271605 \cdot 10^{-6}, x \cdot x, -0.006944444444444444\right) \cdot \left(x \cdot x\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.013333333333333334, x \cdot x, -0.4\right), x \cdot x, -12\right), 1\right), x \cdot x, 2\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  2.0
  (fma
   (fma
    (*
     (fma (* (* x x) 7.71604938271605e-6) (* x x) -0.006944444444444444)
     (* x x))
    (fma (fma -0.013333333333333334 (* x x) -0.4) (* x x) -12.0)
    1.0)
   (* x x)
   2.0)))

C

double code(double x) {
	return 2.0 / fma(fma((fma(((x * x) * 7.71604938271605e-6), (x * x), -0.006944444444444444) * (x * x)), fma(fma(-0.013333333333333334, (x * x), -0.4), (x * x), -12.0), 1.0), (x * x), 2.0);
}

Julia

function code(x)
	return Float64(2.0 / fma(fma(Float64(fma(Float64(Float64(x * x) * 7.71604938271605e-6), Float64(x * x), -0.006944444444444444) * Float64(x * x)), fma(fma(-0.013333333333333334, Float64(x * x), -0.4), Float64(x * x), -12.0), 1.0), Float64(x * x), 2.0))
end

Wolfram

code[x_] := N[(2.0 / N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 7.71604938271605e-6), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.006944444444444444), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.013333333333333334 * N[(x * x), $MachinePrecision] + -0.4), $MachinePrecision] * N[(x * x), $MachinePrecision] + -12.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 90.0

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

5. Applied rewrites 90.0%

   \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot x, 2\right)}} \]
6. Step-by-step derivation

7. Applied rewrites 90.0%

   \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right)}}, \color{blue}{x} \cdot x, 2\right)} \]
8. Step-by-step derivation

9. Applied rewrites 69.3%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 95.2%

    \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(7.71604938271605 \cdot 10^{-6} \cdot \left(x \cdot x\right), x \cdot x, -0.006944444444444444\right) \cdot \left(x \cdot x\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.013333333333333334, x \cdot x, -0.4\right), x \cdot x, -12\right), 1\right), x \cdot x, 2\right)} \]

13. Final simplification 95.2%

    \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 7.71604938271605 \cdot 10^{-6}, x \cdot x, -0.006944444444444444\right) \cdot \left(x \cdot x\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.013333333333333334, x \cdot x, -0.4\right), x \cdot x, -12\right), 1\right), x \cdot x, 2\right)} \]
14. Add Preprocessing

Alternative 5: 95.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 7.71604938271605 \cdot 10^{-6}, x \cdot x, -0.006944444444444444\right) \cdot \left(x \cdot x\right), \mathsf{fma}\left(-0.4, x \cdot x, -12\right), 1\right), x \cdot x, 2\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  2.0
  (fma
   (fma
    (*
     (fma (* (* x x) 7.71604938271605e-6) (* x x) -0.006944444444444444)
     (* x x))
    (fma -0.4 (* x x) -12.0)
    1.0)
   (* x x)
   2.0)))

C

double code(double x) {
	return 2.0 / fma(fma((fma(((x * x) * 7.71604938271605e-6), (x * x), -0.006944444444444444) * (x * x)), fma(-0.4, (x * x), -12.0), 1.0), (x * x), 2.0);
}

Julia

function code(x)
	return Float64(2.0 / fma(fma(Float64(fma(Float64(Float64(x * x) * 7.71604938271605e-6), Float64(x * x), -0.006944444444444444) * Float64(x * x)), fma(-0.4, Float64(x * x), -12.0), 1.0), Float64(x * x), 2.0))
end

Wolfram

code[x_] := N[(2.0 / N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 7.71604938271605e-6), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.006944444444444444), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(-0.4 * N[(x * x), $MachinePrecision] + -12.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 90.0

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

5. Applied rewrites 90.0%

   \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot x, 2\right)}} \]
6. Step-by-step derivation

7. Applied rewrites 90.0%

   \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right)}}, \color{blue}{x} \cdot x, 2\right)} \]
8. Step-by-step derivation

9. Applied rewrites 69.3%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 95.1%

    \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(7.71604938271605 \cdot 10^{-6} \cdot \left(x \cdot x\right), x \cdot x, -0.006944444444444444\right) \cdot \left(x \cdot x\right), \mathsf{fma}\left(-0.4, x \cdot x, -12\right), 1\right), x \cdot x, 2\right)} \]

13. Final simplification 95.1%

    \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 7.71604938271605 \cdot 10^{-6}, x \cdot x, -0.006944444444444444\right) \cdot \left(x \cdot x\right), \mathsf{fma}\left(-0.4, x \cdot x, -12\right), 1\right), x \cdot x, 2\right)} \]
14. Add Preprocessing

Alternative 6: 94.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 7.71604938271605 \cdot 10^{-6}, x \cdot x, -0.006944444444444444\right) \cdot \left(x \cdot x\right), -12, 1\right), x \cdot x, 2\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  2.0
  (fma
   (fma
    (*
     (fma (* (* x x) 7.71604938271605e-6) (* x x) -0.006944444444444444)
     (* x x))
    -12.0
    1.0)
   (* x x)
   2.0)))

C

double code(double x) {
	return 2.0 / fma(fma((fma(((x * x) * 7.71604938271605e-6), (x * x), -0.006944444444444444) * (x * x)), -12.0, 1.0), (x * x), 2.0);
}

Julia

function code(x)
	return Float64(2.0 / fma(fma(Float64(fma(Float64(Float64(x * x) * 7.71604938271605e-6), Float64(x * x), -0.006944444444444444) * Float64(x * x)), -12.0, 1.0), Float64(x * x), 2.0))
end

Wolfram

code[x_] := N[(2.0 / N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 7.71604938271605e-6), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.006944444444444444), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * -12.0 + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 90.0

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

5. Applied rewrites 90.0%

   \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot x, 2\right)}} \]
6. Step-by-step derivation

7. Applied rewrites 90.0%

   \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right)}}, \color{blue}{x} \cdot x, 2\right)} \]
8. Step-by-step derivation

9. Applied rewrites 69.3%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 92.6%

    \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(7.71604938271605 \cdot 10^{-6} \cdot \left(x \cdot x\right), x \cdot x, -0.006944444444444444\right) \cdot \left(x \cdot x\right), -12, 1\right), x \cdot x, 2\right)} \]

13. Final simplification 92.6%

    \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 7.71604938271605 \cdot 10^{-6}, x \cdot x, -0.006944444444444444\right) \cdot \left(x \cdot x\right), -12, 1\right), x \cdot x, 2\right)} \]
14. Add Preprocessing

Alternative 7: 92.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot x, 2\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  2.0
  (fma
   (fma (fma 0.002777777777777778 (* x x) 0.08333333333333333) (* x x) 1.0)
   (* x x)
   2.0)))

C

double code(double x) {
	return 2.0 / fma(fma(fma(0.002777777777777778, (x * x), 0.08333333333333333), (x * x), 1.0), (x * x), 2.0);
}

Julia

function code(x)
	return Float64(2.0 / fma(fma(fma(0.002777777777777778, Float64(x * x), 0.08333333333333333), Float64(x * x), 1.0), Float64(x * x), 2.0))
end

Wolfram

code[x_] := N[(2.0 / N[(N[(N[(0.002777777777777778 * N[(x * x), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 90.0

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

5. Applied rewrites 90.0%

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

Alternative 8: 92.8% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778 \cdot \left(x \cdot x\right), x \cdot x, 1\right), x \cdot x, 2\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 2.0 (fma (fma (* 0.002777777777777778 (* x x)) (* x x) 1.0) (* x x) 2.0)))

C

double code(double x) {
	return 2.0 / fma(fma((0.002777777777777778 * (x * x)), (x * x), 1.0), (x * x), 2.0);
}

Julia

function code(x)
	return Float64(2.0 / fma(fma(Float64(0.002777777777777778 * Float64(x * x)), Float64(x * x), 1.0), Float64(x * x), 2.0))
end

Wolfram

code[x_] := N[(2.0 / N[(N[(N[(0.002777777777777778 * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 90.0

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

5. Applied rewrites 90.0%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 89.8%

   \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778 \cdot \left(x \cdot x\right), x \cdot x, 1\right), x \cdot x, 2\right)} \]
9. Add Preprocessing

Alternative 9: 92.4% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 90.0

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

5. Applied rewrites 90.0%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 89.5%

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

9. Final simplification 89.5%

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

Alternative 10: 89.0% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right) \cdot x, x, 2\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 2.0 (fma (* (fma 0.08333333333333333 (* x x) 1.0) x) x 2.0)))

C

double code(double x) {
	return 2.0 / fma((fma(0.08333333333333333, (x * x), 1.0) * x), x, 2.0);
}

Julia

function code(x)
	return Float64(2.0 / fma(Float64(fma(0.08333333333333333, Float64(x * x), 1.0) * x), x, 2.0))
end

Wolfram

code[x_] := N[(2.0 / N[(N[(N[(0.08333333333333333 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * 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

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 90.0

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

5. Applied rewrites 90.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 88.0%

   \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right), x \cdot x, 2\right)} \]
9. Step-by-step derivation

10. Applied rewrites 88.0%

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

Alternative 11: 63.9% accurate, 9.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 67.0

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

   5. Applied rewrites 67.0%

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

   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

      \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2}}} \]
   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 48.1

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

   5. Applied rewrites 48.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 48.1%

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

Alternative 12: 77.0% 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 74.0

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

5. Applied rewrites 74.0%

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

Alternative 13: 51.4% 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. Applied rewrites 52.5%

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

Reproduce

herbie shell --seed 2024240 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))