Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 10.3s

Alternatives: 14

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: 92.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 4.0], N[(1.0 / N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $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. 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. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 99.7

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

   7. Applied rewrites 99.7%

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

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

   1. Initial program 100.0%

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

      1. 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. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 84.6

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

   7. Applied rewrites 84.6%

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

   9. Applied rewrites 84.6%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 84.6%

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

4. Final simplification 92.3%

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

Reproduce

herbie shell --seed 2024226 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))