Average Error: 0.0 → 0.0
Time: 3.1s
Precision: binary64
\[\frac{2}{e^{x} + e^{-x}} \]
↓
\[\frac{2}{\mathsf{fma}\left(\cosh x, 2, 0\right)} \]
(FPCore (x) :precision binary64 (/ 2.0 (+ (exp x) (exp (- x)))))
↓
(FPCore (x) :precision binary64 (/ 2.0 (fma (cosh x) 2.0 0.0)))
double code(double x) {
	return 2.0 / (exp(x) + exp(-x));
}
↓
double code(double x) {
	return 2.0 / fma(cosh(x), 2.0, 0.0);
}

function code(x)
	return Float64(2.0 / Float64(exp(x) + exp(Float64(-x))))
end
↓
function code(x)
	return Float64(2.0 / fma(cosh(x), 2.0, 0.0))
end

code[x_] := N[(2.0 / N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_] := N[(2.0 / N[(N[Cosh[x], $MachinePrecision] * 2.0 + 0.0), $MachinePrecision]), $MachinePrecision]

\frac{2}{e^{x} + e^{-x}}
↓
\frac{2}{\mathsf{fma}\left(\cosh x, 2, 0\right)}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

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

  2. Applied egg-rr0.5

    \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \cosh x\right)\right)}} \]

  3. Applied egg-rr0.0

    \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\cosh x, 2, 0\right)}} \]

  4. Final simplification0.0

    \[\leadsto \frac{2}{\mathsf{fma}\left(\cosh x, 2, 0\right)} \]

Reproduce

herbie shell --seed 2022155 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))