Average Error: 29.3 → 0.7
Time: 10.6s
Precision: binary64
\[\left(e^{x} - 2\right) + e^{-x} \]
↓
\[\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right) \]
(FPCore (x) :precision binary64 (+ (- (exp x) 2.0) (exp (- x))))
↓
(FPCore (x) :precision binary64 (fma x x (* 0.08333333333333333 (pow x 4.0))))
double code(double x) {
	return (exp(x) - 2.0) + exp(-x);
}
↓
double code(double x) {
	return fma(x, x, (0.08333333333333333 * pow(x, 4.0)));
}

function code(x)
	return Float64(Float64(exp(x) - 2.0) + exp(Float64(-x)))
end
↓
function code(x)
	return fma(x, x, Float64(0.08333333333333333 * (x ^ 4.0)))
end

code[x_] := N[(N[(N[Exp[x], $MachinePrecision] - 2.0), $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]
↓
code[x_] := N[(x * x + N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\left(e^{x} - 2\right) + e^{-x}
↓
\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)

Error

Bits error versus x

Target

Original29.3
Target0.0
Herbie0.7
\[4 \cdot {\sinh \left(\frac{x}{2}\right)}^{2} \]

Derivation

  1. Initial program 29.3

    \[\left(e^{x} - 2\right) + e^{-x} \]

  2. Simplified29.3

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

  3. Taylor expanded in x around 0 0.7

    \[\leadsto \color{blue}{0.08333333333333333 \cdot {x}^{4} + {x}^{2}} \]

  4. Simplified0.7

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)} \]

  5. Final simplification0.7

    \[\leadsto \mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right) \]

Reproduce

herbie shell --seed 2022160 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

  :herbie-target
  (* 4.0 (pow (sinh (/ x 2.0)) 2.0))

  (+ (- (exp x) 2.0) (exp (- x))))