Average Error: 59.5 → 36.7
Time: 12.9s
Precision: binary64
\[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{e^{\sqrt{\cos x}}}\\ \frac{\left(\left(e^{x}\right) \bmod \left(\log \left(t_0 \cdot t_0\right) + \log t_0\right)\right)}{e^{x}} \end{array} \]
(FPCore (x) :precision binary64 (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (exp (sqrt (cos x))))))
   (/ (fmod (exp x) (+ (log (* t_0 t_0)) (log t_0))) (exp x))))
double code(double x) {
	return fmod(exp(x), sqrt(cos(x))) * exp(-x);
}
↓
double code(double x) {
	double t_0 = cbrt(exp(sqrt(cos(x))));
	return fmod(exp(x), (log((t_0 * t_0)) + log(t_0))) / exp(x);
}

function code(x)
	return Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x)))
end
↓
function code(x)
	t_0 = cbrt(exp(sqrt(cos(x))))
	return Float64(rem(exp(x), Float64(log(Float64(t_0 * t_0)) + log(t_0))) / exp(x))
end

code[x_] := N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]
↓
code[x_] := Block[{t$95$0 = N[Power[N[Exp[N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[Log[N[(t$95$0 * t$95$0), $MachinePrecision]], $MachinePrecision] + N[Log[t$95$0], $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]

\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}
↓
\begin{array}{l}
t_0 := \sqrt[3]{e^{\sqrt{\cos x}}}\\
\frac{\left(\left(e^{x}\right) \bmod \left(\log \left(t_0 \cdot t_0\right) + \log t_0\right)\right)}{e^{x}}
\end{array}

Error

Bits error versus x

Derivation

  1. Initial program 59.5

    \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

  2. Simplified59.5

    \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}} \]

  3. Applied egg-rr36.7

    \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\log \left(\sqrt[3]{e^{\sqrt{\cos x}}} \cdot \sqrt[3]{e^{\sqrt{\cos x}}}\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)}\right)}{e^{x}} \]

  4. Final simplification36.7

    \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\log \left(\sqrt[3]{e^{\sqrt{\cos x}}} \cdot \sqrt[3]{e^{\sqrt{\cos x}}}\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

Reproduce

herbie shell --seed 2022150 
(FPCore (x)
  :name "expfmod"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))