Average Error: 39.4 → 0.4
Time: 6.8s
Precision: binary64
\[\cos \left(x + \varepsilon\right) - \cos x \]
↓
\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \varepsilon\right)\\ t_0 \cdot \left(-2 \cdot \mathsf{fma}\left(t_0, \cos x, \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin x\right)\right) \end{array} \]
(FPCore (x eps) :precision binary64 (- (cos (+ x eps)) (cos x)))
↓
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 eps))))
   (* t_0 (* -2.0 (fma t_0 (cos x) (* (cos (* 0.5 eps)) (sin x)))))))
double code(double x, double eps) {
	return cos((x + eps)) - cos(x);
}
↓
double code(double x, double eps) {
	double t_0 = sin((0.5 * eps));
	return t_0 * (-2.0 * fma(t_0, cos(x), (cos((0.5 * eps)) * sin(x))));
}

function code(x, eps)
	return Float64(cos(Float64(x + eps)) - cos(x))
end
↓
function code(x, eps)
	t_0 = sin(Float64(0.5 * eps))
	return Float64(t_0 * Float64(-2.0 * fma(t_0, cos(x), Float64(cos(Float64(0.5 * eps)) * sin(x)))))
end

code[x_, eps_] := N[(N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]
↓
code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision]}, N[(t$95$0 * N[(-2.0 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(N[Cos[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
t_0 := \sin \left(0.5 \cdot \varepsilon\right)\\
t_0 \cdot \left(-2 \cdot \mathsf{fma}\left(t_0, \cos x, \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin x\right)\right)
\end{array}

Error

Derivation

  1. Initial program 39.4

    \[\cos \left(x + \varepsilon\right) - \cos x \]

  2. Applied egg-rr15.3

    \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\left(\varepsilon + \left(x - x\right)\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right)} \]

  3. Taylor expanded in eps around inf 15.3

    \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)} \]

  4. Simplified15.3

    \[\leadsto \color{blue}{\sin \left(0.5 \cdot \varepsilon\right) \cdot \left(-2 \cdot \sin \left(\mathsf{fma}\left(0.5, \varepsilon, x\right)\right)\right)} \]

  5. Applied egg-rr0.4

    \[\leadsto \sin \left(0.5 \cdot \varepsilon\right) \cdot \left(-2 \cdot \color{blue}{\mathsf{fma}\left(\sin \left(0.5 \cdot \varepsilon\right), \cos x, \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin x\right)}\right) \]

  6. Final simplification0.4

    \[\leadsto \sin \left(0.5 \cdot \varepsilon\right) \cdot \left(-2 \cdot \mathsf{fma}\left(\sin \left(0.5 \cdot \varepsilon\right), \cos x, \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin x\right)\right) \]

Reproduce

herbie shell --seed 2022192 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))