?

Average Error: 57.39% → 0.62%
Time: 14.0s
Precision: binary64
Cost: 38912

?

\[\sin \left(x + \varepsilon\right) - \sin x \]
↓
\[\mathsf{fma}\left(\sin x, \cos \varepsilon, -\sin x\right) + \cos x \cdot \sin \varepsilon \]
(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))
↓
(FPCore (x eps)
 :precision binary64
 (+ (fma (sin x) (cos eps) (- (sin x))) (* (cos x) (sin eps))))
double code(double x, double eps) {
	return sin((x + eps)) - sin(x);
}
↓
double code(double x, double eps) {
	return fma(sin(x), cos(eps), -sin(x)) + (cos(x) * sin(eps));
}

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

code[x_, eps_] := N[(N[Sin[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]
↓
code[x_, eps_] := N[(N[(N[Sin[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + (-N[Sin[x], $MachinePrecision])), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\sin \left(x + \varepsilon\right) - \sin x
↓
\mathsf{fma}\left(\sin x, \cos \varepsilon, -\sin x\right) + \cos x \cdot \sin \varepsilon

Error?

Target

Original57.39%
Target23.78%
Herbie0.62%
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \]

Derivation?

  1. Initial program 57.39

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

  2. Applied egg-rr0.65

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

  3. Applied egg-rr0.62

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

  4. Final simplification0.62

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

Alternatives

Alternative 1
Error23.81%
Cost39880
\[\begin{array}{l} t_0 := \sin \left(x + \varepsilon\right) - \sin x\\ \mathbf{if}\;t_0 \leq -0.02:\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;2 \cdot \left(\cos x \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 2
Error0.61%
Cost32448
\[\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \left(\cos \varepsilon + -1\right)\right) \]
Alternative 3
Error0.62%
Cost32448
\[\mathsf{fma}\left(\cos \varepsilon + -1, \sin x, \cos x \cdot \sin \varepsilon\right) \]
Alternative 4
Error0.63%
Cost26176
\[\cos x \cdot \sin \varepsilon + \sin x \cdot \left(\cos \varepsilon + -1\right) \]
Alternative 5
Error22.5%
Cost25920
\[\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot 0\right) \]
Alternative 6
Error23.78%
Cost13632
\[2 \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \left(\varepsilon + x \cdot 2\right)\right)\right) \]
Alternative 7
Error24.11%
Cost13257
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000195 \lor \neg \left(\varepsilon \leq 1.8 \cdot 10^{-33}\right):\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \end{array} \]
Alternative 8
Error25.04%
Cost6856
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -18000000:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.8 \cdot 10^{-33}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]
Alternative 9
Error44.82%
Cost6464
\[\sin \varepsilon \]
Alternative 10
Error69.16%
Cost836
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq 63:\\ \;\;\;\;2 \cdot \frac{1}{\frac{2}{\varepsilon} + \varepsilon \cdot 0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]
Alternative 11
Error69.35%
Cost196
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq 2:\\ \;\;\;\;\varepsilon\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]
Alternative 12
Error71.13%
Cost64
\[\varepsilon \]

Error

Reproduce?

herbie shell --seed 2023121 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

  :herbie-target
  (* 2.0 (* (cos (+ x (/ eps 2.0))) (sin (/ eps 2.0))))

  (- (sin (+ x eps)) (sin x)))