?

Average Error: 57.76% → 0.66%
Time: 14.7s
Precision: binary64
Cost: 32448

?

\[\sin \left(x + \varepsilon\right) - \sin x \]
↓
\[\mathsf{fma}\left(\cos \varepsilon + -1, \sin x, \cos x \cdot \sin \varepsilon\right) \]
(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))
↓
(FPCore (x eps)
 :precision binary64
 (fma (+ (cos eps) -1.0) (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((cos(eps) + -1.0), sin(x), (cos(x) * sin(eps)));
}

function code(x, eps)
	return Float64(sin(Float64(x + eps)) - sin(x))
end
↓
function code(x, eps)
	return fma(Float64(cos(eps) + -1.0), 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[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision] * N[Sin[x], $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\sin \left(x + \varepsilon\right) - \sin x
↓
\mathsf{fma}\left(\cos \varepsilon + -1, \sin x, \cos x \cdot \sin \varepsilon\right)

Error?

Target

Original57.76%
Target23.98%
Herbie0.66%
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \]

Derivation?

  1. Initial program 57.76

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

  2. Applied egg-rr0.69

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

  3. Applied egg-rr46.05

    \[\leadsto \color{blue}{\left(\sin x \cdot \left(\cos \varepsilon - 1\right) + e^{\mathsf{log1p}\left(\cos x \cdot \sin \varepsilon\right)}\right) - 1} \]

  4. Applied egg-rr0.66

    \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \varepsilon + -1, \sin x, \cos x \cdot \sin \varepsilon\right)} \]

  5. Final simplification0.66

    \[\leadsto \mathsf{fma}\left(\cos \varepsilon + -1, \sin x, \cos x \cdot \sin \varepsilon\right) \]

Alternatives

Alternative 1
Error0.65%
Cost32448
\[\mathsf{fma}\left(\sin \varepsilon, \cos x, \left(\cos \varepsilon + -1\right) \cdot \sin x\right) \]
Alternative 2
Error0.67%
Cost26176
\[\cos x \cdot \sin \varepsilon + \left(\cos \varepsilon + -1\right) \cdot \sin x \]
Alternative 3
Error22.73%
Cost26048
\[\cos x \cdot \sin \varepsilon + \left(\sin x - \sin x\right) \]
Alternative 4
Error23.99%
Cost13888
\[\sin \left(\left(\varepsilon + \left(x - x\right)\right) \cdot 0.5\right) \cdot \left(2 \cdot \cos \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right) \]
Alternative 5
Error23.48%
Cost13257
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.5 \cdot 10^{-6} \lor \neg \left(\varepsilon \leq 1.05 \cdot 10^{-5}\right):\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \]
Alternative 6
Error24.06%
Cost6856
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.2 \cdot 10^{-5}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 2.6 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]
Alternative 7
Error44.67%
Cost6464
\[\sin \varepsilon \]
Alternative 8
Error70.36%
Cost64
\[\varepsilon \]

Error

Reproduce?

herbie shell --seed 2023125 
(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)))