Average Error: 36.7 → 0.2
Time: 11.8s
Precision: binary64
Cost: 39040
\[\sin \left(x + \varepsilon\right) - \sin x \]
↓
\[\mathsf{fma}\left(\sin \varepsilon, \cos x, \left(\tan \left(\frac{\varepsilon}{2}\right) \cdot \left(-\sin \varepsilon\right)\right) \cdot \sin x\right) \]
(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))
↓
(FPCore (x eps)
 :precision binary64
 (fma (sin eps) (cos x) (* (* (tan (/ eps 2.0)) (- (sin eps))) (sin x))))
double code(double x, double eps) {
	return sin((x + eps)) - sin(x);
}
↓
double code(double x, double eps) {
	return fma(sin(eps), cos(x), ((tan((eps / 2.0)) * -sin(eps)) * sin(x)));
}

function code(x, eps)
	return Float64(sin(Float64(x + eps)) - sin(x))
end
↓
function code(x, eps)
	return fma(sin(eps), cos(x), Float64(Float64(tan(Float64(eps / 2.0)) * Float64(-sin(eps))) * sin(x)))
end

code[x_, eps_] := N[(N[Sin[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]
↓
code[x_, eps_] := N[(N[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(N[Tan[N[(eps / 2.0), $MachinePrecision]], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision])), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\sin \left(x + \varepsilon\right) - \sin x
↓
\mathsf{fma}\left(\sin \varepsilon, \cos x, \left(\tan \left(\frac{\varepsilon}{2}\right) \cdot \left(-\sin \varepsilon\right)\right) \cdot \sin x\right)

Error

Target

Original36.7
Target14.4
Herbie0.2
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \]

Derivation

  1. Initial program 36.7

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

  2. Applied egg-rr22.1

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

  3. Taylor expanded in x around inf 22.1

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

  4. Simplified0.4

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

  5. Applied egg-rr0.3

    \[\leadsto \mathsf{fma}\left(\sin \varepsilon, \cos x, \color{blue}{\left(\left(-{\sin \varepsilon}^{2}\right) \cdot \frac{1}{\cos \varepsilon + 1}\right)} \cdot \sin x\right) \]

  6. Taylor expanded in eps around inf 0.3

    \[\leadsto \mathsf{fma}\left(\sin \varepsilon, \cos x, \color{blue}{\left(-1 \cdot \frac{{\sin \varepsilon}^{2}}{1 + \cos \varepsilon}\right)} \cdot \sin x\right) \]

  7. Simplified0.2

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

  8. Final simplification0.2

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

Alternatives

Alternative 1
Error0.4
Cost32448
\[\mathsf{fma}\left(\sin x, \cos \varepsilon + -1, \sin \varepsilon \cdot \cos x\right) \]
Alternative 2
Error0.4
Cost32448
\[\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \left(\cos \varepsilon + -1\right)\right) \]
Alternative 3
Error13.6
Cost26440
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -58520974586.495186:\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{elif}\;\varepsilon \leq 1096.8183850469568:\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, \cos x, \left(\varepsilon \cdot \sin x\right) \cdot \left(\varepsilon \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x + \left(\sin \varepsilon \cdot \cos x - \sin x\right)\\ \end{array} \]
Alternative 4
Error14.0
Cost26312
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -58520974586.495186:\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{elif}\;\varepsilon \leq 2.0808353193995894 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(\cos x + \sin x \cdot \left(\varepsilon \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x + \left(\sin \varepsilon \cdot \cos x - \sin x\right)\\ \end{array} \]
Alternative 5
Error0.4
Cost26176
\[\sin x \cdot \left(\cos \varepsilon + -1\right) + \sin \varepsilon \cdot \cos x \]
Alternative 6
Error14.3
Cost13640
\[\begin{array}{l} t_0 := \sin \varepsilon - \sin x\\ \mathbf{if}\;\varepsilon \leq -58520974586.495186:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 2.0808353193995894 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(\cos x + \sin x \cdot \left(\varepsilon \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 7
Error14.1
Cost13256
\[\begin{array}{l} t_0 := \sin \varepsilon - \sin x\\ \mathbf{if}\;\varepsilon \leq -1.3915928499252286:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 2.0808353193995894 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 8
Error14.5
Cost6856
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.3915928499252286:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 2.0808353193995894 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]
Alternative 9
Error28.7
Cost6464
\[\sin \varepsilon \]
Alternative 10
Error45.8
Cost64
\[\varepsilon \]

Error

Reproduce

herbie shell --seed 2022217 
(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)))