Average Error: 36.9 → 0.4
Time: 19.0s
Precision: binary64
Cost: 45504
\[\sin \left(x + \varepsilon\right) - \sin x \]
↓
\[\mathsf{fma}\left(\sin x, \frac{-{\sin \varepsilon}^{2}}{\cos \varepsilon + 1}, \sin \varepsilon \cdot \cos x\right) \]
(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))
↓
(FPCore (x eps)
 :precision binary64
 (fma
  (sin x)
  (/ (- (pow (sin eps) 2.0)) (+ (cos eps) 1.0))
  (* (sin eps) (cos x))))
double code(double x, double eps) {
	return sin((x + eps)) - sin(x);
}
↓
double code(double x, double eps) {
	return fma(sin(x), (-pow(sin(eps), 2.0) / (cos(eps) + 1.0)), (sin(eps) * cos(x)));
}

function code(x, eps)
	return Float64(sin(Float64(x + eps)) - sin(x))
end
↓
function code(x, eps)
	return fma(sin(x), Float64(Float64(-(sin(eps) ^ 2.0)) / Float64(cos(eps) + 1.0)), Float64(sin(eps) * cos(x)))
end

code[x_, eps_] := N[(N[Sin[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]
↓
code[x_, eps_] := N[(N[Sin[x], $MachinePrecision] * N[((-N[Power[N[Sin[eps], $MachinePrecision], 2.0], $MachinePrecision]) / N[(N[Cos[eps], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\sin \left(x + \varepsilon\right) - \sin x
↓
\mathsf{fma}\left(\sin x, \frac{-{\sin \varepsilon}^{2}}{\cos \varepsilon + 1}, \sin \varepsilon \cdot \cos x\right)

Error

Target

Original36.9
Target15.0
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \]

Derivation

  1. Initial program 36.9

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

  2. Applied egg-rr21.6

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

  3. Applied egg-rr0.4

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

  4. Applied egg-rr0.4

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

  5. Applied egg-rr0.4

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

  6. Final simplification0.4

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

Alternatives

Alternative 1
Error0.4
Cost32576
\[\sin \varepsilon \cdot \cos x + \left(\sin x \cdot \cos \varepsilon - \sin x\right) \]
Alternative 2
Error0.4
Cost32448
\[\mathsf{fma}\left(\sin x, \cos \varepsilon + -1, \sin \varepsilon \cdot \cos x\right) \]
Alternative 3
Error0.4
Cost26176
\[\sin \varepsilon \cdot \cos x + \sin x \cdot \left(\cos \varepsilon + -1\right) \]
Alternative 4
Error14.3
Cost26048
\[\sin \varepsilon \cdot \cos x + \left(\sin x - \sin x\right) \]
Alternative 5
Error14.0
Cost20168
\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \cos x\\ t_1 := t_0 - \sin x\\ \mathbf{if}\;\varepsilon \leq -8536231994861.184:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 5.573415221579436 \cdot 10^{-13}:\\ \;\;\;\;t_0 + \varepsilon \cdot \left(\sin x \cdot \left(\varepsilon \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 6
Error14.0
Cost20040
\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \cos x - \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0052071745123544975:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 5.573415221579436 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\sin x, \varepsilon \cdot \left(\varepsilon \cdot -0.5\right), \varepsilon \cdot \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 7
Error14.0
Cost19784
\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \cos x - \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0052071745123544975:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 5.573415221579436 \cdot 10^{-13}:\\ \;\;\;\;\varepsilon \cdot \left(\cos x + \sin x \cdot \left(\varepsilon \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 8
Error14.8
Cost13640
\[\begin{array}{l} t_0 := \sin \varepsilon - \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0052071745123544975:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 5.573415221579436 \cdot 10^{-13}:\\ \;\;\;\;\varepsilon \cdot \left(\cos x + \sin x \cdot \left(\varepsilon \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 9
Error14.9
Cost13256
\[\begin{array}{l} t_0 := \sin \varepsilon - \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0052071745123544975:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 5.573415221579436 \cdot 10^{-13}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 10
Error15.3
Cost6856
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0052071745123544975:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 5.573415221579436 \cdot 10^{-13}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]
Alternative 11
Error28.5
Cost6464
\[\sin \varepsilon \]
Alternative 12
Error45.2
Cost64
\[\varepsilon \]

Error

Reproduce

herbie shell --seed 2022317 
(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)))