Average Error: 36.9 → 0.4
Time: 13.5s
Precision: binary64
Cost: 45248
\[\sin \left(x + \varepsilon\right) - \sin x \]
↓
\[\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \log \left(e^{\cos \varepsilon + -1}\right)\right) \]
(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))
↓
(FPCore (x eps)
 :precision binary64
 (fma (sin eps) (cos x) (* (sin x) (log (exp (+ (cos eps) -1.0))))))
double code(double x, double eps) {
	return sin((x + eps)) - sin(x);
}
↓
double code(double x, double eps) {
	return fma(sin(eps), cos(x), (sin(x) * log(exp((cos(eps) + -1.0)))));
}

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

\sin \left(x + \varepsilon\right) - \sin x
↓
\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \log \left(e^{\cos \varepsilon + -1}\right)\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-rr21.8

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

  3. Simplified0.4

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

    [Start]21.8

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

    associate-+r+ [=>]0.4

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

    +-commutative [<=]0.4

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

    *-commutative [=>]0.4

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

    fma-def [=>]0.4

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

    *-commutative [=>]0.4

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

    neg-mul-1 [=>]0.4

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

    distribute-rgt-out [=>]0.4

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

  4. Applied egg-rr0.4

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

  5. Final simplification0.4

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

Alternatives

Alternative 1
Error0.4
Cost32448
\[\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \left(\cos \varepsilon + -1\right)\right) \]
Alternative 2
Error0.4
Cost26176
\[\sin x \cdot \left(\cos \varepsilon + -1\right) + \sin \varepsilon \cdot \cos x \]
Alternative 3
Error15.0
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 4
Error14.5
Cost13769
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00085 \lor \neg \left(\varepsilon \leq 0.00145\right):\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\right) + \varepsilon \cdot \cos x\\ \end{array} \]
Alternative 5
Error14.5
Cost13641
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00245 \lor \neg \left(\varepsilon \leq 0.00029\right):\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\cos x + -0.5 \cdot \left(\varepsilon \cdot \sin x\right)\right)\\ \end{array} \]
Alternative 6
Error14.6
Cost13257
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00175 \lor \neg \left(\varepsilon \leq 0.0021\right):\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \]
Alternative 7
Error15.2
Cost6856
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3000:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.15 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]
Alternative 8
Error28.7
Cost6464
\[\sin \varepsilon \]
Alternative 9
Error44.9
Cost64
\[\varepsilon \]

Error

Reproduce

herbie shell --seed 2023016 
(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)))