Average Error: 37.3 → 0.4
Time: 7.2s
Precision: binary64
\[\sin \left(x + \varepsilon\right) - \sin x \]
↓
\[\mathsf{fma}\left(\log \left(e^{\cos \varepsilon + -1}\right), \sin x, \cos x \cdot \sin \varepsilon\right) \]
(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))
↓
(FPCore (x eps)
 :precision binary64
 (fma (log (exp (+ (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(log(exp((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(log(exp(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[Log[N[Exp[N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]], $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(\log \left(e^{\cos \varepsilon + -1}\right), \sin x, \cos x \cdot \sin \varepsilon\right)

Error

Bits error versus x

Bits error versus eps

Target

Original37.3
Target14.4
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \]

Derivation

  1. Initial program 37.3

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

  2. Applied egg-rr22.8

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

  3. Taylor expanded in x around inf 22.8

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

  4. Simplified0.4

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

  5. Taylor expanded in eps around inf 22.8

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

  6. Simplified0.4

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

  7. Applied egg-rr0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2022150 
(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)))