Average Error: 37.0 → 0.4
Time: 8.8s
Precision: binary64
\[\sin \left(x + \varepsilon\right) - \sin x \]
↓
\[\mathsf{fma}\left(\cos x, \sin \varepsilon, \frac{{\sin \varepsilon}^{2} \cdot \left(-\sin x\right)}{\frac{1 + {\cos \varepsilon}^{3}}{1 + \left({\cos \varepsilon}^{2} - \cos \varepsilon\right)}}\right) \]
(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))
↓
(FPCore (x eps)
 :precision binary64
 (fma
  (cos x)
  (sin eps)
  (/
   (* (pow (sin eps) 2.0) (- (sin x)))
   (/ (+ 1.0 (pow (cos eps) 3.0)) (+ 1.0 (- (pow (cos eps) 2.0) (cos eps)))))))
double code(double x, double eps) {
	return sin((x + eps)) - sin(x);
}
↓
double code(double x, double eps) {
	return fma(cos(x), sin(eps), ((pow(sin(eps), 2.0) * -sin(x)) / ((1.0 + pow(cos(eps), 3.0)) / (1.0 + (pow(cos(eps), 2.0) - cos(eps))))));
}

function code(x, eps)
	return Float64(sin(Float64(x + eps)) - sin(x))
end
↓
function code(x, eps)
	return fma(cos(x), sin(eps), Float64(Float64((sin(eps) ^ 2.0) * Float64(-sin(x))) / Float64(Float64(1.0 + (cos(eps) ^ 3.0)) / Float64(1.0 + Float64((cos(eps) ^ 2.0) - cos(eps))))))
end

code[x_, eps_] := N[(N[Sin[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]
↓
code[x_, eps_] := N[(N[Cos[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision] + N[(N[(N[Power[N[Sin[eps], $MachinePrecision], 2.0], $MachinePrecision] * (-N[Sin[x], $MachinePrecision])), $MachinePrecision] / N[(N[(1.0 + N[Power[N[Cos[eps], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[N[Cos[eps], $MachinePrecision], 2.0], $MachinePrecision] - N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\sin \left(x + \varepsilon\right) - \sin x
↓
\mathsf{fma}\left(\cos x, \sin \varepsilon, \frac{{\sin \varepsilon}^{2} \cdot \left(-\sin x\right)}{\frac{1 + {\cos \varepsilon}^{3}}{1 + \left({\cos \varepsilon}^{2} - \cos \varepsilon\right)}}\right)

Error

Target

Original37.0
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 37.0

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

  2. Applied egg-rr21.8

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

  3. Taylor expanded in eps around inf 21.8

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

  4. Simplified0.4

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

  5. Applied egg-rr0.4

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

  6. Applied egg-rr0.4

    \[\leadsto \mathsf{fma}\left(\cos x, \sin \varepsilon, \frac{\left(-{\sin \varepsilon}^{2}\right) \cdot \sin x}{\color{blue}{\frac{1 + {\cos \varepsilon}^{3}}{1 + \left({\cos \varepsilon}^{2} - \cos \varepsilon\right)}}}\right) \]

  7. Final simplification0.4

    \[\leadsto \mathsf{fma}\left(\cos x, \sin \varepsilon, \frac{{\sin \varepsilon}^{2} \cdot \left(-\sin x\right)}{\frac{1 + {\cos \varepsilon}^{3}}{1 + \left({\cos \varepsilon}^{2} - \cos \varepsilon\right)}}\right) \]

Reproduce

herbie shell --seed 2022210 
(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)))