Average Error: 36.8 → 0.4
Time: 7.2s
Precision: binary64
\[\sin \left(x + \varepsilon\right) - \sin x \]
↓
\[\mathsf{fma}\left(\sin \varepsilon, \cos x, \frac{{\sin \varepsilon}^{2} \cdot \sin x}{-1 - \cos \varepsilon}\right) \]
(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))
↓
(FPCore (x eps)
 :precision binary64
 (fma
  (sin eps)
  (cos x)
  (/ (* (pow (sin eps) 2.0) (sin x)) (- -1.0 (cos eps)))))
double code(double x, double eps) {
	return sin((x + eps)) - sin(x);
}
↓
double code(double x, double eps) {
	return fma(sin(eps), cos(x), ((pow(sin(eps), 2.0) * sin(x)) / (-1.0 - cos(eps))));
}

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((sin(eps) ^ 2.0) * sin(x)) / Float64(-1.0 - cos(eps))))
end

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

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

Error

Bits error versus x

Bits error versus eps

Target

Original36.8
Target14.6
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \]

Derivation

  1. Initial program 36.8

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

  2. Applied egg-rr22.0

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

  3. Taylor expanded in x around inf 22.0

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

  4. Simplified0.4

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

  5. Applied egg-rr0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2022153 
(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)))