Average Error: 37.3 → 0.4
Time: 6.9s
Precision: binary64
\[\sin \left(x + \varepsilon\right) - \sin x \]
↓
\[\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \frac{1}{\frac{1 + \cos \varepsilon}{-{\sin \varepsilon}^{2}}}\right) \]
(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))
↓
(FPCore (x eps)
 :precision binary64
 (fma
  (sin eps)
  (cos x)
  (* (sin x) (/ 1.0 (/ (+ 1.0 (cos eps)) (- (pow (sin eps) 2.0)))))))
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) * (1.0 / ((1.0 + cos(eps)) / -pow(sin(eps), 2.0)))));
}

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) * Float64(1.0 / Float64(Float64(1.0 + cos(eps)) / Float64(-(sin(eps) ^ 2.0))))))
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[(1.0 / N[(N[(1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision] / (-N[Power[N[Sin[eps], $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

Bits error versus x

Bits error versus eps

Target

Original37.3
Target14.7
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-rr22.5

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

  3. Taylor expanded in x around inf 22.5

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

  4. Simplified0.4

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

  5. Applied egg-rr0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2022166 
(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)))