Average Error: 37.3 → 0.4
Time: 9.8s
Precision: binary64
\[\sin \left(x + \varepsilon\right) - \sin x \]
↓
\[\mathsf{fma}\left(\cos x, \sin \varepsilon, \frac{\sin x \cdot \mathsf{fma}\left(\cos \varepsilon, \cos \varepsilon, -1\right)}{\cos \varepsilon + 1}\right) \]
(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))
↓
(FPCore (x eps)
 :precision binary64
 (fma
  (cos x)
  (sin eps)
  (/ (* (sin x) (fma (cos eps) (cos eps) -1.0)) (+ (cos eps) 1.0))))
double code(double x, double eps) {
	return sin((x + eps)) - sin(x);
}
↓
double code(double x, double eps) {
	return fma(cos(x), sin(eps), ((sin(x) * fma(cos(eps), cos(eps), -1.0)) / (cos(eps) + 1.0)));
}

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(x) * fma(cos(eps), cos(eps), -1.0)) / 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[Cos[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision] + N[(N[(N[Sin[x], $MachinePrecision] * N[(N[Cos[eps], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[eps], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

Bits error versus x

Bits error versus eps

Target

Original37.3
Target15.4
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 sin-sum_binary6421.8

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

  3. Applied associate--l+_binary6421.8

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

  4. Applied add-cube-cbrt_binary6422.5

    \[\leadsto \color{blue}{\left(\sqrt[3]{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)} \cdot \sqrt[3]{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\right) \cdot \sqrt[3]{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}} \]

  5. Simplified22.4

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

  6. Simplified1.6

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

  7. Applied pow1_binary641.6

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

  8. Applied pow1_binary641.6

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

  9. Applied pow1_binary641.6

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

  10. Applied pow-prod-down_binary641.6

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

  11. Applied pow-prod-down_binary641.6

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

  12. Simplified0.4

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

  13. Applied flip--_binary640.5

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

  14. Applied associate-*l/_binary640.5

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

  15. Simplified0.4

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

  16. Final simplification0.4

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

Reproduce

herbie shell --seed 2022131 
(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)))