Average Error: 36.8 → 1.6
Time: 5.9s
Precision: binary64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\frac{\cos x \cdot \sin \varepsilon + \sin x \cdot \left(\cos \varepsilon + -1\right)}{\sqrt[3]{\sin x \cdot \left(\cos \varepsilon + 1\right) - \cos x \cdot \sin \varepsilon} \cdot \sqrt[3]{\sin x \cdot \left(\cos \varepsilon + 1\right) - \cos x \cdot \sin \varepsilon}} \cdot \frac{\sin x \cdot \left(\cos \varepsilon + 1\right) - \cos x \cdot \sin \varepsilon}{\sqrt[3]{\sin x \cdot \left(\cos \varepsilon + 1\right) - \cos x \cdot \sin \varepsilon}}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\frac{\cos x \cdot \sin \varepsilon + \sin x \cdot \left(\cos \varepsilon + -1\right)}{\sqrt[3]{\sin x \cdot \left(\cos \varepsilon + 1\right) - \cos x \cdot \sin \varepsilon} \cdot \sqrt[3]{\sin x \cdot \left(\cos \varepsilon + 1\right) - \cos x \cdot \sin \varepsilon}} \cdot \frac{\sin x \cdot \left(\cos \varepsilon + 1\right) - \cos x \cdot \sin \varepsilon}{\sqrt[3]{\sin x \cdot \left(\cos \varepsilon + 1\right) - \cos x \cdot \sin \varepsilon}}
(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))
↓
(FPCore (x eps)
 :precision binary64
 (*
  (/
   (+ (* (cos x) (sin eps)) (* (sin x) (+ (cos eps) -1.0)))
   (*
    (cbrt (- (* (sin x) (+ (cos eps) 1.0)) (* (cos x) (sin eps))))
    (cbrt (- (* (sin x) (+ (cos eps) 1.0)) (* (cos x) (sin eps))))))
  (/
   (- (* (sin x) (+ (cos eps) 1.0)) (* (cos x) (sin eps)))
   (cbrt (- (* (sin x) (+ (cos eps) 1.0)) (* (cos x) (sin eps)))))))
double code(double x, double eps) {
	return ((double) (((double) sin(((double) (x + eps)))) - ((double) sin(x))));
}
↓
double code(double x, double eps) {
	return ((double) ((((double) (((double) (((double) cos(x)) * ((double) sin(eps)))) + ((double) (((double) sin(x)) * ((double) (((double) cos(eps)) + -1.0)))))) / ((double) (((double) cbrt(((double) (((double) (((double) sin(x)) * ((double) (((double) cos(eps)) + 1.0)))) - ((double) (((double) cos(x)) * ((double) sin(eps)))))))) * ((double) cbrt(((double) (((double) (((double) sin(x)) * ((double) (((double) cos(eps)) + 1.0)))) - ((double) (((double) cos(x)) * ((double) sin(eps))))))))))) * (((double) (((double) (((double) sin(x)) * ((double) (((double) cos(eps)) + 1.0)))) - ((double) (((double) cos(x)) * ((double) sin(eps)))))) / ((double) cbrt(((double) (((double) (((double) sin(x)) * ((double) (((double) cos(eps)) + 1.0)))) - ((double) (((double) cos(x)) * ((double) sin(eps)))))))))));
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.8
Target14.9
Herbie1.6
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program Error: 36.8 bits

    \[\sin \left(x + \varepsilon\right) - \sin x\]
  2. Using strategy rm

  3. Applied sin-sumError: 21.9 bits

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

  4. Applied associate--l+Error: 21.9 bits

    \[\leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
  5. Using strategy rm

  6. Applied flip-+Error: 24.0 bits

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

  7. SimplifiedError: 6.1 bits

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

  8. SimplifiedError: 6.0 bits

    \[\leadsto \frac{\left(\cos x \cdot \sin \varepsilon + \sin x \cdot \left(\cos \varepsilon + -1\right)\right) \cdot \left(\sin x \cdot \left(\cos \varepsilon + 1\right) - \cos x \cdot \sin \varepsilon\right)}{\color{blue}{\sin x \cdot \left(\cos \varepsilon + 1\right) - \cos x \cdot \sin \varepsilon}}\]
  9. Using strategy rm

  10. Applied add-cube-cbrtError: 6.9 bits

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

  11. Applied times-fracError: 1.6 bits

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

  12. Final simplificationError: 1.6 bits

    \[\leadsto \frac{\cos x \cdot \sin \varepsilon + \sin x \cdot \left(\cos \varepsilon + -1\right)}{\sqrt[3]{\sin x \cdot \left(\cos \varepsilon + 1\right) - \cos x \cdot \sin \varepsilon} \cdot \sqrt[3]{\sin x \cdot \left(\cos \varepsilon + 1\right) - \cos x \cdot \sin \varepsilon}} \cdot \frac{\sin x \cdot \left(\cos \varepsilon + 1\right) - \cos x \cdot \sin \varepsilon}{\sqrt[3]{\sin x \cdot \left(\cos \varepsilon + 1\right) - \cos x \cdot \sin \varepsilon}}\]

Reproduce

herbie shell --seed 2020204 
(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)))