Average Error: 14.4 → 0.3
Time: 5.3s
Precision: binary64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\frac{\sin \left(x \cdot 0.5\right)}{\sin x} \cdot \frac{\sin \left(x \cdot 0.5\right)}{0.375}\]
\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}
↓
\frac{\sin \left(x \cdot 0.5\right)}{\sin x} \cdot \frac{\sin \left(x \cdot 0.5\right)}{0.375}
(FPCore (x)
 :precision binary64
 (/ (* (* (/ 8.0 3.0) (sin (* x 0.5))) (sin (* x 0.5))) (sin x)))
↓
(FPCore (x)
 :precision binary64
 (* (/ (sin (* x 0.5)) (sin x)) (/ (sin (* x 0.5)) 0.375)))
double code(double x) {
	return (((8.0 / 3.0) * sin(x * 0.5)) * sin(x * 0.5)) / sin(x);
}
↓
double code(double x) {
	return (sin(x * 0.5) / sin(x)) * (sin(x * 0.5) / 0.375);
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.4
Target0.3
Herbie0.3
\[\frac{\frac{8 \cdot \sin \left(x \cdot 0.5\right)}{3}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}\]

Derivation

  1. Initial program 14.4

    \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]

  2. Simplified14.4

    \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot \left(2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)\right)}{\sin x}}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity_binary6414.4

    \[\leadsto \frac{\sin \left(x \cdot 0.5\right) \cdot \left(2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)\right)}{\color{blue}{1 \cdot \sin x}}\]

  5. Applied times-frac_binary640.5

    \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{1} \cdot \frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\sin x}}\]

  6. Simplified0.5

    \[\leadsto \color{blue}{\sin \left(x \cdot 0.5\right)} \cdot \frac{2.6666666666666665 \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]

  7. Simplified0.5

    \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \color{blue}{\frac{\sin \left(x \cdot 0.5\right) \cdot 2.6666666666666665}{\sin x}}\]
  8. Using strategy rm

  9. Applied associate-/l*_binary640.5

    \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\frac{\sin x}{2.6666666666666665}}}\]
  10. Using strategy rm

  11. Applied div-inv_binary640.5

    \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{\sin x \cdot \frac{1}{2.6666666666666665}}}\]

  12. Applied *-un-lft-identity_binary640.5

    \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \frac{\color{blue}{1 \cdot \sin \left(x \cdot 0.5\right)}}{\sin x \cdot \frac{1}{2.6666666666666665}}\]

  13. Applied times-frac_binary640.5

    \[\leadsto \sin \left(x \cdot 0.5\right) \cdot \color{blue}{\left(\frac{1}{\sin x} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\frac{1}{2.6666666666666665}}\right)}\]

  14. Applied associate-*r*_binary640.4

    \[\leadsto \color{blue}{\left(\sin \left(x \cdot 0.5\right) \cdot \frac{1}{\sin x}\right) \cdot \frac{\sin \left(x \cdot 0.5\right)}{\frac{1}{2.6666666666666665}}}\]

  15. Simplified0.3

    \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\sin x}} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\frac{1}{2.6666666666666665}}\]

  16. Final simplification0.3

    \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\sin x} \cdot \frac{\sin \left(x \cdot 0.5\right)}{0.375}\]

Reproduce

herbie shell --seed 2020219 
(FPCore (x)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, A"
  :precision binary64

  :herbie-target
  (/ (/ (* 8.0 (sin (* x 0.5))) 3.0) (/ (sin x) (sin (* x 0.5))))

  (/ (* (* (/ 8.0 3.0) (sin (* x 0.5))) (sin (* x 0.5))) (sin x)))