Average Error: 14.4 → 0.4
Time: 5.4s
Precision: binary64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\frac{\sin \left(x \cdot 0.5\right)}{\sqrt[3]{{\left(\frac{\sin x}{\sin \left(x \cdot 0.5\right)}\right)}^{3} \cdot 0.052734375}}\]
\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)}{\sqrt[3]{{\left(\frac{\sin x}{\sin \left(x \cdot 0.5\right)}\right)}^{3} \cdot 0.052734375}}
(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))
  (cbrt (* (pow (/ (sin x) (sin (* x 0.5))) 3.0) 0.052734375))))
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) / cbrt(pow((sin(x) / sin(x * 0.5)), 3.0) * 0.052734375);
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

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 associate-/l*_binary640.5

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

  5. Simplified0.5

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

  7. Applied add-cbrt-cube_binary640.4

    \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\color{blue}{\sqrt[3]{\left(\frac{\sin x}{\sin \left(x \cdot 0.5\right) \cdot 2.6666666666666665} \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right) \cdot 2.6666666666666665}\right) \cdot \frac{\sin x}{\sin \left(x \cdot 0.5\right) \cdot 2.6666666666666665}}}}\]

  8. Simplified0.4

    \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\sqrt[3]{\color{blue}{{\left(\frac{\sin x}{\sin \left(x \cdot 0.5\right)}\right)}^{3} \cdot 0.052734375}}}\]

  9. Final simplification0.4

    \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\sqrt[3]{{\left(\frac{\sin x}{\sin \left(x \cdot 0.5\right)}\right)}^{3} \cdot 0.052734375}}\]

Reproduce

herbie shell --seed 2020260 
(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)))