Average Error: 14.8 → 0.4
Time: 5.4s
Precision: 64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\frac{8 \cdot \sin \left(x \cdot 0.5\right)}{\sqrt[3]{{\left(\frac{\sin x}{\sin \left(0.5 \cdot x\right)}\right)}^{3}} \cdot 3}\]
\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{8 \cdot \sin \left(x \cdot 0.5\right)}{\sqrt[3]{{\left(\frac{\sin x}{\sin \left(0.5 \cdot x\right)}\right)}^{3}} \cdot 3}
double code(double x) {
	return ((((8.0 / 3.0) * sin((x * 0.5))) * sin((x * 0.5))) / sin(x));
}
↓
double code(double x) {
	return ((8.0 * sin((x * 0.5))) / (cbrt(pow((sin(x) / sin((0.5 * x))), 3.0)) * 3.0));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.8
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 14.8

    \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
  2. Using strategy rm

  3. Applied associate-/l*0.5

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

  4. Simplified0.5

    \[\leadsto \frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{\color{blue}{\frac{\sin x}{\sin \left(0.5 \cdot x\right)}}}\]
  5. Using strategy rm

  6. Applied associate-*l/0.3

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

  7. Applied associate-/l/0.3

    \[\leadsto \color{blue}{\frac{8 \cdot \sin \left(x \cdot 0.5\right)}{\frac{\sin x}{\sin \left(0.5 \cdot x\right)} \cdot 3}}\]
  8. Using strategy rm

  9. Applied add-cbrt-cube20.0

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

  10. Applied add-cbrt-cube21.3

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

  11. Applied cbrt-undiv21.2

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

  12. Simplified0.4

    \[\leadsto \frac{8 \cdot \sin \left(x \cdot 0.5\right)}{\sqrt[3]{\color{blue}{{\left(\frac{\sin x}{\sin \left(0.5 \cdot x\right)}\right)}^{3}}} \cdot 3}\]

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (x)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, A"
  :precision binary64

  :herbie-target
  (/ (/ (* 8 (sin (* x 0.5))) 3) (/ (sin x) (sin (* x 0.5))))

  (/ (* (* (/ 8 3) (sin (* x 0.5))) (sin (* x 0.5))) (sin x)))