Average Error: 14.7 → 0.3
Time: 5.2s
Precision: binary64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\left(8 \cdot \frac{\sin \left(0.5 \cdot x\right)}{3}\right) \cdot \frac{\sin \left(0.5 \cdot x\right)}{\sin x}\]
\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}
↓
\left(8 \cdot \frac{\sin \left(0.5 \cdot x\right)}{3}\right) \cdot \frac{\sin \left(0.5 \cdot x\right)}{\sin x}
double code(double x) {
	return ((double) (((double) (((double) (((double) (8.0 / 3.0)) * ((double) sin(((double) (x * 0.5)))))) * ((double) sin(((double) (x * 0.5)))))) / ((double) sin(x))));
}
↓
double code(double x) {
	return ((double) (((double) (8.0 * ((double) (((double) sin(((double) (0.5 * x)))) / 3.0)))) * ((double) (((double) sin(((double) (0.5 * x)))) / ((double) sin(x))))));
}

Error

Bits error versus x

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Results

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Target

Original14.7
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 14.7

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

  3. Applied *-un-lft-identity14.7

    \[\leadsto \frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\color{blue}{1 \cdot \sin x}}\]

  4. Applied times-frac0.5

    \[\leadsto \color{blue}{\frac{\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)}{1} \cdot \frac{\sin \left(x \cdot 0.5\right)}{\sin x}}\]

  5. Simplified0.5

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

  6. Simplified0.5

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

  8. Applied div-inv0.5

    \[\leadsto \left(\color{blue}{\left(8 \cdot \frac{1}{3}\right)} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \frac{\sin \left(0.5 \cdot x\right)}{\sin x}\]

  9. Applied associate-*l*0.5

    \[\leadsto \color{blue}{\left(8 \cdot \left(\frac{1}{3} \cdot \sin \left(x \cdot 0.5\right)\right)\right)} \cdot \frac{\sin \left(0.5 \cdot x\right)}{\sin x}\]

  10. Simplified0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2020150 
(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)))