Average Error: 15.4 → 0.4
Time: 5.4s
Precision: binary64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[8 \cdot \left(\frac{\sin \left(x \cdot 0.5\right)}{3} \cdot \log \left(e^{\frac{\sin \left(x \cdot 0.5\right)}{\sin x}}\right)\right)\]
\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}
↓
8 \cdot \left(\frac{\sin \left(x \cdot 0.5\right)}{3} \cdot \log \left(e^{\frac{\sin \left(x \cdot 0.5\right)}{\sin x}}\right)\right)
double code(double x) {
	return (((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) (8.0 * ((double) ((((double) sin(((double) (x * 0.5)))) / 3.0) * ((double) log(((double) exp((((double) sin(((double) (x * 0.5)))) / ((double) sin(x)))))))))));
}

Error

Bits error versus x

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Results

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Target

Original15.4
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 15.4

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

  2. Simplified0.5

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

  4. Applied *-un-lft-identity0.5

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

  5. Applied times-frac0.5

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

  6. Applied associate-*r*0.5

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

  7. Simplified0.3

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

  9. Applied add-log-exp0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2020196 
(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)))