Average Error: 14.7 → 0.4
Time: 1.1m
Precision: 64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\sqrt{\frac{8}{3}} \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{\sqrt{\frac{8}{3}}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}\right)\]
\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}
↓
\sqrt{\frac{8}{3}} \cdot \left(\sin \left(x \cdot 0.5\right) \cdot \frac{\sqrt{\frac{8}{3}}}{\frac{\sin x}{\sin \left(x \cdot 0.5\right)}}\right)
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) sqrt(((double) (8.0 / 3.0)))) * ((double) (((double) sin(((double) (x * 0.5)))) * ((double) (((double) sqrt(((double) (8.0 / 3.0)))) / ((double) (((double) sin(x)) / ((double) sin(((double) (x * 0.5))))))))))));
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.7
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 14.7

    \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
  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 add-sqr-sqrt38.9

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

  5. Applied add-sqr-sqrt39.0

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

  6. Applied add-sqr-sqrt39.0

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

  7. Applied unswap-sqr39.0

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

  8. Applied unswap-sqr39.0

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

  9. Applied times-frac31.9

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

  10. Simplified31.8

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

  11. Simplified0.4

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

  13. Applied *-commutative0.4

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

  14. Applied associate-*l*0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2020113 +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)))