Average Error: 15.0 → 0.3
Time: 1.3min
Precision: binary64
\[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}\]
↓
\[\left(8 \cdot \frac{\sin \left(0.5 \cdot x\right)}{\sin x}\right) \cdot \frac{\sin \left(0.5 \cdot x\right)}{3}\]
\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x}
↓
\left(8 \cdot \frac{\sin \left(0.5 \cdot x\right)}{\sin x}\right) \cdot \frac{\sin \left(0.5 \cdot x\right)}{3}
(FPCore (x)
 :precision binary64
 (/ (* (* (/ 8.0 3.0) (sin (* x 0.5))) (sin (* x 0.5))) (sin x)))
↓
(FPCore (x)
 :precision binary64
 (* (* 8.0 (/ (sin (* 0.5 x)) (sin x))) (/ (sin (* 0.5 x)) 3.0)))
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(0.5 * x) / sin(x))) * (sin(0.5 * x) / 3.0);
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.0
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 15.0

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

  2. Simplified0.3

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

  4. Applied associate-/r/_binary64_266220.3

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

  5. Applied *-un-lft-identity_binary64_266760.3

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

  6. Applied times-frac_binary64_266820.3

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

  7. Simplified0.3

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

  8. Taylor expanded around inf 0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2021176 
(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)))