Average Error: 5.8 → 0.1
Time: 2.7s
Precision: binary64
\[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
↓
\[\frac{\frac{1 - x}{y}}{\frac{3}{3 - x}} \]
\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3}
↓
\frac{\frac{1 - x}{y}}{\frac{3}{3 - x}}

(FPCore (x y) :precision binary64 (/ (* (- 1.0 x) (- 3.0 x)) (* y 3.0)))
↓
(FPCore (x y) :precision binary64 (/ (/ (- 1.0 x) y) (/ 3.0 (- 3.0 x))))
double code(double x, double y) {
	return ((1.0 - x) * (3.0 - x)) / (y * 3.0);
}
↓
double code(double x, double y) {
	return ((1.0 - x) / y) / (3.0 / (3.0 - x));
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original5.8
Target0.1
Herbie0.1
\[\frac{1 - x}{y} \cdot \frac{3 - x}{3} \]

Derivation

  1. Initial program 5.8

    \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]

  2. Applied associate-/l*_binary640.3

    \[\leadsto \color{blue}{\frac{1 - x}{\frac{y \cdot 3}{3 - x}}} \]

  3. Applied *-un-lft-identity_binary640.3

    \[\leadsto \frac{1 - x}{\frac{y \cdot 3}{\color{blue}{1 \cdot \left(3 - x\right)}}} \]

  4. Applied times-frac_binary640.1

    \[\leadsto \frac{1 - x}{\color{blue}{\frac{y}{1} \cdot \frac{3}{3 - x}}} \]

  5. Applied associate-/r*_binary640.1

    \[\leadsto \color{blue}{\frac{\frac{1 - x}{\frac{y}{1}}}{\frac{3}{3 - x}}} \]

  6. Simplified0.1

    \[\leadsto \frac{\color{blue}{\frac{1 - x}{y}}}{\frac{3}{3 - x}} \]

  7. Final simplification0.1

    \[\leadsto \frac{\frac{1 - x}{y}}{\frac{3}{3 - x}} \]

Reproduce

herbie shell --seed 2021274 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:bezierFromSweepQ1 from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (* (/ (- 1.0 x) y) (/ (- 3.0 x) 3.0))

  (/ (* (- 1.0 x) (- 3.0 x)) (* y 3.0)))