Average Error: 5.8 → 0.1
Time: 11.8s
Precision: 64
\[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3}\]
↓
\[\frac{1 - x}{y \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{3}{3 - x}\right)\right)}\]
\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3}
↓
\frac{1 - x}{y \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{3}{3 - x}\right)\right)}
double f(double x, double y) {
        double r716103 = 1.0;
        double r716104 = x;
        double r716105 = r716103 - r716104;
        double r716106 = 3.0;
        double r716107 = r716106 - r716104;
        double r716108 = r716105 * r716107;
        double r716109 = y;
        double r716110 = r716109 * r716106;
        double r716111 = r716108 / r716110;
        return r716111;
}
↓
double f(double x, double y) {
        double r716112 = 1.0;
        double r716113 = x;
        double r716114 = r716112 - r716113;
        double r716115 = y;
        double r716116 = 3.0;
        double r716117 = r716116 - r716113;
        double r716118 = r716116 / r716117;
        double r716119 = expm1(r716118);
        double r716120 = log1p(r716119);
        double r716121 = r716115 * r716120;
        double r716122 = r716114 / r716121;
        return r716122;
}

Error

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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. Using strategy rm

  3. Applied associate-/l*0.3

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

  4. Simplified0.1

    \[\leadsto \frac{1 - x}{\color{blue}{y \cdot \frac{3}{3 - x}}}\]
  5. Using strategy rm

  6. Applied expm1-log1p-u0.1

    \[\leadsto \frac{1 - x}{y \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{3}{3 - x}\right)\right)}}\]
  7. Using strategy rm

  8. Applied log1p-expm1-u0.1

    \[\leadsto \frac{1 - x}{y \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{3}{3 - x}\right)\right)\right)\right)}}\]

  9. Simplified0.1

    \[\leadsto \frac{1 - x}{y \cdot \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\frac{3}{3 - x}\right)}\right)}\]

  10. Final simplification0.1

    \[\leadsto \frac{1 - x}{y \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{3}{3 - x}\right)\right)}\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:bezierFromSweepQ1 from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (* (/ (- 1 x) y) (/ (- 3 x) 3))

  (/ (* (- 1 x) (- 3 x)) (* y 3)))