Average Error: 1.4 → 1.1
Time: 7.5s
Precision: 64
\[x + y \cdot \frac{z - t}{z - a}\]
↓
\[x + \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z - a}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z - a}} \cdot \frac{z - t}{\sqrt[3]{z - a}}\right)\]
x + y \cdot \frac{z - t}{z - a}
↓
x + \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z - a}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z - a}} \cdot \frac{z - t}{\sqrt[3]{z - a}}\right)
double f(double x, double y, double z, double t, double a) {
        double r833148 = x;
        double r833149 = y;
        double r833150 = z;
        double r833151 = t;
        double r833152 = r833150 - r833151;
        double r833153 = a;
        double r833154 = r833150 - r833153;
        double r833155 = r833152 / r833154;
        double r833156 = r833149 * r833155;
        double r833157 = r833148 + r833156;
        return r833157;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r833158 = x;
        double r833159 = y;
        double r833160 = cbrt(r833159);
        double r833161 = r833160 * r833160;
        double r833162 = z;
        double r833163 = a;
        double r833164 = r833162 - r833163;
        double r833165 = cbrt(r833164);
        double r833166 = r833161 / r833165;
        double r833167 = r833160 / r833165;
        double r833168 = t;
        double r833169 = r833162 - r833168;
        double r833170 = r833169 / r833165;
        double r833171 = r833167 * r833170;
        double r833172 = r833166 * r833171;
        double r833173 = r833158 + r833172;
        return r833173;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

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Results

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Target

Original1.4
Target1.3
Herbie1.1
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Initial program 1.4

    \[x + y \cdot \frac{z - t}{z - a}\]
  2. Using strategy rm

  3. Applied add-cube-cbrt1.9

    \[\leadsto x + y \cdot \frac{z - t}{\color{blue}{\left(\sqrt[3]{z - a} \cdot \sqrt[3]{z - a}\right) \cdot \sqrt[3]{z - a}}}\]

  4. Applied *-un-lft-identity1.9

    \[\leadsto x + y \cdot \frac{\color{blue}{1 \cdot \left(z - t\right)}}{\left(\sqrt[3]{z - a} \cdot \sqrt[3]{z - a}\right) \cdot \sqrt[3]{z - a}}\]

  5. Applied times-frac1.9

    \[\leadsto x + y \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{z - a} \cdot \sqrt[3]{z - a}} \cdot \frac{z - t}{\sqrt[3]{z - a}}\right)}\]

  6. Applied associate-*r*2.3

    \[\leadsto x + \color{blue}{\left(y \cdot \frac{1}{\sqrt[3]{z - a} \cdot \sqrt[3]{z - a}}\right) \cdot \frac{z - t}{\sqrt[3]{z - a}}}\]

  7. Simplified2.3

    \[\leadsto x + \color{blue}{\frac{y}{\sqrt[3]{z - a} \cdot \sqrt[3]{z - a}}} \cdot \frac{z - t}{\sqrt[3]{z - a}}\]
  8. Using strategy rm

  9. Applied add-cube-cbrt2.4

    \[\leadsto x + \frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{\sqrt[3]{z - a} \cdot \sqrt[3]{z - a}} \cdot \frac{z - t}{\sqrt[3]{z - a}}\]

  10. Applied times-frac2.4

    \[\leadsto x + \color{blue}{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z - a}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z - a}}\right)} \cdot \frac{z - t}{\sqrt[3]{z - a}}\]

  11. Applied associate-*l*1.1

    \[\leadsto x + \color{blue}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z - a}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z - a}} \cdot \frac{z - t}{\sqrt[3]{z - a}}\right)}\]

  12. Final simplification1.1

    \[\leadsto x + \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z - a}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z - a}} \cdot \frac{z - t}{\sqrt[3]{z - a}}\right)\]

Reproduce

herbie shell --seed 2020035 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (+ x (/ y (/ (- z a) (- z t))))

  (+ x (* y (/ (- z t) (- z a)))))