Average Error: 1.4 → 0.8
Time: 15.7s
Precision: 64
\[x + y \cdot \frac{z - t}{z - a}\]
↓
\[x + \frac{\frac{\frac{z - t}{\frac{\sqrt[3]{z - a}}{\sqrt[3]{y}}}}{\frac{\sqrt[3]{z - a}}{\sqrt[3]{y}}}}{\frac{\sqrt[3]{z - a}}{\sqrt[3]{y}}}\]
x + y \cdot \frac{z - t}{z - a}
↓
x + \frac{\frac{\frac{z - t}{\frac{\sqrt[3]{z - a}}{\sqrt[3]{y}}}}{\frac{\sqrt[3]{z - a}}{\sqrt[3]{y}}}}{\frac{\sqrt[3]{z - a}}{\sqrt[3]{y}}}
double f(double x, double y, double z, double t, double a) {
        double r488055 = x;
        double r488056 = y;
        double r488057 = z;
        double r488058 = t;
        double r488059 = r488057 - r488058;
        double r488060 = a;
        double r488061 = r488057 - r488060;
        double r488062 = r488059 / r488061;
        double r488063 = r488056 * r488062;
        double r488064 = r488055 + r488063;
        return r488064;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r488065 = x;
        double r488066 = z;
        double r488067 = t;
        double r488068 = r488066 - r488067;
        double r488069 = a;
        double r488070 = r488066 - r488069;
        double r488071 = cbrt(r488070);
        double r488072 = y;
        double r488073 = cbrt(r488072);
        double r488074 = r488071 / r488073;
        double r488075 = r488068 / r488074;
        double r488076 = r488075 / r488074;
        double r488077 = r488076 / r488074;
        double r488078 = r488065 + r488077;
        return r488078;
}

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
Target1.4
Herbie0.8
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Initial program 1.4

    \[x + y \cdot \frac{z - t}{z - a}\]

  2. Simplified3.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(z - t, \frac{y}{z - a}, x\right)}\]
  3. Using strategy rm

  4. Applied fma-udef3.1

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

  5. Simplified3.3

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

  7. Applied add-cube-cbrt3.7

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

  8. Applied add-cube-cbrt3.8

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

  9. Applied times-frac3.8

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

  10. Applied associate-/r*1.1

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

  11. Simplified0.8

    \[\leadsto \frac{\color{blue}{\frac{\frac{z - t}{\frac{\sqrt[3]{z - a}}{\sqrt[3]{y}}}}{\frac{\sqrt[3]{z - a}}{\sqrt[3]{y}}}}}{\frac{\sqrt[3]{z - a}}{\sqrt[3]{y}}} + x\]

  12. Final simplification0.8

    \[\leadsto x + \frac{\frac{\frac{z - t}{\frac{\sqrt[3]{z - a}}{\sqrt[3]{y}}}}{\frac{\sqrt[3]{z - a}}{\sqrt[3]{y}}}}{\frac{\sqrt[3]{z - a}}{\sqrt[3]{y}}}\]

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A"

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

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