Average Error: 10.9 → 1.0
Time: 25.9s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{z - a}\]
↓
\[x + \left(\left(\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}\right) \cdot \frac{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z - a}}}{\sqrt[3]{z - a}}\right) \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z - a}} \cdot \sqrt[3]{z - t}\right)\]
x + \frac{y \cdot \left(z - t\right)}{z - a}
↓
x + \left(\left(\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}\right) \cdot \frac{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z - a}}}{\sqrt[3]{z - a}}\right) \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z - a}} \cdot \sqrt[3]{z - t}\right)
double f(double x, double y, double z, double t, double a) {
        double r489114 = x;
        double r489115 = y;
        double r489116 = z;
        double r489117 = t;
        double r489118 = r489116 - r489117;
        double r489119 = r489115 * r489118;
        double r489120 = a;
        double r489121 = r489116 - r489120;
        double r489122 = r489119 / r489121;
        double r489123 = r489114 + r489122;
        return r489123;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r489124 = x;
        double r489125 = z;
        double r489126 = t;
        double r489127 = r489125 - r489126;
        double r489128 = cbrt(r489127);
        double r489129 = r489128 * r489128;
        double r489130 = y;
        double r489131 = cbrt(r489130);
        double r489132 = r489131 * r489131;
        double r489133 = a;
        double r489134 = r489125 - r489133;
        double r489135 = cbrt(r489134);
        double r489136 = r489132 / r489135;
        double r489137 = r489136 / r489135;
        double r489138 = r489129 * r489137;
        double r489139 = r489131 / r489135;
        double r489140 = r489139 * r489128;
        double r489141 = r489138 * r489140;
        double r489142 = r489124 + r489141;
        return r489142;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.9
Target1.4
Herbie1.0
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Initial program 10.9

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

  2. Simplified1.4

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

  4. Applied fma-udef1.4

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

  5. Simplified1.4

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

  7. Applied add-cube-cbrt1.9

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

  8. Applied add-cube-cbrt1.7

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

  9. Applied times-frac1.7

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

  10. Applied add-cube-cbrt2.0

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

  11. Applied times-frac0.9

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

  12. Simplified1.0

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

  13. Simplified1.0

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

  14. Final simplification1.0

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

Reproduce

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

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

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