Average Error: 10.5 → 1.4
Time: 27.9s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{z - a}\]
↓
\[x + \left(\frac{\frac{\sqrt[3]{y}}{\frac{\sqrt[3]{z - a}}{\sqrt[3]{y}}}}{\sqrt[3]{\sqrt[3]{z - a}}} \cdot \frac{z - t}{\sqrt[3]{\sqrt[3]{z - a}}}\right) \cdot \frac{\frac{\sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{z - a}}}}{\sqrt[3]{z - a}}\]
x + \frac{y \cdot \left(z - t\right)}{z - a}
↓
x + \left(\frac{\frac{\sqrt[3]{y}}{\frac{\sqrt[3]{z - a}}{\sqrt[3]{y}}}}{\sqrt[3]{\sqrt[3]{z - a}}} \cdot \frac{z - t}{\sqrt[3]{\sqrt[3]{z - a}}}\right) \cdot \frac{\frac{\sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{z - a}}}}{\sqrt[3]{z - a}}
double f(double x, double y, double z, double t, double a) {
        double r25497047 = x;
        double r25497048 = y;
        double r25497049 = z;
        double r25497050 = t;
        double r25497051 = r25497049 - r25497050;
        double r25497052 = r25497048 * r25497051;
        double r25497053 = a;
        double r25497054 = r25497049 - r25497053;
        double r25497055 = r25497052 / r25497054;
        double r25497056 = r25497047 + r25497055;
        return r25497056;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r25497057 = x;
        double r25497058 = y;
        double r25497059 = cbrt(r25497058);
        double r25497060 = z;
        double r25497061 = a;
        double r25497062 = r25497060 - r25497061;
        double r25497063 = cbrt(r25497062);
        double r25497064 = r25497063 / r25497059;
        double r25497065 = r25497059 / r25497064;
        double r25497066 = cbrt(r25497063);
        double r25497067 = r25497065 / r25497066;
        double r25497068 = t;
        double r25497069 = r25497060 - r25497068;
        double r25497070 = r25497069 / r25497066;
        double r25497071 = r25497067 * r25497070;
        double r25497072 = r25497059 / r25497066;
        double r25497073 = r25497072 / r25497063;
        double r25497074 = r25497071 * r25497073;
        double r25497075 = r25497057 + r25497074;
        return r25497075;
}

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

Original10.5
Target1.3
Herbie1.4
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Initial program 10.5

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

  2. Simplified2.9

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

  4. Applied fma-udef2.9

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

  6. Applied add-cube-cbrt3.4

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

  7. Applied associate-/r*3.4

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

  9. Applied *-un-lft-identity3.4

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

  10. Applied cbrt-prod3.4

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

  11. Applied add-cube-cbrt3.5

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

  12. Applied times-frac3.5

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

  13. Applied times-frac3.5

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

  14. Applied associate-*r*3.3

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

  15. Simplified3.3

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

  17. Applied *-un-lft-identity3.3

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

  18. Applied cbrt-prod3.3

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

  19. Applied add-cube-cbrt3.4

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

  20. Applied *-un-lft-identity3.4

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

  21. Applied cbrt-prod3.4

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

  22. Applied times-frac3.4

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

  23. Applied times-frac3.4

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

  24. Applied associate-*r*3.4

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

  25. Simplified1.4

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

  26. Final simplification1.4

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

Reproduce

herbie shell --seed 2019168 +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))))