Average Error: 10.5 → 1.7
Time: 6.3s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{z - a}\]
↓
\[\frac{\frac{z - t}{\sqrt[3]{z - a} \cdot \sqrt[3]{z - a}}}{\frac{\sqrt[3]{z - a}}{y}} + x\]
x + \frac{y \cdot \left(z - t\right)}{z - a}
↓
\frac{\frac{z - t}{\sqrt[3]{z - a} \cdot \sqrt[3]{z - a}}}{\frac{\sqrt[3]{z - a}}{y}} + x
double f(double x, double y, double z, double t, double a) {
        double r557947 = x;
        double r557948 = y;
        double r557949 = z;
        double r557950 = t;
        double r557951 = r557949 - r557950;
        double r557952 = r557948 * r557951;
        double r557953 = a;
        double r557954 = r557949 - r557953;
        double r557955 = r557952 / r557954;
        double r557956 = r557947 + r557955;
        return r557956;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r557957 = z;
        double r557958 = t;
        double r557959 = r557957 - r557958;
        double r557960 = a;
        double r557961 = r557957 - r557960;
        double r557962 = cbrt(r557961);
        double r557963 = r557962 * r557962;
        double r557964 = r557959 / r557963;
        double r557965 = y;
        double r557966 = r557962 / r557965;
        double r557967 = r557964 / r557966;
        double r557968 = x;
        double r557969 = r557967 + r557968;
        return r557969;
}

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
Target1.2
Herbie1.7
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Initial program 10.5

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

  2. Simplified3.3

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

  4. Applied clear-num3.5

    \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{z - a}{y}}}, z - t, x\right)\]
  5. Using strategy rm

  6. Applied fma-udef3.5

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

  7. Simplified3.3

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

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

    \[\leadsto \frac{z - t}{\frac{z - a}{\color{blue}{1 \cdot y}}} + x\]

  10. Applied add-cube-cbrt3.8

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

  11. Applied times-frac3.8

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

  12. Applied associate-/r*1.7

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

  13. Simplified1.7

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

  14. Final simplification1.7

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

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A"
  :precision binary64

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

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