Average Error: 10.4 → 1.4
Time: 4.7s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{z - a}\]
↓
\[\frac{\frac{z - t}{z - a}}{\frac{1}{y}} + x\]
x + \frac{y \cdot \left(z - t\right)}{z - a}
↓
\frac{\frac{z - t}{z - a}}{\frac{1}{y}} + x
double f(double x, double y, double z, double t, double a) {
        double r718114 = x;
        double r718115 = y;
        double r718116 = z;
        double r718117 = t;
        double r718118 = r718116 - r718117;
        double r718119 = r718115 * r718118;
        double r718120 = a;
        double r718121 = r718116 - r718120;
        double r718122 = r718119 / r718121;
        double r718123 = r718114 + r718122;
        return r718123;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r718124 = z;
        double r718125 = t;
        double r718126 = r718124 - r718125;
        double r718127 = a;
        double r718128 = r718124 - r718127;
        double r718129 = r718126 / r718128;
        double r718130 = 1.0;
        double r718131 = y;
        double r718132 = r718130 / r718131;
        double r718133 = r718129 / r718132;
        double r718134 = x;
        double r718135 = r718133 + r718134;
        return r718135;
}

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

Original10.4
Target1.2
Herbie1.4
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Initial program 10.4

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

  2. Simplified2.9

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

  4. Applied div-inv3.0

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

  6. Applied add-cube-cbrt3.4

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

  7. Applied associate-*l*3.4

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

  8. Simplified3.4

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

  10. Applied fma-udef3.4

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

  11. Simplified3.0

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

  13. Applied div-inv3.1

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

  14. Applied associate-/r*1.4

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

  15. Final simplification1.4

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

Reproduce

herbie shell --seed 2019362 +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))))