Average Error: 10.9 → 1.3
Time: 11.4s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{z - a}\]
↓
\[x + y \cdot \frac{z - t}{z - a}\]
x + \frac{y \cdot \left(z - t\right)}{z - a}
↓
x + y \cdot \frac{z - t}{z - a}
double f(double x, double y, double z, double t, double a) {
        double r611934 = x;
        double r611935 = y;
        double r611936 = z;
        double r611937 = t;
        double r611938 = r611936 - r611937;
        double r611939 = r611935 * r611938;
        double r611940 = a;
        double r611941 = r611936 - r611940;
        double r611942 = r611939 / r611941;
        double r611943 = r611934 + r611942;
        return r611943;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r611944 = x;
        double r611945 = y;
        double r611946 = z;
        double r611947 = t;
        double r611948 = r611946 - r611947;
        double r611949 = a;
        double r611950 = r611946 - r611949;
        double r611951 = r611948 / r611950;
        double r611952 = r611945 * r611951;
        double r611953 = r611944 + r611952;
        return r611953;
}

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
Target1.2
Herbie1.3
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Initial program 10.9

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

  2. Simplified2.9

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

  4. Applied clear-num3.1

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

  6. Applied fma-udef3.1

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

  7. Simplified3.0

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

  9. Applied *-un-lft-identity3.0

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

  10. Applied *-un-lft-identity3.0

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

  11. Applied times-frac3.0

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

  12. Applied *-un-lft-identity3.0

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

  13. Applied times-frac3.0

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

  14. Simplified3.0

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

  15. Simplified1.3

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

  16. Final simplification1.3

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

Reproduce

herbie shell --seed 2020047 +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))))