Average Error: 11.2 → 1.3
Time: 17.7s
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 r369220 = x;
        double r369221 = y;
        double r369222 = z;
        double r369223 = t;
        double r369224 = r369222 - r369223;
        double r369225 = r369221 * r369224;
        double r369226 = a;
        double r369227 = r369222 - r369226;
        double r369228 = r369225 / r369227;
        double r369229 = r369220 + r369228;
        return r369229;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r369230 = x;
        double r369231 = y;
        double r369232 = z;
        double r369233 = t;
        double r369234 = r369232 - r369233;
        double r369235 = a;
        double r369236 = r369232 - r369235;
        double r369237 = r369234 / r369236;
        double r369238 = r369231 * r369237;
        double r369239 = r369230 + r369238;
        return r369239;
}

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

Original11.2
Target1.1
Herbie1.3
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Initial program 11.2

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

  2. Simplified3.0

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

  4. Applied clear-num3.2

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

  6. Applied fma-udef3.2

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

  7. Simplified3.1

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

  9. Applied *-un-lft-identity3.1

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

  10. Applied *-un-lft-identity3.1

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

  11. Applied times-frac3.1

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

  12. Applied *-un-lft-identity3.1

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

  13. Applied times-frac3.1

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

  14. Simplified3.1

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

  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 2019322 +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))))