Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, B

Average Error: 10.9 → 0.7

Time: 10.8s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a - t} \le -7.619609112581202273765869446605205908105 \cdot 10^{120}:\\ \;\;\;\;\frac{z - t}{a - t} \cdot y + x\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a - t} \le 2.141643850967665421502672143203672013512 \cdot 10^{280}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{z - t}{\frac{a - t}{y}} + x\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r650054 = x;
        double r650055 = y;
        double r650056 = z;
        double r650057 = t;
        double r650058 = r650056 - r650057;
        double r650059 = r650055 * r650058;
        double r650060 = a;
        double r650061 = r650060 - r650057;
        double r650062 = r650059 / r650061;
        double r650063 = r650054 + r650062;
        return r650063;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r650064 = y;
        double r650065 = z;
        double r650066 = t;
        double r650067 = r650065 - r650066;
        double r650068 = r650064 * r650067;
        double r650069 = a;
        double r650070 = r650069 - r650066;
        double r650071 = r650068 / r650070;
        double r650072 = -7.619609112581202e+120;
        bool r650073 = r650071 <= r650072;
        double r650074 = r650067 / r650070;
        double r650075 = r650074 * r650064;
        double r650076 = x;
        double r650077 = r650075 + r650076;
        double r650078 = 2.1416438509676654e+280;
        bool r650079 = r650071 <= r650078;
        double r650080 = r650076 + r650071;
        double r650081 = r650070 / r650064;
        double r650082 = r650067 / r650081;
        double r650083 = r650082 + r650076;
        double r650084 = r650079 ? r650080 : r650083;
        double r650085 = r650073 ? r650077 : r650084;
        return r650085;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	10.9
Target	1.3
Herbie	0.7

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

Derivation

1. Split input into 3 regimes

2. if (/ (* y (- z t)) (- a t)) < -7.619609112581202e+120

   1. Initial program 35.7

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

   2. Simplified 3.0

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

   4. Applied div-inv 3.1

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

   6. Applied fma-udef 3.1

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

   7. Simplified 3.0

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

   9. Applied clear-num 3.1

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

   11. Applied associate-/r/ 3.1

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

   12. Applied associate-*r* 3.1

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

   13. Simplified 3.0

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

   if -7.619609112581202e+120 < (/ (* y (- z t)) (- a t)) < 2.1416438509676654e+280

   1. Initial program 0.2

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

   if 2.1416438509676654e+280 < (/ (* y (- z t)) (- a t))

   1. Initial program 59.8

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

   2. Simplified 1.3

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

   4. Applied div-inv 1.4

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

   6. Applied fma-udef 1.4

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

   7. Simplified 1.3

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

   9. Applied clear-num 1.4

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

   11. Applied pow1 1.4

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

   12. Applied pow1 1.4

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

   13. Applied pow-prod-down 1.4

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

   14. Simplified 1.2

       \[\leadsto {\color{blue}{\left(\frac{z - t}{\frac{a - t}{y}}\right)}}^{1} + x\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a - t} \le -7.619609112581202273765869446605205908105 \cdot 10^{120}:\\ \;\;\;\;\frac{z - t}{a - t} \cdot y + x\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a - t} \le 2.141643850967665421502672143203672013512 \cdot 10^{280}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{z - t}{\frac{a - t}{y}} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, B"
  :precision binary64

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

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