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

Average Error: 16.6 → 7.4

Time: 20.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -3.007208788506131572892149446918939079375 \cdot 10^{-140} \lor \neg \left(a \le 4.588832024847483261932699198337709983398 \cdot 10^{-287}\right):\\ \;\;\;\;\left(y - \frac{1}{\frac{1}{y \cdot \frac{z - t}{a - t}}}\right) + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r459938 = x;
        double r459939 = y;
        double r459940 = r459938 + r459939;
        double r459941 = z;
        double r459942 = t;
        double r459943 = r459941 - r459942;
        double r459944 = r459943 * r459939;
        double r459945 = a;
        double r459946 = r459945 - r459942;
        double r459947 = r459944 / r459946;
        double r459948 = r459940 - r459947;
        return r459948;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r459949 = a;
        double r459950 = -3.0072087885061316e-140;
        bool r459951 = r459949 <= r459950;
        double r459952 = 4.588832024847483e-287;
        bool r459953 = r459949 <= r459952;
        double r459954 = !r459953;
        bool r459955 = r459951 || r459954;
        double r459956 = y;
        double r459957 = 1.0;
        double r459958 = z;
        double r459959 = t;
        double r459960 = r459958 - r459959;
        double r459961 = r459949 - r459959;
        double r459962 = r459960 / r459961;
        double r459963 = r459956 * r459962;
        double r459964 = r459957 / r459963;
        double r459965 = r459957 / r459964;
        double r459966 = r459956 - r459965;
        double r459967 = x;
        double r459968 = r459966 + r459967;
        double r459969 = r459959 / r459958;
        double r459970 = r459956 / r459969;
        double r459971 = r459967 + r459970;
        double r459972 = r459955 ? r459968 : r459971;
        return r459972;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	16.6
Target	8.3
Herbie	7.4

\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.366497088939072697550672266103566343531 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.475429344457723334351036314450840066235 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if a < -3.0072087885061316e-140 or 4.588832024847483e-287 < a

   1. Initial program 16.0

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

   2. Simplified 16.0

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

   4. Applied associate--l+ 14.2

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

   5. Simplified 6.6

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

   7. Applied clear-num 6.7

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

   9. Applied clear-num 7.3

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

   10. Simplified 7.3

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

   if -3.0072087885061316e-140 < a < 4.588832024847483e-287

   1. Initial program 20.1

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

   2. Simplified 20.1

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

   4. Applied associate--l+ 15.7

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

   5. Simplified 11.9

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

   7. Applied clear-num 12.3

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

   9. Applied clear-num 14.1

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

   10. Simplified 14.1

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

   11. Taylor expanded around inf 9.1

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

   12. Simplified 8.5

       \[\leadsto x + \color{blue}{\frac{y}{\frac{t}{z}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 7.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -3.007208788506131572892149446918939079375 \cdot 10^{-140} \lor \neg \left(a \le 4.588832024847483261932699198337709983398 \cdot 10^{-287}\right):\\ \;\;\;\;\left(y - \frac{1}{\frac{1}{y \cdot \frac{z - t}{a - t}}}\right) + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"

  :herbie-target
  (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) -1.3664970889390727e-07) (- (+ y x) (* (* (- z t) (/ 1.0 (- a t))) y)) (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) 1.4754293444577233e-239) (/ (- (* y (- a z)) (* x t)) (- a t)) (- (+ y x) (* (* (- z t) (/ 1.0 (- a t))) y))))

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