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

Average Error: 15.8 → 7.4

Time: 1.4m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{elif}\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le -1.3435650229451346 \cdot 10^{-276}:\\ \;\;\;\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{elif}\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 0.0:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{elif}\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 2.447401890545968 \cdot 10^{+295}:\\ \;\;\;\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r25767981 = x;
        double r25767982 = y;
        double r25767983 = r25767981 + r25767982;
        double r25767984 = z;
        double r25767985 = t;
        double r25767986 = r25767984 - r25767985;
        double r25767987 = r25767986 * r25767982;
        double r25767988 = a;
        double r25767989 = r25767988 - r25767985;
        double r25767990 = r25767987 / r25767989;
        double r25767991 = r25767983 - r25767990;
        return r25767991;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r25767992 = y;
        double r25767993 = x;
        double r25767994 = r25767992 + r25767993;
        double r25767995 = z;
        double r25767996 = t;
        double r25767997 = r25767995 - r25767996;
        double r25767998 = r25767997 * r25767992;
        double r25767999 = a;
        double r25768000 = r25767999 - r25767996;
        double r25768001 = r25767998 / r25768000;
        double r25768002 = r25767994 - r25768001;
        double r25768003 = -inf.0;
        bool r25768004 = r25768002 <= r25768003;
        double r25768005 = r25767995 / r25767996;
        double r25768006 = fma(r25768005, r25767992, r25767993);
        double r25768007 = -1.3435650229451346e-276;
        bool r25768008 = r25768002 <= r25768007;
        double r25768009 = 0.0;
        bool r25768010 = r25768002 <= r25768009;
        double r25768011 = 2.447401890545968e+295;
        bool r25768012 = r25768002 <= r25768011;
        double r25768013 = r25768012 ? r25768002 : r25768006;
        double r25768014 = r25768010 ? r25768006 : r25768013;
        double r25768015 = r25768008 ? r25768002 : r25768014;
        double r25768016 = r25768004 ? r25768006 : r25768015;
        return r25768016;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	15.8
Target	8.3
Herbie	7.4

\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.3664970889390727 \cdot 10^{-07}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.4754293444577233 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (- (+ x y) (/ (* (- z t) y) (- a t))) < -inf.0 or -1.3435650229451346e-276 < (- (+ x y) (/ (* (- z t) y) (- a t))) < 0.0 or 2.447401890545968e+295 < (- (+ x y) (/ (* (- z t) y) (- a t)))

   1. Initial program 58.4

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

   2. Simplified 36.5

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

   3. Taylor expanded around inf 33.7

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

   4. Simplified 25.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}\]

   if -inf.0 < (- (+ x y) (/ (* (- z t) y) (- a t))) < -1.3435650229451346e-276 or 0.0 < (- (+ x y) (/ (* (- z t) y) (- a t))) < 2.447401890545968e+295

   1. Initial program 1.4

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
3. Recombined 2 regimes into one program.

4. Final simplification 7.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{elif}\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le -1.3435650229451346 \cdot 10^{-276}:\\ \;\;\;\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{elif}\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 0.0:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{elif}\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 2.447401890545968 \cdot 10^{+295}:\\ \;\;\;\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(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 (- 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 (- a t))) y))))

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