Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, A

Average Error: 7.2 → 2.7

Time: 5.2s

Precision: 64

Math

\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1.396688144658496768166786539590339685249 \cdot 10^{153}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;z \le -5.843152401362096536261056797293777370498 \cdot 10^{-92}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)}{\left(x + 1\right) \cdot 1} - \frac{\frac{x}{t \cdot z - x}}{x + 1}\\ \mathbf{elif}\;z \le 4.324240404418594692132041898102529823429 \cdot 10^{145}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{\mathsf{fma}\left(t, z, -x\right)}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r663768 = x;
        double r663769 = y;
        double r663770 = z;
        double r663771 = r663769 * r663770;
        double r663772 = r663771 - r663768;
        double r663773 = t;
        double r663774 = r663773 * r663770;
        double r663775 = r663774 - r663768;
        double r663776 = r663772 / r663775;
        double r663777 = r663768 + r663776;
        double r663778 = 1.0;
        double r663779 = r663768 + r663778;
        double r663780 = r663777 / r663779;
        return r663780;
}

↓

double f(double x, double y, double z, double t) {
        double r663781 = z;
        double r663782 = -1.3966881446584968e+153;
        bool r663783 = r663781 <= r663782;
        double r663784 = x;
        double r663785 = y;
        double r663786 = t;
        double r663787 = r663785 / r663786;
        double r663788 = r663784 + r663787;
        double r663789 = 1.0;
        double r663790 = r663784 + r663789;
        double r663791 = r663788 / r663790;
        double r663792 = -5.8431524013620965e-92;
        bool r663793 = r663781 <= r663792;
        double r663794 = r663786 * r663781;
        double r663795 = r663794 - r663784;
        double r663796 = r663785 / r663795;
        double r663797 = fma(r663796, r663781, r663784);
        double r663798 = 1.0;
        double r663799 = r663790 * r663798;
        double r663800 = r663797 / r663799;
        double r663801 = r663784 / r663795;
        double r663802 = r663801 / r663790;
        double r663803 = r663800 - r663802;
        double r663804 = 4.3242404044185947e+145;
        bool r663805 = r663781 <= r663804;
        double r663806 = r663785 * r663781;
        double r663807 = r663806 - r663784;
        double r663808 = -r663784;
        double r663809 = fma(r663786, r663781, r663808);
        double r663810 = r663807 / r663809;
        double r663811 = r663784 + r663810;
        double r663812 = r663811 / r663790;
        double r663813 = r663805 ? r663812 : r663791;
        double r663814 = r663793 ? r663803 : r663813;
        double r663815 = r663783 ? r663791 : r663814;
        return r663815;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.2
Target	0.4
Herbie	2.7

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

Derivation

1. Split input into 3 regimes

2. if z < -1.3966881446584968e+153 or 4.3242404044185947e+145 < z

   1. Initial program 21.4

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]

   2. Taylor expanded around inf 6.0

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1}\]

   if -1.3966881446584968e+153 < z < -5.8431524013620965e-92

   1. Initial program 5.2

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
   2. Using strategy rm

   3. Applied div-sub 5.2

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

   4. Applied associate-+r- 5.2

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

   5. Applied div-sub 5.2

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

   6. Simplified 1.8

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

   if -5.8431524013620965e-92 < z < 4.3242404044185947e+145

   1. Initial program 1.5

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
   2. Using strategy rm

   3. Applied fma-neg 1.5

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

4. Final simplification 2.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.396688144658496768166786539590339685249 \cdot 10^{153}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;z \le -5.843152401362096536261056797293777370498 \cdot 10^{-92}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)}{\left(x + 1\right) \cdot 1} - \frac{\frac{x}{t \cdot z - x}}{x + 1}\\ \mathbf{elif}\;z \le 4.324240404418594692132041898102529823429 \cdot 10^{145}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{\mathsf{fma}\left(t, z, -x\right)}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :herbie-target
  (/ (+ x (- (/ y (- t (/ x z))) (/ x (- (* t z) x)))) (+ x 1))

  (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1)))