Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, I

Average Error: 7.5 → 4.8

Time: 9.2s

Precision: 64

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

Math

\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -3.13725196086471922 \cdot 10^{210}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le 5.09131880507002992 \cdot 10^{115}:\\ \;\;\;\;\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \frac{t \cdot z}{a}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r777095 = x;
        double r777096 = y;
        double r777097 = r777095 * r777096;
        double r777098 = z;
        double r777099 = 9.0;
        double r777100 = r777098 * r777099;
        double r777101 = t;
        double r777102 = r777100 * r777101;
        double r777103 = r777097 - r777102;
        double r777104 = a;
        double r777105 = 2.0;
        double r777106 = r777104 * r777105;
        double r777107 = r777103 / r777106;
        return r777107;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r777108 = x;
        double r777109 = y;
        double r777110 = r777108 * r777109;
        double r777111 = z;
        double r777112 = 9.0;
        double r777113 = r777111 * r777112;
        double r777114 = t;
        double r777115 = r777113 * r777114;
        double r777116 = r777110 - r777115;
        double r777117 = -3.137251960864719e+210;
        bool r777118 = r777116 <= r777117;
        double r777119 = 0.5;
        double r777120 = a;
        double r777121 = r777110 / r777120;
        double r777122 = r777119 * r777121;
        double r777123 = 4.5;
        double r777124 = r777120 / r777111;
        double r777125 = r777114 / r777124;
        double r777126 = r777123 * r777125;
        double r777127 = r777122 - r777126;
        double r777128 = 5.09131880507003e+115;
        bool r777129 = r777116 <= r777128;
        double r777130 = 1.0;
        double r777131 = 2.0;
        double r777132 = r777120 * r777131;
        double r777133 = r777130 / r777132;
        double r777134 = r777116 * r777133;
        double r777135 = r777120 / r777109;
        double r777136 = r777108 / r777135;
        double r777137 = r777119 * r777136;
        double r777138 = r777114 * r777111;
        double r777139 = r777138 / r777120;
        double r777140 = r777123 * r777139;
        double r777141 = r777137 - r777140;
        double r777142 = r777129 ? r777134 : r777141;
        double r777143 = r777118 ? r777127 : r777142;
        return r777143;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	7.5
Target	5.5
Herbie	4.8

\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.14403070783397609 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (- (* x y) (* (* z 9.0) t)) < -3.137251960864719e+210

   1. Initial program 29.3

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]

   2. Taylor expanded around 0 28.9

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}}\]
   3. Using strategy rm

   4. Applied associate-/l* 16.2

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \color{blue}{\frac{t}{\frac{a}{z}}}\]

   if -3.137251960864719e+210 < (- (* x y) (* (* z 9.0) t)) < 5.09131880507003e+115

   1. Initial program 0.8

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]
   2. Using strategy rm

   3. Applied div-inv 0.8

      \[\leadsto \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \frac{1}{a \cdot 2}}\]

   if 5.09131880507003e+115 < (- (* x y) (* (* z 9.0) t))

   1. Initial program 18.5

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]

   2. Taylor expanded around 0 18.5

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}}\]
   3. Using strategy rm

   4. Applied associate-/l* 12.2

      \[\leadsto 0.5 \cdot \color{blue}{\frac{x}{\frac{a}{y}}} - 4.5 \cdot \frac{t \cdot z}{a}\]
3. Recombined 3 regimes into one program.

4. Final simplification 4.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -3.13725196086471922 \cdot 10^{210}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le 5.09131880507002992 \cdot 10^{115}:\\ \;\;\;\;\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \frac{t \cdot z}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 +o rules:numerics
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64

  :herbie-target
  (if (< a -2.090464557976709e+86) (- (* 0.5 (/ (* y x) a)) (* 4.5 (/ t (/ a z)))) (if (< a 2.144030707833976e+99) (/ (- (* x y) (* z (* 9 t))) (* a 2)) (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5)))))

  (/ (- (* x y) (* (* z 9) t)) (* a 2)))