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

Average Error: 7.8 → 1.5

Time: 29.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -5.8428229986270315 \cdot 10^{139} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 4.93643854715891061 \cdot 10^{207}\right):\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot \frac{y}{a} - \left(4.5 \cdot t\right) \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \frac{1}{a \cdot 2}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r574115 = x;
        double r574116 = y;
        double r574117 = r574115 * r574116;
        double r574118 = z;
        double r574119 = 9.0;
        double r574120 = r574118 * r574119;
        double r574121 = t;
        double r574122 = r574120 * r574121;
        double r574123 = r574117 - r574122;
        double r574124 = a;
        double r574125 = 2.0;
        double r574126 = r574124 * r574125;
        double r574127 = r574123 / r574126;
        return r574127;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r574128 = x;
        double r574129 = y;
        double r574130 = r574128 * r574129;
        double r574131 = z;
        double r574132 = 9.0;
        double r574133 = r574131 * r574132;
        double r574134 = t;
        double r574135 = r574133 * r574134;
        double r574136 = r574130 - r574135;
        double r574137 = -5.842822998627031e+139;
        bool r574138 = r574136 <= r574137;
        double r574139 = 4.9364385471589106e+207;
        bool r574140 = r574136 <= r574139;
        double r574141 = !r574140;
        bool r574142 = r574138 || r574141;
        double r574143 = 0.5;
        double r574144 = r574143 * r574128;
        double r574145 = a;
        double r574146 = r574129 / r574145;
        double r574147 = r574144 * r574146;
        double r574148 = 4.5;
        double r574149 = r574148 * r574134;
        double r574150 = r574131 / r574145;
        double r574151 = r574149 * r574150;
        double r574152 = r574147 - r574151;
        double r574153 = 1.0;
        double r574154 = 2.0;
        double r574155 = r574145 * r574154;
        double r574156 = r574153 / r574155;
        double r574157 = r574136 * r574156;
        double r574158 = r574142 ? r574152 : r574157;
        return r574158;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	7.8
Target	5.4
Herbie	1.5

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

Derivation

1. Split input into 2 regimes

2. if (- (* x y) (* (* z 9.0) t)) < -5.842822998627031e+139 or 4.9364385471589106e+207 < (- (* x y) (* (* z 9.0) t))

   1. Initial program 24.2

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

   2. Taylor expanded around 0 23.9

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

   4. Applied *-un-lft-identity 23.9

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{\color{blue}{1 \cdot a}}\]

   5. Applied times-frac 13.9

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \color{blue}{\left(\frac{t}{1} \cdot \frac{z}{a}\right)}\]

   6. Applied associate-*r* 14.0

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - \color{blue}{\left(4.5 \cdot \frac{t}{1}\right) \cdot \frac{z}{a}}\]

   7. Simplified 14.0

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - \color{blue}{\left(4.5 \cdot t\right)} \cdot \frac{z}{a}\]
   8. Using strategy rm

   9. Applied *-un-lft-identity 14.0

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{\color{blue}{1 \cdot a}} - \left(4.5 \cdot t\right) \cdot \frac{z}{a}\]

   10. Applied times-frac 2.9

       \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{1} \cdot \frac{y}{a}\right)} - \left(4.5 \cdot t\right) \cdot \frac{z}{a}\]

   11. Applied associate-*r* 2.8

       \[\leadsto \color{blue}{\left(0.5 \cdot \frac{x}{1}\right) \cdot \frac{y}{a}} - \left(4.5 \cdot t\right) \cdot \frac{z}{a}\]

   12. Simplified 2.8

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

   if -5.842822998627031e+139 < (- (* x y) (* (* z 9.0) t)) < 4.9364385471589106e+207

   1. Initial program 0.9

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

   3. Applied div-inv 1.0

      \[\leadsto \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \frac{1}{a \cdot 2}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -5.8428229986270315 \cdot 10^{139} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 4.93643854715891061 \cdot 10^{207}\right):\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot \frac{y}{a} - \left(4.5 \cdot t\right) \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \frac{1}{a \cdot 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"

  :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 (* 9.0 t))) (* a 2.0)) (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5)))))

  (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))