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

Average Error: 20.8 → 8.0

Time: 5.2s

Precision: 64

Math

\[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -2.9319548093872591 \cdot 10^{-9} \lor \neg \left(z \le 6.2299952706301125 \cdot 10^{117}\right):\\ \;\;\;\;\left(\frac{\frac{b}{z}}{c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{a}{\frac{c}{t}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r730609 = x;
        double r730610 = 9.0;
        double r730611 = r730609 * r730610;
        double r730612 = y;
        double r730613 = r730611 * r730612;
        double r730614 = z;
        double r730615 = 4.0;
        double r730616 = r730614 * r730615;
        double r730617 = t;
        double r730618 = r730616 * r730617;
        double r730619 = a;
        double r730620 = r730618 * r730619;
        double r730621 = r730613 - r730620;
        double r730622 = b;
        double r730623 = r730621 + r730622;
        double r730624 = c;
        double r730625 = r730614 * r730624;
        double r730626 = r730623 / r730625;
        return r730626;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r730627 = z;
        double r730628 = -2.931954809387259e-09;
        bool r730629 = r730627 <= r730628;
        double r730630 = 6.229995270630113e+117;
        bool r730631 = r730627 <= r730630;
        double r730632 = !r730631;
        bool r730633 = r730629 || r730632;
        double r730634 = b;
        double r730635 = r730634 / r730627;
        double r730636 = c;
        double r730637 = r730635 / r730636;
        double r730638 = 9.0;
        double r730639 = x;
        double r730640 = r730627 * r730636;
        double r730641 = y;
        double r730642 = r730640 / r730641;
        double r730643 = r730639 / r730642;
        double r730644 = r730638 * r730643;
        double r730645 = r730637 + r730644;
        double r730646 = 4.0;
        double r730647 = a;
        double r730648 = t;
        double r730649 = r730647 * r730648;
        double r730650 = r730649 / r730636;
        double r730651 = r730646 * r730650;
        double r730652 = r730645 - r730651;
        double r730653 = r730634 / r730640;
        double r730654 = r730639 * r730641;
        double r730655 = r730654 / r730640;
        double r730656 = r730638 * r730655;
        double r730657 = r730653 + r730656;
        double r730658 = r730636 / r730648;
        double r730659 = r730647 / r730658;
        double r730660 = r730646 * r730659;
        double r730661 = r730657 - r730660;
        double r730662 = r730633 ? r730652 : r730661;
        return r730662;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	20.8
Target	14.7
Herbie	8.0

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -1.10015674080410512 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -0.0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.17088779117474882 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 2.8768236795461372 \cdot 10^{130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.3838515042456319 \cdot 10^{158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < -2.931954809387259e-09 or 6.229995270630113e+117 < z

   1. Initial program 31.8

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

   2. Taylor expanded around 0 13.9

      \[\leadsto \color{blue}{\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{a \cdot t}{c}}\]
   3. Using strategy rm

   4. Applied associate-/l* 11.2

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \color{blue}{\frac{x}{\frac{z \cdot c}{y}}}\right) - 4 \cdot \frac{a \cdot t}{c}\]
   5. Using strategy rm

   6. Applied associate-/r* 8.6

      \[\leadsto \left(\color{blue}{\frac{\frac{b}{z}}{c}} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \frac{a \cdot t}{c}\]

   if -2.931954809387259e-09 < z < 6.229995270630113e+117

   1. Initial program 9.1

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

   2. Taylor expanded around 0 10.0

      \[\leadsto \color{blue}{\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{a \cdot t}{c}}\]
   3. Using strategy rm

   4. Applied associate-/l* 7.4

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \color{blue}{\frac{a}{\frac{c}{t}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 8.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.9319548093872591 \cdot 10^{-9} \lor \neg \left(z \le 6.2299952706301125 \cdot 10^{117}\right):\\ \;\;\;\;\left(\frac{\frac{b}{z}}{c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{a}{\frac{c}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, J"
  :precision binary64

  :herbie-target
  (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) -1.1001567408041051e-171) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) -0.0) (/ (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) z) c) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 1.1708877911747488e-53) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 2.876823679546137e+130) (- (+ (* (* 9 (/ y c)) (/ x z)) (/ b (* c z))) (* 4 (/ (* a t) c))) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 1.3838515042456319e+158) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (- (+ (* 9 (* (/ y (* c z)) x)) (/ b (* c z))) (* 4 (/ (* a t) c))))))))

  (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)))