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

Average Error: 20.7 → 6.2

Time: 26.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -1.314207964319466632441523825508139952412 \cdot 10^{-29}:\\ \;\;\;\;\left(9 \cdot \frac{x}{\frac{z \cdot c}{y}} + \frac{b}{z \cdot c}\right) - 4 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 3.30967174159267506940238545153318712189 \cdot 10^{-58}:\\ \;\;\;\;\frac{1}{\frac{c}{\frac{b + \left(x \cdot 9\right) \cdot y}{z} - 4 \cdot \left(t \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \frac{x}{\frac{z \cdot c}{y}} + \frac{b}{z \cdot c}\right) - 4 \cdot \frac{a}{\frac{c}{t}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r198174664 = x;
        double r198174665 = 9.0;
        double r198174666 = r198174664 * r198174665;
        double r198174667 = y;
        double r198174668 = r198174666 * r198174667;
        double r198174669 = z;
        double r198174670 = 4.0;
        double r198174671 = r198174669 * r198174670;
        double r198174672 = t;
        double r198174673 = r198174671 * r198174672;
        double r198174674 = a;
        double r198174675 = r198174673 * r198174674;
        double r198174676 = r198174668 - r198174675;
        double r198174677 = b;
        double r198174678 = r198174676 + r198174677;
        double r198174679 = c;
        double r198174680 = r198174669 * r198174679;
        double r198174681 = r198174678 / r198174680;
        return r198174681;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r198174682 = x;
        double r198174683 = 9.0;
        double r198174684 = r198174682 * r198174683;
        double r198174685 = y;
        double r198174686 = r198174684 * r198174685;
        double r198174687 = z;
        double r198174688 = 4.0;
        double r198174689 = r198174687 * r198174688;
        double r198174690 = t;
        double r198174691 = r198174689 * r198174690;
        double r198174692 = a;
        double r198174693 = r198174691 * r198174692;
        double r198174694 = r198174686 - r198174693;
        double r198174695 = b;
        double r198174696 = r198174694 + r198174695;
        double r198174697 = c;
        double r198174698 = r198174687 * r198174697;
        double r198174699 = r198174696 / r198174698;
        double r198174700 = -1.3142079643194666e-29;
        bool r198174701 = r198174699 <= r198174700;
        double r198174702 = r198174698 / r198174685;
        double r198174703 = r198174682 / r198174702;
        double r198174704 = r198174683 * r198174703;
        double r198174705 = r198174695 / r198174698;
        double r198174706 = r198174704 + r198174705;
        double r198174707 = r198174697 / r198174690;
        double r198174708 = r198174692 / r198174707;
        double r198174709 = r198174688 * r198174708;
        double r198174710 = r198174706 - r198174709;
        double r198174711 = 3.309671741592675e-58;
        bool r198174712 = r198174699 <= r198174711;
        double r198174713 = 1.0;
        double r198174714 = r198174695 + r198174686;
        double r198174715 = r198174714 / r198174687;
        double r198174716 = r198174690 * r198174692;
        double r198174717 = r198174688 * r198174716;
        double r198174718 = r198174715 - r198174717;
        double r198174719 = r198174697 / r198174718;
        double r198174720 = r198174713 / r198174719;
        double r198174721 = r198174712 ? r198174720 : r198174710;
        double r198174722 = r198174701 ? r198174710 : r198174721;
        return r198174722;
}

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	20.7
Target	14.9
Herbie	6.2

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -1.100156740804104887233830094663413900721 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -0.0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.170887791174748819600820354912645756062 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 2.876823679546137226963937101710277849382 \cdot 10^{130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.383851504245631860711731716196098366993 \cdot 10^{158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -1.3142079643194666e-29 or 3.309671741592675e-58 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))

   1. Initial program 21.8

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

   2. Simplified 17.8

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

   3. Taylor expanded around 0 12.4

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

   5. Applied associate-/l* 10.9

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

   7. Applied associate-/l* 7.8

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

   if -1.3142079643194666e-29 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 3.309671741592675e-58

   1. Initial program 17.7

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

   2. Simplified 0.8

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

   4. Applied clear-num 1.5

      \[\leadsto \color{blue}{\frac{1}{\frac{c}{\frac{b + \left(x \cdot 9\right) \cdot y}{z} - 4 \cdot \left(t \cdot a\right)}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 6.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -1.314207964319466632441523825508139952412 \cdot 10^{-29}:\\ \;\;\;\;\left(9 \cdot \frac{x}{\frac{z \cdot c}{y}} + \frac{b}{z \cdot c}\right) - 4 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 3.30967174159267506940238545153318712189 \cdot 10^{-58}:\\ \;\;\;\;\frac{1}{\frac{c}{\frac{b + \left(x \cdot 9\right) \cdot y}{z} - 4 \cdot \left(t \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \frac{x}{\frac{z \cdot c}{y}} + \frac{b}{z \cdot c}\right) - 4 \cdot \frac{a}{\frac{c}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, J"

  :herbie-target
  (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) -1.100156740804105e-171) (/ (+ (- (* (* x 9.0) y) (* (* z 4.0) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) -0.0) (/ (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) z) c) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) 1.1708877911747488e-53) (/ (+ (- (* (* x 9.0) y) (* (* z 4.0) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) 2.876823679546137e+130) (- (+ (* (* 9.0 (/ y c)) (/ x z)) (/ b (* c z))) (* 4.0 (/ (* a t) c))) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) 1.3838515042456319e+158) (/ (+ (- (* (* x 9.0) y) (* (* z 4.0) (* t a))) b) (* z c)) (- (+ (* 9.0 (* (/ y (* c z)) x)) (/ b (* c z))) (* 4.0 (/ (* a t) c))))))))

  (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)))