Average Error: 20.8 → 10.9
Time: 15.2s
Precision: 64
\[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(t \le -3.7783325311094971158369023264124989667 \cdot 10^{-126} \lor t \le -5.034624607544583773811584986668675123642 \cdot 10^{-288}\right) \lor t \le 1.598096783313447022186703298780515686584 \cdot 10^{-291}\right) \lor \neg \left(t \le 1.173627343261278622871730108352946289209 \cdot 10^{-111}\right):\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \end{array}\]
\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}\;\left(\left(t \le -3.7783325311094971158369023264124989667 \cdot 10^{-126} \lor t \le -5.034624607544583773811584986668675123642 \cdot 10^{-288}\right) \lor t \le 1.598096783313447022186703298780515686584 \cdot 10^{-291}\right) \lor \neg \left(t \le 1.173627343261278622871730108352946289209 \cdot 10^{-111}\right):\\
\;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\

\mathbf{else}:\\
\;\;\;\;\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}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r516551 = x;
        double r516552 = 9.0;
        double r516553 = r516551 * r516552;
        double r516554 = y;
        double r516555 = r516553 * r516554;
        double r516556 = z;
        double r516557 = 4.0;
        double r516558 = r516556 * r516557;
        double r516559 = t;
        double r516560 = r516558 * r516559;
        double r516561 = a;
        double r516562 = r516560 * r516561;
        double r516563 = r516555 - r516562;
        double r516564 = b;
        double r516565 = r516563 + r516564;
        double r516566 = c;
        double r516567 = r516556 * r516566;
        double r516568 = r516565 / r516567;
        return r516568;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r516569 = t;
        double r516570 = -3.778332531109497e-126;
        bool r516571 = r516569 <= r516570;
        double r516572 = -5.034624607544584e-288;
        bool r516573 = r516569 <= r516572;
        bool r516574 = r516571 || r516573;
        double r516575 = 1.598096783313447e-291;
        bool r516576 = r516569 <= r516575;
        bool r516577 = r516574 || r516576;
        double r516578 = 1.1736273432612786e-111;
        bool r516579 = r516569 <= r516578;
        double r516580 = !r516579;
        bool r516581 = r516577 || r516580;
        double r516582 = b;
        double r516583 = z;
        double r516584 = c;
        double r516585 = r516583 * r516584;
        double r516586 = r516582 / r516585;
        double r516587 = 9.0;
        double r516588 = x;
        double r516589 = y;
        double r516590 = r516588 * r516589;
        double r516591 = r516590 / r516585;
        double r516592 = r516587 * r516591;
        double r516593 = r516586 + r516592;
        double r516594 = 4.0;
        double r516595 = a;
        double r516596 = r516569 / r516584;
        double r516597 = r516595 * r516596;
        double r516598 = r516594 * r516597;
        double r516599 = r516593 - r516598;
        double r516600 = r516588 * r516587;
        double r516601 = r516600 * r516589;
        double r516602 = r516583 * r516594;
        double r516603 = r516602 * r516569;
        double r516604 = r516603 * r516595;
        double r516605 = r516601 - r516604;
        double r516606 = r516605 + r516582;
        double r516607 = r516606 / r516583;
        double r516608 = r516607 / r516584;
        double r516609 = r516581 ? r516599 : r516608;
        return r516609;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.8
Target14.5
Herbie10.9
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -1.100156740804104887233830094663413900721 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -0.0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.170887791174748819600820354912645756062 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 2.876823679546137226963937101710277849382 \cdot 10^{130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.383851504245631860711731716196098366993 \cdot 10^{158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if t < -3.778332531109497e-126 or 3.486647258756959e+245 < t

    1. Initial program 26.6

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

    2. Taylor expanded around 0 13.6

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

    4. Applied associate-/l*10.9

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

    6. Applied associate-/r/10.2

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

    if -3.778332531109497e-126 < t < -5.034624607544584e-288 or 1.1736273432612786e-111 < t < 1.2672421849506084e-71

    1. Initial program 12.2

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

    2. Taylor expanded around 0 8.4

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

    4. Applied *-un-lft-identity8.4

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{a \cdot t}{\color{blue}{1 \cdot c}}\]

    5. Applied times-frac10.4

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \color{blue}{\left(\frac{a}{1} \cdot \frac{t}{c}\right)}\]

    6. Simplified10.4

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

    if -5.034624607544584e-288 < t < 1.598096783313447e-291

    1. Initial program 10.4

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

    2. Taylor expanded around 0 8.3

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

    4. Applied *-un-lft-identity8.3

      \[\leadsto \left(\frac{\color{blue}{1 \cdot b}}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{a \cdot t}{c}\]

    5. Applied times-frac9.5

      \[\leadsto \left(\color{blue}{\frac{1}{z} \cdot \frac{b}{c}} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{a \cdot t}{c}\]

    if 1.598096783313447e-291 < t < 1.1736273432612786e-111

    1. Initial program 14.0

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

    3. Applied associate-/r*11.1

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

    if 1.2672421849506084e-71 < t < 3.486647258756959e+245

    1. Initial program 24.4

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

    2. Taylor expanded around 0 12.3

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

    4. Applied associate-/l*10.1

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

    6. Applied associate-/l*8.8

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \color{blue}{\frac{x}{\frac{z \cdot c}{y}}}\right) - 4 \cdot \frac{a}{\frac{c}{t}}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification10.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t \le -3.7783325311094971158369023264124989667 \cdot 10^{-126} \lor t \le -5.034624607544583773811584986668675123642 \cdot 10^{-288}\right) \lor t \le 1.598096783313447022186703298780515686584 \cdot 10^{-291}\right) \lor \neg \left(t \le 1.173627343261278622871730108352946289209 \cdot 10^{-111}\right):\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \end{array}\]

Reproduce

herbie shell --seed 1978988140 
(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)) -1.1001567408041049e-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)) 1.17088779117474882e-53) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 2.8768236795461372e130) (- (+ (* (* 9 (/ y c)) (/ x z)) (/ b (* c z))) (* 4 (/ (* a t) c))) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 1.3838515042456319e158) (/ (+ (- (* (* 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)))