Average Error: 20.8 → 10.6
Time: 5.7s
Precision: 64
\[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.731646499562622 \cdot 10^{125}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{\frac{x}{z}}{\frac{c}{y}}\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{elif}\;y \le -2.1267767531027279 \cdot 10^{-249}:\\ \;\;\;\;\left(\frac{1}{z} \cdot \frac{b}{c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{elif}\;y \le 2.890065698620191 \cdot 10^{-286}:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;y \le 67868620829.5429611:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{z \cdot \frac{c}{y}}\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \end{array}\]
\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}\;y \le -1.731646499562622 \cdot 10^{125}:\\
\;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{\frac{x}{z}}{\frac{c}{y}}\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\

\mathbf{elif}\;y \le -2.1267767531027279 \cdot 10^{-249}:\\
\;\;\;\;\left(\frac{1}{z} \cdot \frac{b}{c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\

\mathbf{elif}\;y \le 2.890065698620191 \cdot 10^{-286}:\\
\;\;\;\;\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}\;y \le 67868620829.5429611:\\
\;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{a}{\frac{c}{t}}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{z \cdot \frac{c}{y}}\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r914577 = x;
        double r914578 = 9.0;
        double r914579 = r914577 * r914578;
        double r914580 = y;
        double r914581 = r914579 * r914580;
        double r914582 = z;
        double r914583 = 4.0;
        double r914584 = r914582 * r914583;
        double r914585 = t;
        double r914586 = r914584 * r914585;
        double r914587 = a;
        double r914588 = r914586 * r914587;
        double r914589 = r914581 - r914588;
        double r914590 = b;
        double r914591 = r914589 + r914590;
        double r914592 = c;
        double r914593 = r914582 * r914592;
        double r914594 = r914591 / r914593;
        return r914594;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r914595 = y;
        double r914596 = -1.731646499562622e+125;
        bool r914597 = r914595 <= r914596;
        double r914598 = b;
        double r914599 = z;
        double r914600 = c;
        double r914601 = r914599 * r914600;
        double r914602 = r914598 / r914601;
        double r914603 = 9.0;
        double r914604 = x;
        double r914605 = r914604 / r914599;
        double r914606 = r914600 / r914595;
        double r914607 = r914605 / r914606;
        double r914608 = r914603 * r914607;
        double r914609 = r914602 + r914608;
        double r914610 = 4.0;
        double r914611 = a;
        double r914612 = t;
        double r914613 = r914612 / r914600;
        double r914614 = r914611 * r914613;
        double r914615 = r914610 * r914614;
        double r914616 = r914609 - r914615;
        double r914617 = -2.1267767531027279e-249;
        bool r914618 = r914595 <= r914617;
        double r914619 = 1.0;
        double r914620 = r914619 / r914599;
        double r914621 = r914598 / r914600;
        double r914622 = r914620 * r914621;
        double r914623 = r914601 / r914595;
        double r914624 = r914604 / r914623;
        double r914625 = r914603 * r914624;
        double r914626 = r914622 + r914625;
        double r914627 = r914626 - r914615;
        double r914628 = 2.8900656986201915e-286;
        bool r914629 = r914595 <= r914628;
        double r914630 = r914604 * r914603;
        double r914631 = r914630 * r914595;
        double r914632 = r914599 * r914610;
        double r914633 = r914632 * r914612;
        double r914634 = r914633 * r914611;
        double r914635 = r914631 - r914634;
        double r914636 = r914635 + r914598;
        double r914637 = r914636 / r914599;
        double r914638 = r914637 / r914600;
        double r914639 = 67868620829.54296;
        bool r914640 = r914595 <= r914639;
        double r914641 = r914604 * r914595;
        double r914642 = r914641 / r914601;
        double r914643 = r914603 * r914642;
        double r914644 = r914602 + r914643;
        double r914645 = r914600 / r914612;
        double r914646 = r914611 / r914645;
        double r914647 = r914610 * r914646;
        double r914648 = r914644 - r914647;
        double r914649 = r914599 * r914606;
        double r914650 = r914604 / r914649;
        double r914651 = r914603 * r914650;
        double r914652 = r914602 + r914651;
        double r914653 = r914652 - r914615;
        double r914654 = r914640 ? r914648 : r914653;
        double r914655 = r914629 ? r914638 : r914654;
        double r914656 = r914618 ? r914627 : r914655;
        double r914657 = r914597 ? r914616 : r914656;
        return r914657;
}

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
Target14.9
Herbie10.6
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -1.10015674080410512 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -0.0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.17088779117474882 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 2.8768236795461372 \cdot 10^{130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.3838515042456319 \cdot 10^{158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if y < -1.731646499562622e+125

    1. Initial program 30.9

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

    2. Taylor expanded around 0 23.5

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

    4. Applied *-un-lft-identity23.5

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

    5. Applied times-frac22.2

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

    6. Simplified22.2

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

    8. Applied associate-/l*16.6

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

    10. Applied *-un-lft-identity16.6

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

    11. Applied times-frac14.3

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

    12. Applied associate-/r*11.9

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

    13. Simplified11.9

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

    if -1.731646499562622e+125 < y < -2.1267767531027279e-249

    1. Initial program 17.9

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

    2. Taylor expanded around 0 9.1

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

    4. Applied *-un-lft-identity9.1

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

    5. Applied times-frac9.1

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

    6. Simplified9.1

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

    8. Applied associate-/l*9.0

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

    10. Applied *-un-lft-identity9.0

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

    11. Applied times-frac10.2

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

    if -2.1267767531027279e-249 < y < 2.8900656986201915e-286

    1. Initial program 17.6

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

    3. Applied associate-/r*17.7

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

    if 2.8900656986201915e-286 < y < 67868620829.54296

    1. Initial program 17.2

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

    2. Taylor expanded around 0 7.2

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

    4. Applied associate-/l*7.0

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

    if 67868620829.54296 < y

    1. Initial program 25.1

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

    2. Taylor expanded around 0 17.4

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

    4. Applied *-un-lft-identity17.4

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

    5. Applied times-frac16.6

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

    6. Simplified16.6

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

    8. Applied associate-/l*13.2

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

    10. Applied *-un-lft-identity13.2

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

    11. Applied times-frac12.3

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

    12. Simplified12.3

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.731646499562622 \cdot 10^{125}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{\frac{x}{z}}{\frac{c}{y}}\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{elif}\;y \le -2.1267767531027279 \cdot 10^{-249}:\\ \;\;\;\;\left(\frac{1}{z} \cdot \frac{b}{c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{elif}\;y \le 2.890065698620191 \cdot 10^{-286}:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;y \le 67868620829.5429611:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{z \cdot \frac{c}{y}}\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(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)))