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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} = -\infty:\\ \;\;\;\;\left(\frac{b + \left(x \cdot 9\right) \cdot y}{z} - \left(a \cdot 4\right) \cdot t\right) \cdot \frac{1}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -66822341220707467817254601621504:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 3.418447849634938614215250112165796864162 \cdot 10^{-57}:\\ \;\;\;\;\frac{\left(b + \left(x \cdot 9\right) \cdot y\right) \cdot \frac{1}{z} - \left(a \cdot 4\right) \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 6.077745527002132601131494112606614000821 \cdot 10^{298}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{z} - \left(a \cdot 4\right) \cdot t}{c}\\ \end{array}\]

\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}\;\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} = -\infty:\\
\;\;\;\;\left(\frac{b + \left(x \cdot 9\right) \cdot y}{z} - \left(a \cdot 4\right) \cdot t\right) \cdot \frac{1}{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} \le -66822341220707467817254601621504:\\
\;\;\;\;\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}\\

\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 3.418447849634938614215250112165796864162 \cdot 10^{-57}:\\
\;\;\;\;\frac{\left(b + \left(x \cdot 9\right) \cdot y\right) \cdot \frac{1}{z} - \left(a \cdot 4\right) \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} \le 6.077745527002132601131494112606614000821 \cdot 10^{298}:\\
\;\;\;\;\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}\\

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

\end{array}

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r488595 = x;
        double r488596 = 9.0;
        double r488597 = r488595 * r488596;
        double r488598 = y;
        double r488599 = r488597 * r488598;
        double r488600 = z;
        double r488601 = 4.0;
        double r488602 = r488600 * r488601;
        double r488603 = t;
        double r488604 = r488602 * r488603;
        double r488605 = a;
        double r488606 = r488604 * r488605;
        double r488607 = r488599 - r488606;
        double r488608 = b;
        double r488609 = r488607 + r488608;
        double r488610 = c;
        double r488611 = r488600 * r488610;
        double r488612 = r488609 / r488611;
        return r488612;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r488613 = x;
        double r488614 = 9.0;
        double r488615 = r488613 * r488614;
        double r488616 = y;
        double r488617 = r488615 * r488616;
        double r488618 = z;
        double r488619 = 4.0;
        double r488620 = r488618 * r488619;
        double r488621 = t;
        double r488622 = r488620 * r488621;
        double r488623 = a;
        double r488624 = r488622 * r488623;
        double r488625 = r488617 - r488624;
        double r488626 = b;
        double r488627 = r488625 + r488626;
        double r488628 = c;
        double r488629 = r488618 * r488628;
        double r488630 = r488627 / r488629;
        double r488631 = -inf.0;
        bool r488632 = r488630 <= r488631;
        double r488633 = r488626 + r488617;
        double r488634 = r488633 / r488618;
        double r488635 = r488623 * r488619;
        double r488636 = r488635 * r488621;
        double r488637 = r488634 - r488636;
        double r488638 = 1.0;
        double r488639 = r488638 / r488628;
        double r488640 = r488637 * r488639;
        double r488641 = -6.682234122070747e+31;
        bool r488642 = r488630 <= r488641;
        double r488643 = 3.4184478496349386e-57;
        bool r488644 = r488630 <= r488643;
        double r488645 = r488638 / r488618;
        double r488646 = r488633 * r488645;
        double r488647 = r488646 - r488636;
        double r488648 = r488647 / r488628;
        double r488649 = 6.077745527002133e+298;
        bool r488650 = r488630 <= r488649;
        double r488651 = r488626 / r488618;
        double r488652 = r488651 - r488636;
        double r488653 = r488652 / r488628;
        double r488654 = r488650 ? r488630 : r488653;
        double r488655 = r488644 ? r488648 : r488654;
        double r488656 = r488642 ? r488630 : r488655;
        double r488657 = r488632 ? r488640 : r488656;
        return r488657;
}

Target

Original	20.5
Target	14.6
Herbie	7.5

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

Derivation

Split input into 4 regimes
if (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -inf.0
1. Initial program 64.0
  \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]
2. Simplified24.0
  \[\leadsto \color{blue}{\frac{\frac{b + \left(x \cdot 9\right) \cdot y}{z} - \left(a \cdot 4\right) \cdot t}{c}}\]
3. Using strategy rm
4. Applied div-inv24.1
  \[\leadsto \color{blue}{\left(\frac{b + \left(x \cdot 9\right) \cdot y}{z} - \left(a \cdot 4\right) \cdot t\right) \cdot \frac{1}{c}}\]
if -inf.0 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -6.682234122070747e+31 or 3.4184478496349386e-57 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 6.077745527002133e+298
1. Initial program 0.6
  \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]
if -6.682234122070747e+31 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 3.4184478496349386e-57
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. Simplified1.0
  \[\leadsto \color{blue}{\frac{\frac{b + \left(x \cdot 9\right) \cdot y}{z} - \left(a \cdot 4\right) \cdot t}{c}}\]
3. Using strategy rm
4. Applied div-inv1.1
  \[\leadsto \frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) \cdot \frac{1}{z}} - \left(a \cdot 4\right) \cdot t}{c}\]
if 6.077745527002133e+298 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))
1. Initial program 61.2
  \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]
2. Simplified26.8
  \[\leadsto \color{blue}{\frac{\frac{b + \left(x \cdot 9\right) \cdot y}{z} - \left(a \cdot 4\right) \cdot t}{c}}\]
3. Using strategy rm
4. Applied clear-num26.8
  \[\leadsto \frac{\color{blue}{\frac{1}{\frac{z}{b + \left(x \cdot 9\right) \cdot y}}} - \left(a \cdot 4\right) \cdot t}{c}\]
5. Taylor expanded around 0 27.7
  \[\leadsto \frac{\color{blue}{\frac{b}{z}} - \left(a \cdot 4\right) \cdot t}{c}\]
Recombined 4 regimes into one program.
Final simplification7.5
\[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} = -\infty:\\ \;\;\;\;\left(\frac{b + \left(x \cdot 9\right) \cdot y}{z} - \left(a \cdot 4\right) \cdot t\right) \cdot \frac{1}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -66822341220707467817254601621504:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 3.418447849634938614215250112165796864162 \cdot 10^{-57}:\\ \;\;\;\;\frac{\left(b + \left(x \cdot 9\right) \cdot y\right) \cdot \frac{1}{z} - \left(a \cdot 4\right) \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 6.077745527002132601131494112606614000821 \cdot 10^{298}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{z} - \left(a \cdot 4\right) \cdot t}{c}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(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)) -1.100156740804105e-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)))

Error

Try it out

Target

Derivation

`if (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -inf.0`

`if -inf.0 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -6.682234122070747e+31 or 3.4184478496349386e-57 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 6.077745527002133e+298`

`if -6.682234122070747e+31 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 3.4184478496349386e-57`

`if 6.077745527002133e+298 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))`

Reproduce

In
Out