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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, \frac{y \cdot 9}{c}, \frac{\frac{b}{c}}{z} - \frac{a \cdot 4}{c} \cdot t\right)\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le -48321584405561035694420036419584:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 1.238318805906636100034210185302784297183 \cdot 10^{-165}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, \frac{x}{\frac{z}{y}}, \frac{b}{z}\right) - t \cdot \left(a \cdot 4\right)}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 7.468702005485942468554359890686701640407 \cdot 10^{289}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, \frac{y \cdot 9}{c}, \frac{\frac{b}{c}}{z} - \frac{a \cdot 4}{c} \cdot t\right)\\ \end{array}\]

\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}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} = -\infty:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{z}, \frac{y \cdot 9}{c}, \frac{\frac{b}{c}}{z} - \frac{a \cdot 4}{c} \cdot t\right)\\

\mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le -48321584405561035694420036419584:\\
\;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z}\\

\mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 1.238318805906636100034210185302784297183 \cdot 10^{-165}:\\
\;\;\;\;\frac{\mathsf{fma}\left(9, \frac{x}{\frac{z}{y}}, \frac{b}{z}\right) - t \cdot \left(a \cdot 4\right)}{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}{c \cdot z} \le 7.468702005485942468554359890686701640407 \cdot 10^{289}:\\
\;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z}\\

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

\end{array}

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r611482 = x;
        double r611483 = 9.0;
        double r611484 = r611482 * r611483;
        double r611485 = y;
        double r611486 = r611484 * r611485;
        double r611487 = z;
        double r611488 = 4.0;
        double r611489 = r611487 * r611488;
        double r611490 = t;
        double r611491 = r611489 * r611490;
        double r611492 = a;
        double r611493 = r611491 * r611492;
        double r611494 = r611486 - r611493;
        double r611495 = b;
        double r611496 = r611494 + r611495;
        double r611497 = c;
        double r611498 = r611487 * r611497;
        double r611499 = r611496 / r611498;
        return r611499;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r611500 = x;
        double r611501 = 9.0;
        double r611502 = r611500 * r611501;
        double r611503 = y;
        double r611504 = r611502 * r611503;
        double r611505 = z;
        double r611506 = 4.0;
        double r611507 = r611505 * r611506;
        double r611508 = t;
        double r611509 = r611507 * r611508;
        double r611510 = a;
        double r611511 = r611509 * r611510;
        double r611512 = r611504 - r611511;
        double r611513 = b;
        double r611514 = r611512 + r611513;
        double r611515 = c;
        double r611516 = r611515 * r611505;
        double r611517 = r611514 / r611516;
        double r611518 = -inf.0;
        bool r611519 = r611517 <= r611518;
        double r611520 = r611500 / r611505;
        double r611521 = r611503 * r611501;
        double r611522 = r611521 / r611515;
        double r611523 = r611513 / r611515;
        double r611524 = r611523 / r611505;
        double r611525 = r611510 * r611506;
        double r611526 = r611525 / r611515;
        double r611527 = r611526 * r611508;
        double r611528 = r611524 - r611527;
        double r611529 = fma(r611520, r611522, r611528);
        double r611530 = -4.832158440556104e+31;
        bool r611531 = r611517 <= r611530;
        double r611532 = 1.238318805906636e-165;
        bool r611533 = r611517 <= r611532;
        double r611534 = r611505 / r611503;
        double r611535 = r611500 / r611534;
        double r611536 = r611513 / r611505;
        double r611537 = fma(r611501, r611535, r611536);
        double r611538 = r611508 * r611525;
        double r611539 = r611537 - r611538;
        double r611540 = r611539 / r611515;
        double r611541 = 7.4687020054859425e+289;
        bool r611542 = r611517 <= r611541;
        double r611543 = r611542 ? r611517 : r611529;
        double r611544 = r611533 ? r611540 : r611543;
        double r611545 = r611531 ? r611517 : r611544;
        double r611546 = r611519 ? r611529 : r611545;
        return r611546;
}

Target

Original	20.8
Target	14.8
Herbie	3.6

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

Derivation

Split input into 3 regimes
if (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -inf.0 or 7.4687020054859425e+289 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))
1. Initial program 61.4
  \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]
2. Simplified26.6
  \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x \cdot 9, y, b\right)}{z} - \left(a \cdot 4\right) \cdot t}{c}}\]
3. Using strategy rm
4. Applied div-inv26.6
  \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot 9, y, b\right) \cdot \frac{1}{z}} - \left(a \cdot 4\right) \cdot t}{c}\]
5. Taylor expanded around 0 30.4
  \[\leadsto \color{blue}{\left(9 \cdot \frac{x \cdot y}{z \cdot c} + \frac{b}{z \cdot c}\right) - 4 \cdot \frac{t \cdot a}{c}}\]
6. Simplified10.0
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{z}, \frac{y \cdot 9}{c}, \frac{\frac{b}{c}}{z} - \frac{a \cdot 4}{c} \cdot t\right)}\]
if -inf.0 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -4.832158440556104e+31 or 1.238318805906636e-165 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 7.4687020054859425e+289
1. Initial program 0.7
  \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]
if -4.832158440556104e+31 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 1.238318805906636e-165
1. Initial program 16.6
  \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]
2. Simplified0.7
  \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x \cdot 9, y, b\right)}{z} - \left(a \cdot 4\right) \cdot t}{c}}\]
3. Using strategy rm
4. Applied div-inv0.7
  \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot 9, y, b\right) \cdot \frac{1}{z}} - \left(a \cdot 4\right) \cdot t}{c}\]
5. Taylor expanded around 0 0.7
  \[\leadsto \frac{\color{blue}{\left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)} - \left(a \cdot 4\right) \cdot t}{c}\]
6. Simplified2.5
  \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(9, \frac{x}{\frac{z}{y}}, \frac{b}{z}\right)} - \left(a \cdot 4\right) \cdot t}{c}\]
Recombined 3 regimes into one program.
Final simplification3.6
\[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, \frac{y \cdot 9}{c}, \frac{\frac{b}{c}}{z} - \frac{a \cdot 4}{c} \cdot t\right)\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le -48321584405561035694420036419584:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 1.238318805906636100034210185302784297183 \cdot 10^{-165}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, \frac{x}{\frac{z}{y}}, \frac{b}{z}\right) - t \cdot \left(a \cdot 4\right)}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 7.468702005485942468554359890686701640407 \cdot 10^{289}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, \frac{y \cdot 9}{c}, \frac{\frac{b}{c}}{z} - \frac{a \cdot 4}{c} \cdot t\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(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)))

Error

Target

Derivation

`if (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -inf.0 or 7.4687020054859425e+289 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))`

`if -inf.0 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -4.832158440556104e+31 or 1.238318805906636e-165 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 7.4687020054859425e+289`

`if -4.832158440556104e+31 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 1.238318805906636e-165`

Reproduce