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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} = -\infty:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{c \cdot \frac{z}{y}}\right) - 4 \cdot \left(\frac{a}{c} \cdot t\right)\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -3.86575567753969965 \cdot 10^{-184}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -0.0:\\ \;\;\;\;\left(\frac{b + \left(x \cdot 9\right) \cdot y}{z} - 4 \cdot \left(t \cdot a\right)\right) \cdot \frac{1}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 2.5630952637815053 \cdot 10^{303}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{c \cdot \frac{z}{y}}\right) - 4 \cdot \left(\frac{a}{c} \cdot t\right)\\ \end{array}\]

\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}{z \cdot c} = -\infty:\\
\;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{c \cdot \frac{z}{y}}\right) - 4 \cdot \left(\frac{a}{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}{z \cdot c} \le -3.86575567753969965 \cdot 10^{-184}:\\
\;\;\;\;\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 -0.0:\\
\;\;\;\;\left(\frac{b + \left(x \cdot 9\right) \cdot y}{z} - 4 \cdot \left(t \cdot a\right)\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 2.5630952637815053 \cdot 10^{303}:\\
\;\;\;\;\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}:\\
\;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{c \cdot \frac{z}{y}}\right) - 4 \cdot \left(\frac{a}{c} \cdot t\right)\\

\end{array}

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r866936 = x;
        double r866937 = 9.0;
        double r866938 = r866936 * r866937;
        double r866939 = y;
        double r866940 = r866938 * r866939;
        double r866941 = z;
        double r866942 = 4.0;
        double r866943 = r866941 * r866942;
        double r866944 = t;
        double r866945 = r866943 * r866944;
        double r866946 = a;
        double r866947 = r866945 * r866946;
        double r866948 = r866940 - r866947;
        double r866949 = b;
        double r866950 = r866948 + r866949;
        double r866951 = c;
        double r866952 = r866941 * r866951;
        double r866953 = r866950 / r866952;
        return r866953;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r866954 = x;
        double r866955 = 9.0;
        double r866956 = r866954 * r866955;
        double r866957 = y;
        double r866958 = r866956 * r866957;
        double r866959 = z;
        double r866960 = 4.0;
        double r866961 = r866959 * r866960;
        double r866962 = t;
        double r866963 = r866961 * r866962;
        double r866964 = a;
        double r866965 = r866963 * r866964;
        double r866966 = r866958 - r866965;
        double r866967 = b;
        double r866968 = r866966 + r866967;
        double r866969 = c;
        double r866970 = r866959 * r866969;
        double r866971 = r866968 / r866970;
        double r866972 = -inf.0;
        bool r866973 = r866971 <= r866972;
        double r866974 = r866967 / r866970;
        double r866975 = r866959 / r866957;
        double r866976 = r866969 * r866975;
        double r866977 = r866954 / r866976;
        double r866978 = r866955 * r866977;
        double r866979 = r866974 + r866978;
        double r866980 = r866964 / r866969;
        double r866981 = r866980 * r866962;
        double r866982 = r866960 * r866981;
        double r866983 = r866979 - r866982;
        double r866984 = -3.8657556775396997e-184;
        bool r866985 = r866971 <= r866984;
        double r866986 = -0.0;
        bool r866987 = r866971 <= r866986;
        double r866988 = r866967 + r866958;
        double r866989 = r866988 / r866959;
        double r866990 = r866962 * r866964;
        double r866991 = r866960 * r866990;
        double r866992 = r866989 - r866991;
        double r866993 = 1.0;
        double r866994 = r866993 / r866969;
        double r866995 = r866992 * r866994;
        double r866996 = 2.5630952637815053e+303;
        bool r866997 = r866971 <= r866996;
        double r866998 = r866997 ? r866971 : r866983;
        double r866999 = r866987 ? r866995 : r866998;
        double r867000 = r866985 ? r866971 : r866999;
        double r867001 = r866973 ? r866983 : r867000;
        return r867001;
}

Target

Original	20.8
Target	14.9
Herbie	3.4

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

Derivation

Split input into 3 regimes
if (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -inf.0 or 2.5630952637815053e+303 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))
1. Initial program 63.1
  \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]
2. Simplified26.8
  \[\leadsto \color{blue}{\frac{\frac{b + \left(x \cdot 9\right) \cdot y}{z} - 4 \cdot \left(t \cdot a\right)}{c}}\]
3. Taylor expanded around 0 30.7
  \[\leadsto \color{blue}{\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{a \cdot t}{c}}\]
4. Using strategy rm
5. Applied associate-/l*26.0
  \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \color{blue}{\frac{a}{\frac{c}{t}}}\]
6. Using strategy rm
7. Applied associate-/r/25.6
  \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \color{blue}{\left(\frac{a}{c} \cdot t\right)}\]
8. Using strategy rm
9. Applied associate-/l*14.3
  \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \color{blue}{\frac{x}{\frac{z \cdot c}{y}}}\right) - 4 \cdot \left(\frac{a}{c} \cdot t\right)\]
10. Simplified11.1
  \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\color{blue}{c \cdot \frac{z}{y}}}\right) - 4 \cdot \left(\frac{a}{c} \cdot t\right)\]
if -inf.0 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -3.8657556775396997e-184 or -0.0 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 2.5630952637815053e+303
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 -3.8657556775396997e-184 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -0.0
1. Initial program 33.9
  \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]
2. Simplified0.6
  \[\leadsto \color{blue}{\frac{\frac{b + \left(x \cdot 9\right) \cdot y}{z} - 4 \cdot \left(t \cdot a\right)}{c}}\]
3. Using strategy rm
4. Applied div-inv0.6
  \[\leadsto \color{blue}{\left(\frac{b + \left(x \cdot 9\right) \cdot y}{z} - 4 \cdot \left(t \cdot a\right)\right) \cdot \frac{1}{c}}\]
Recombined 3 regimes into one program.
Final simplification3.4
\[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} = -\infty:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{c \cdot \frac{z}{y}}\right) - 4 \cdot \left(\frac{a}{c} \cdot t\right)\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -3.86575567753969965 \cdot 10^{-184}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -0.0:\\ \;\;\;\;\left(\frac{b + \left(x \cdot 9\right) \cdot y}{z} - 4 \cdot \left(t \cdot a\right)\right) \cdot \frac{1}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 2.5630952637815053 \cdot 10^{303}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{c \cdot \frac{z}{y}}\right) - 4 \cdot \left(\frac{a}{c} \cdot t\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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)))

Falling back on racket egraph because egg-herbie package not installed (more)

Error

Try it out

Target

Derivation

`if (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -inf.0 or 2.5630952637815053e+303 < (/ (+ (- (* (* 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)) < -3.8657556775396997e-184 or -0.0 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 2.5630952637815053e+303`

`if -3.8657556775396997e-184 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -0.0`

Reproduce

In
Out