Average Error: 20.4 → 10.5
Time: 5.1s
Precision: 64
\[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(x \cdot 9\right) \cdot y \le -1.01280399873340739 \cdot 10^{-88}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \left(\left(a \cdot t\right) \cdot \frac{1}{c}\right)\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \le -9.44653292186014332 \cdot 10^{-285}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \le 1.80429930440544056 \cdot 10^{-135}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{x}{z \cdot c} \cdot y\right)\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \le 2.50481645443219275 \cdot 10^{244}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{1}{\frac{c}{a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{\frac{x}{z}}{\frac{c}{y}}\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \end{array}\]
\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(x \cdot 9\right) \cdot y \le -1.01280399873340739 \cdot 10^{-88}:\\
\;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \left(\left(a \cdot t\right) \cdot \frac{1}{c}\right)\\

\mathbf{elif}\;\left(x \cdot 9\right) \cdot y \le -9.44653292186014332 \cdot 10^{-285}:\\
\;\;\;\;\frac{1}{z} \cdot \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}\\

\mathbf{elif}\;\left(x \cdot 9\right) \cdot y \le 1.80429930440544056 \cdot 10^{-135}:\\
\;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{x}{z \cdot c} \cdot y\right)\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\

\mathbf{elif}\;\left(x \cdot 9\right) \cdot y \le 2.50481645443219275 \cdot 10^{244}:\\
\;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{1}{\frac{c}{a \cdot t}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r766237 = x;
        double r766238 = 9.0;
        double r766239 = r766237 * r766238;
        double r766240 = y;
        double r766241 = r766239 * r766240;
        double r766242 = z;
        double r766243 = 4.0;
        double r766244 = r766242 * r766243;
        double r766245 = t;
        double r766246 = r766244 * r766245;
        double r766247 = a;
        double r766248 = r766246 * r766247;
        double r766249 = r766241 - r766248;
        double r766250 = b;
        double r766251 = r766249 + r766250;
        double r766252 = c;
        double r766253 = r766242 * r766252;
        double r766254 = r766251 / r766253;
        return r766254;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r766255 = x;
        double r766256 = 9.0;
        double r766257 = r766255 * r766256;
        double r766258 = y;
        double r766259 = r766257 * r766258;
        double r766260 = -1.0128039987334074e-88;
        bool r766261 = r766259 <= r766260;
        double r766262 = b;
        double r766263 = z;
        double r766264 = c;
        double r766265 = r766263 * r766264;
        double r766266 = r766262 / r766265;
        double r766267 = r766265 / r766258;
        double r766268 = r766255 / r766267;
        double r766269 = r766256 * r766268;
        double r766270 = r766266 + r766269;
        double r766271 = 4.0;
        double r766272 = a;
        double r766273 = t;
        double r766274 = r766272 * r766273;
        double r766275 = 1.0;
        double r766276 = r766275 / r766264;
        double r766277 = r766274 * r766276;
        double r766278 = r766271 * r766277;
        double r766279 = r766270 - r766278;
        double r766280 = -9.446532921860143e-285;
        bool r766281 = r766259 <= r766280;
        double r766282 = r766275 / r766263;
        double r766283 = r766263 * r766271;
        double r766284 = r766283 * r766273;
        double r766285 = r766284 * r766272;
        double r766286 = r766259 - r766285;
        double r766287 = r766286 + r766262;
        double r766288 = r766287 / r766264;
        double r766289 = r766282 * r766288;
        double r766290 = 1.8042993044054406e-135;
        bool r766291 = r766259 <= r766290;
        double r766292 = r766255 / r766265;
        double r766293 = r766292 * r766258;
        double r766294 = r766256 * r766293;
        double r766295 = r766266 + r766294;
        double r766296 = r766273 / r766264;
        double r766297 = r766272 * r766296;
        double r766298 = r766271 * r766297;
        double r766299 = r766295 - r766298;
        double r766300 = 2.5048164544321928e+244;
        bool r766301 = r766259 <= r766300;
        double r766302 = r766255 * r766258;
        double r766303 = r766302 / r766265;
        double r766304 = r766256 * r766303;
        double r766305 = r766266 + r766304;
        double r766306 = r766264 / r766274;
        double r766307 = r766275 / r766306;
        double r766308 = r766271 * r766307;
        double r766309 = r766305 - r766308;
        double r766310 = r766255 / r766263;
        double r766311 = r766264 / r766258;
        double r766312 = r766310 / r766311;
        double r766313 = r766256 * r766312;
        double r766314 = r766266 + r766313;
        double r766315 = r766314 - r766298;
        double r766316 = r766301 ? r766309 : r766315;
        double r766317 = r766291 ? r766299 : r766316;
        double r766318 = r766281 ? r766289 : r766317;
        double r766319 = r766261 ? r766279 : r766318;
        return r766319;
}

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

Original20.4
Target14.3
Herbie10.5
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -1.10015674080410512 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -0.0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.17088779117474882 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 2.8768236795461372 \cdot 10^{130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.3838515042456319 \cdot 10^{158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if (* (* x 9.0) y) < -1.0128039987334074e-88

    1. Initial program 23.2

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

    2. Taylor expanded around 0 15.2

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

    4. Applied associate-/l*13.0

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

    6. Applied div-inv13.0

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

    if -1.0128039987334074e-88 < (* (* x 9.0) y) < -9.446532921860143e-285

    1. Initial program 15.7

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

    3. Applied *-un-lft-identity15.7

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

    4. Applied times-frac15.9

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

    if -9.446532921860143e-285 < (* (* x 9.0) y) < 1.8042993044054406e-135

    1. Initial program 17.0

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

    2. Taylor expanded around 0 8.2

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

    4. Applied associate-/l*8.3

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

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

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

    7. Applied times-frac8.6

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

    8. Simplified8.6

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

    10. Applied associate-/r/8.6

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

    if 1.8042993044054406e-135 < (* (* x 9.0) y) < 2.5048164544321928e+244

    1. Initial program 17.4

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

    2. Taylor expanded around 0 7.4

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

    4. Applied clear-num7.5

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

    if 2.5048164544321928e+244 < (* (* x 9.0) y)

    1. Initial program 45.8

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

    2. Taylor expanded around 0 42.1

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

    4. Applied associate-/l*18.0

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

    6. Applied *-un-lft-identity18.0

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

    7. Applied times-frac15.0

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

    8. Simplified15.0

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

    10. Applied *-un-lft-identity15.0

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

    11. Applied times-frac6.8

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

    12. Applied associate-/r*9.0

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

    13. Simplified9.0

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot 9\right) \cdot y \le -1.01280399873340739 \cdot 10^{-88}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \left(\left(a \cdot t\right) \cdot \frac{1}{c}\right)\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \le -9.44653292186014332 \cdot 10^{-285}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \le 1.80429930440544056 \cdot 10^{-135}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{x}{z \cdot c} \cdot y\right)\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{elif}\;\left(x \cdot 9\right) \cdot y \le 2.50481645443219275 \cdot 10^{244}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{1}{\frac{c}{a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{\frac{x}{z}}{\frac{c}{y}}\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020003 
(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)))