Average Error: 20.8 → 10.8
Time: 5.4s
Precision: 64
\[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -6.36732518833848025 \cdot 10^{-192}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{1}{z} \cdot \frac{x}{\frac{c}{y}}\right)\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{elif}\;t \le 3.507527873864621 \cdot 10^{-285}:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;t \le 7.9555437739864164 \cdot 10^{-71}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{x}{z \cdot c} \cdot y\right)\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{elif}\;t \le 2.24947951631066874 \cdot 10^{61}:\\ \;\;\;\;\left(\frac{\sqrt[3]{b} \cdot \sqrt[3]{b}}{z} \cdot \frac{\sqrt[3]{b}}{c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{1}{z} \cdot \frac{x}{\frac{c}{y}}\right)\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \end{array}\]
\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}\;t \le -6.36732518833848025 \cdot 10^{-192}:\\
\;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{1}{z} \cdot \frac{x}{\frac{c}{y}}\right)\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\

\mathbf{elif}\;t \le 3.507527873864621 \cdot 10^{-285}:\\
\;\;\;\;\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}\;t \le 7.9555437739864164 \cdot 10^{-71}:\\
\;\;\;\;\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}\;t \le 2.24947951631066874 \cdot 10^{61}:\\
\;\;\;\;\left(\frac{\sqrt[3]{b} \cdot \sqrt[3]{b}}{z} \cdot \frac{\sqrt[3]{b}}{c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{a \cdot t}{c}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{1}{z} \cdot \frac{x}{\frac{c}{y}}\right)\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 r837246 = x;
        double r837247 = 9.0;
        double r837248 = r837246 * r837247;
        double r837249 = y;
        double r837250 = r837248 * r837249;
        double r837251 = z;
        double r837252 = 4.0;
        double r837253 = r837251 * r837252;
        double r837254 = t;
        double r837255 = r837253 * r837254;
        double r837256 = a;
        double r837257 = r837255 * r837256;
        double r837258 = r837250 - r837257;
        double r837259 = b;
        double r837260 = r837258 + r837259;
        double r837261 = c;
        double r837262 = r837251 * r837261;
        double r837263 = r837260 / r837262;
        return r837263;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r837264 = t;
        double r837265 = -6.36732518833848e-192;
        bool r837266 = r837264 <= r837265;
        double r837267 = b;
        double r837268 = z;
        double r837269 = c;
        double r837270 = r837268 * r837269;
        double r837271 = r837267 / r837270;
        double r837272 = 9.0;
        double r837273 = 1.0;
        double r837274 = r837273 / r837268;
        double r837275 = x;
        double r837276 = y;
        double r837277 = r837269 / r837276;
        double r837278 = r837275 / r837277;
        double r837279 = r837274 * r837278;
        double r837280 = r837272 * r837279;
        double r837281 = r837271 + r837280;
        double r837282 = 4.0;
        double r837283 = a;
        double r837284 = r837264 / r837269;
        double r837285 = r837283 * r837284;
        double r837286 = r837282 * r837285;
        double r837287 = r837281 - r837286;
        double r837288 = 3.507527873864621e-285;
        bool r837289 = r837264 <= r837288;
        double r837290 = r837275 * r837272;
        double r837291 = r837290 * r837276;
        double r837292 = r837268 * r837282;
        double r837293 = r837292 * r837264;
        double r837294 = r837293 * r837283;
        double r837295 = r837291 - r837294;
        double r837296 = r837295 + r837267;
        double r837297 = r837296 / r837268;
        double r837298 = r837297 / r837269;
        double r837299 = 7.955543773986416e-71;
        bool r837300 = r837264 <= r837299;
        double r837301 = r837275 / r837270;
        double r837302 = r837301 * r837276;
        double r837303 = r837272 * r837302;
        double r837304 = r837271 + r837303;
        double r837305 = r837304 - r837286;
        double r837306 = 2.2494795163106687e+61;
        bool r837307 = r837264 <= r837306;
        double r837308 = cbrt(r837267);
        double r837309 = r837308 * r837308;
        double r837310 = r837309 / r837268;
        double r837311 = r837308 / r837269;
        double r837312 = r837310 * r837311;
        double r837313 = r837275 * r837276;
        double r837314 = r837313 / r837270;
        double r837315 = r837272 * r837314;
        double r837316 = r837312 + r837315;
        double r837317 = r837283 * r837264;
        double r837318 = r837317 / r837269;
        double r837319 = r837282 * r837318;
        double r837320 = r837316 - r837319;
        double r837321 = r837307 ? r837320 : r837287;
        double r837322 = r837300 ? r837305 : r837321;
        double r837323 = r837289 ? r837298 : r837322;
        double r837324 = r837266 ? r837287 : r837323;
        return r837324;
}

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
Herbie10.8
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -1.10015674080410512 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -0.0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.17088779117474882 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 2.8768236795461372 \cdot 10^{130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.3838515042456319 \cdot 10^{158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if t < -6.36732518833848e-192 or 2.2494795163106687e+61 < t

    1. Initial program 25.4

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

    2. Taylor expanded around 0 14.0

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

    4. Applied *-un-lft-identity14.0

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

    5. Applied times-frac11.7

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

    6. Simplified11.7

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

    8. Applied associate-/l*9.7

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

    10. Applied *-un-lft-identity9.7

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

    11. Applied times-frac10.0

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

    12. Applied *-un-lft-identity10.0

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

    13. Applied times-frac10.6

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

    14. Simplified10.6

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

    if -6.36732518833848e-192 < t < 3.507527873864621e-285

    1. Initial program 12.6

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

    3. Applied associate-/r*10.9

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

    if 3.507527873864621e-285 < t < 7.955543773986416e-71

    1. Initial program 13.7

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

    2. Taylor expanded around 0 10.1

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

    4. Applied *-un-lft-identity10.1

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

    5. Applied times-frac12.2

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

    6. Simplified12.2

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

    8. Applied associate-/l*11.4

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

    10. Applied associate-/r/12.2

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

    if 7.955543773986416e-71 < t < 2.2494795163106687e+61

    1. Initial program 18.8

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

    2. Taylor expanded around 0 9.6

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

    4. Applied add-cube-cbrt9.9

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

    5. Applied times-frac9.5

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

  4. Final simplification10.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -6.36732518833848025 \cdot 10^{-192}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{1}{z} \cdot \frac{x}{\frac{c}{y}}\right)\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{elif}\;t \le 3.507527873864621 \cdot 10^{-285}:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;t \le 7.9555437739864164 \cdot 10^{-71}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{x}{z \cdot c} \cdot y\right)\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{elif}\;t \le 2.24947951631066874 \cdot 10^{61}:\\ \;\;\;\;\left(\frac{\sqrt[3]{b} \cdot \sqrt[3]{b}}{z} \cdot \frac{\sqrt[3]{b}}{c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{1}{z} \cdot \frac{x}{\frac{c}{y}}\right)\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(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)))