Average Error: 20.6 → 10.7
Time: 16.5s
Precision: 64
\[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -973061476927037835836472021745664:\\ \;\;\;\;\left(\frac{1}{z} \cdot \frac{b}{c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{elif}\;t \le -2.226563407147601316749883762410333966005 \cdot 10^{-92}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{\sqrt[3]{x}}{\frac{z}{\sqrt[3]{x}}} \cdot \frac{\sqrt[3]{x}}{\frac{c}{y}}\right)\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;t \le -6.830485081088784589874484482755074534336 \cdot 10^{-153}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \left(\frac{a}{c} \cdot t\right)\\ \mathbf{elif}\;t \le 6.013601878928024270390563910015845543276 \cdot 10^{-223}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{z \cdot \frac{c}{y}}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;t \le 1.317316851856465580338075111163575233918 \cdot 10^{-130}:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;t \le 1.015852938385750261467669937211472798025 \cdot 10^{-51}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{z \cdot \frac{c}{y}}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{z} \cdot \frac{b}{c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{a}{\frac{c}{t}}\\ \end{array}\]
\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 -973061476927037835836472021745664:\\
\;\;\;\;\left(\frac{1}{z} \cdot \frac{b}{c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{a}{\frac{c}{t}}\\

\mathbf{elif}\;t \le -2.226563407147601316749883762410333966005 \cdot 10^{-92}:\\
\;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{\sqrt[3]{x}}{\frac{z}{\sqrt[3]{x}}} \cdot \frac{\sqrt[3]{x}}{\frac{c}{y}}\right)\right) - 4 \cdot \frac{a \cdot t}{c}\\

\mathbf{elif}\;t \le -6.830485081088784589874484482755074534336 \cdot 10^{-153}:\\
\;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \left(\frac{a}{c} \cdot t\right)\\

\mathbf{elif}\;t \le 6.013601878928024270390563910015845543276 \cdot 10^{-223}:\\
\;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{z \cdot \frac{c}{y}}\right) - 4 \cdot \frac{a \cdot t}{c}\\

\mathbf{elif}\;t \le 1.317316851856465580338075111163575233918 \cdot 10^{-130}:\\
\;\;\;\;\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 1.015852938385750261467669937211472798025 \cdot 10^{-51}:\\
\;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{z \cdot \frac{c}{y}}\right) - 4 \cdot \frac{a \cdot t}{c}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{z} \cdot \frac{b}{c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{a}{\frac{c}{t}}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r584170 = x;
        double r584171 = 9.0;
        double r584172 = r584170 * r584171;
        double r584173 = y;
        double r584174 = r584172 * r584173;
        double r584175 = z;
        double r584176 = 4.0;
        double r584177 = r584175 * r584176;
        double r584178 = t;
        double r584179 = r584177 * r584178;
        double r584180 = a;
        double r584181 = r584179 * r584180;
        double r584182 = r584174 - r584181;
        double r584183 = b;
        double r584184 = r584182 + r584183;
        double r584185 = c;
        double r584186 = r584175 * r584185;
        double r584187 = r584184 / r584186;
        return r584187;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r584188 = t;
        double r584189 = -9.730614769270378e+32;
        bool r584190 = r584188 <= r584189;
        double r584191 = 1.0;
        double r584192 = z;
        double r584193 = r584191 / r584192;
        double r584194 = b;
        double r584195 = c;
        double r584196 = r584194 / r584195;
        double r584197 = r584193 * r584196;
        double r584198 = 9.0;
        double r584199 = x;
        double r584200 = y;
        double r584201 = r584199 * r584200;
        double r584202 = r584192 * r584195;
        double r584203 = r584201 / r584202;
        double r584204 = r584198 * r584203;
        double r584205 = r584197 + r584204;
        double r584206 = 4.0;
        double r584207 = a;
        double r584208 = r584195 / r584188;
        double r584209 = r584207 / r584208;
        double r584210 = r584206 * r584209;
        double r584211 = r584205 - r584210;
        double r584212 = -2.2265634071476013e-92;
        bool r584213 = r584188 <= r584212;
        double r584214 = r584194 / r584202;
        double r584215 = cbrt(r584199);
        double r584216 = r584192 / r584215;
        double r584217 = r584215 / r584216;
        double r584218 = r584195 / r584200;
        double r584219 = r584215 / r584218;
        double r584220 = r584217 * r584219;
        double r584221 = r584198 * r584220;
        double r584222 = r584214 + r584221;
        double r584223 = r584207 * r584188;
        double r584224 = r584223 / r584195;
        double r584225 = r584206 * r584224;
        double r584226 = r584222 - r584225;
        double r584227 = -6.830485081088785e-153;
        bool r584228 = r584188 <= r584227;
        double r584229 = r584214 + r584204;
        double r584230 = r584207 / r584195;
        double r584231 = r584230 * r584188;
        double r584232 = r584206 * r584231;
        double r584233 = r584229 - r584232;
        double r584234 = 6.013601878928024e-223;
        bool r584235 = r584188 <= r584234;
        double r584236 = r584192 * r584218;
        double r584237 = r584199 / r584236;
        double r584238 = r584198 * r584237;
        double r584239 = r584214 + r584238;
        double r584240 = r584239 - r584225;
        double r584241 = 1.3173168518564656e-130;
        bool r584242 = r584188 <= r584241;
        double r584243 = r584199 * r584198;
        double r584244 = r584243 * r584200;
        double r584245 = r584192 * r584206;
        double r584246 = r584245 * r584188;
        double r584247 = r584246 * r584207;
        double r584248 = r584244 - r584247;
        double r584249 = r584248 + r584194;
        double r584250 = r584249 / r584192;
        double r584251 = r584250 / r584195;
        double r584252 = 1.0158529383857503e-51;
        bool r584253 = r584188 <= r584252;
        double r584254 = r584253 ? r584240 : r584211;
        double r584255 = r584242 ? r584251 : r584254;
        double r584256 = r584235 ? r584240 : r584255;
        double r584257 = r584228 ? r584233 : r584256;
        double r584258 = r584213 ? r584226 : r584257;
        double r584259 = r584190 ? r584211 : r584258;
        return r584259;
}

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

Original20.6
Target14.5
Herbie10.7
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -1.100156740804104887233830094663413900721 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -0.0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.170887791174748819600820354912645756062 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 2.876823679546137226963937101710277849382 \cdot 10^{130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.383851504245631860711731716196098366993 \cdot 10^{158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if t < -9.730614769270378e+32 or 1.0158529383857503e-51 < t

    1. Initial program 28.7

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

    2. Taylor expanded around 0 14.9

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

    4. Applied associate-/l*11.0

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

    6. Applied *-un-lft-identity11.0

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

    7. Applied times-frac11.2

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

    if -9.730614769270378e+32 < t < -2.2265634071476013e-92

    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. Taylor expanded around 0 8.4

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

    4. Applied associate-/l*7.6

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

    6. Applied *-un-lft-identity7.6

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

    7. Applied times-frac8.5

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

    8. Applied add-cube-cbrt8.8

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\frac{z}{1} \cdot \frac{c}{y}}\right) - 4 \cdot \frac{a \cdot t}{c}\]

    9. Applied times-frac9.2

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \color{blue}{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{z}{1}} \cdot \frac{\sqrt[3]{x}}{\frac{c}{y}}\right)}\right) - 4 \cdot \frac{a \cdot t}{c}\]

    10. Simplified9.2

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

    if -2.2265634071476013e-92 < t < -6.830485081088785e-153

    1. Initial program 14.3

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

    2. Taylor expanded around 0 8.0

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

    4. Applied associate-/l*10.8

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

    6. Applied associate-/r/11.0

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

    if -6.830485081088785e-153 < t < 6.013601878928024e-223 or 1.3173168518564656e-130 < t < 1.0158529383857503e-51

    1. Initial program 12.3

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

    2. Taylor expanded around 0 8.9

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

    4. Applied associate-/l*9.1

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

    6. Applied *-un-lft-identity9.1

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

    7. Applied times-frac10.0

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

    8. Simplified10.0

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

    if 6.013601878928024e-223 < t < 1.3173168518564656e-130

    1. Initial program 14.3

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

    3. Applied associate-/r*11.9

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

  4. Final simplification10.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -973061476927037835836472021745664:\\ \;\;\;\;\left(\frac{1}{z} \cdot \frac{b}{c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{elif}\;t \le -2.226563407147601316749883762410333966005 \cdot 10^{-92}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{\sqrt[3]{x}}{\frac{z}{\sqrt[3]{x}}} \cdot \frac{\sqrt[3]{x}}{\frac{c}{y}}\right)\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;t \le -6.830485081088784589874484482755074534336 \cdot 10^{-153}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \left(\frac{a}{c} \cdot t\right)\\ \mathbf{elif}\;t \le 6.013601878928024270390563910015845543276 \cdot 10^{-223}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{z \cdot \frac{c}{y}}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;t \le 1.317316851856465580338075111163575233918 \cdot 10^{-130}:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;t \le 1.015852938385750261467669937211472798025 \cdot 10^{-51}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{z \cdot \frac{c}{y}}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{z} \cdot \frac{b}{c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{a}{\frac{c}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019294 
(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)) -1.1001567408041049e-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)) 1.17088779117474882e-53) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 2.8768236795461372e130) (- (+ (* (* 9 (/ y c)) (/ x z)) (/ b (* c z))) (* 4 (/ (* a t) c))) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 1.3838515042456319e158) (/ (+ (- (* (* 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)))