Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, J

Average Error: 20.7 → 3.5

Time: 27.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} = -\infty:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \frac{t}{\frac{c}{a}}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -4.8105757363233579 \cdot 10^{-275}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 0.0:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y + b\right) \cdot \frac{1}{z} - \left(a \cdot 4\right) \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 4.18129635218716863 \cdot 10^{304}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + \left(\left(9 \cdot \frac{x}{z}\right) \cdot \left(\sqrt[3]{\frac{y}{c}} \cdot \sqrt[3]{\frac{y}{c}}\right)\right) \cdot \sqrt[3]{\frac{y}{c}}\right) - 4 \cdot \frac{t}{\frac{c}{a}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r539188 = x;
        double r539189 = 9.0;
        double r539190 = r539188 * r539189;
        double r539191 = y;
        double r539192 = r539190 * r539191;
        double r539193 = z;
        double r539194 = 4.0;
        double r539195 = r539193 * r539194;
        double r539196 = t;
        double r539197 = r539195 * r539196;
        double r539198 = a;
        double r539199 = r539197 * r539198;
        double r539200 = r539192 - r539199;
        double r539201 = b;
        double r539202 = r539200 + r539201;
        double r539203 = c;
        double r539204 = r539193 * r539203;
        double r539205 = r539202 / r539204;
        return r539205;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r539206 = x;
        double r539207 = 9.0;
        double r539208 = r539206 * r539207;
        double r539209 = y;
        double r539210 = r539208 * r539209;
        double r539211 = z;
        double r539212 = 4.0;
        double r539213 = r539211 * r539212;
        double r539214 = t;
        double r539215 = r539213 * r539214;
        double r539216 = a;
        double r539217 = r539215 * r539216;
        double r539218 = r539210 - r539217;
        double r539219 = b;
        double r539220 = r539218 + r539219;
        double r539221 = c;
        double r539222 = r539211 * r539221;
        double r539223 = r539220 / r539222;
        double r539224 = -inf.0;
        bool r539225 = r539223 <= r539224;
        double r539226 = r539219 / r539222;
        double r539227 = r539222 / r539209;
        double r539228 = r539206 / r539227;
        double r539229 = r539207 * r539228;
        double r539230 = r539226 + r539229;
        double r539231 = r539221 / r539216;
        double r539232 = r539214 / r539231;
        double r539233 = r539212 * r539232;
        double r539234 = r539230 - r539233;
        double r539235 = -4.810575736323358e-275;
        bool r539236 = r539223 <= r539235;
        double r539237 = 0.0;
        bool r539238 = r539223 <= r539237;
        double r539239 = r539210 + r539219;
        double r539240 = 1.0;
        double r539241 = r539240 / r539211;
        double r539242 = r539239 * r539241;
        double r539243 = r539216 * r539212;
        double r539244 = r539243 * r539214;
        double r539245 = r539242 - r539244;
        double r539246 = r539245 / r539221;
        double r539247 = 4.1812963521871686e+304;
        bool r539248 = r539223 <= r539247;
        double r539249 = r539206 / r539211;
        double r539250 = r539207 * r539249;
        double r539251 = r539209 / r539221;
        double r539252 = cbrt(r539251);
        double r539253 = r539252 * r539252;
        double r539254 = r539250 * r539253;
        double r539255 = r539254 * r539252;
        double r539256 = r539226 + r539255;
        double r539257 = r539256 - r539233;
        double r539258 = r539248 ? r539223 : r539257;
        double r539259 = r539238 ? r539246 : r539258;
        double r539260 = r539236 ? r539223 : r539259;
        double r539261 = r539225 ? r539234 : r539260;
        return r539261;
}

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

Target

Original	20.7
Target	15.0
Herbie	3.5

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -1.1001567408041049 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -0.0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.17088779117474882 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 2.8768236795461372 \cdot 10^{130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.3838515042456319 \cdot 10^{158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 25.5

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

   3. Taylor expanded around 0 32.3

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

   5. Applied associate-/l* 27.2

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

   7. Applied associate-/l* 10.0

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

   if -inf.0 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -4.810575736323358e-275 or 0.0 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 4.1812963521871686e+304

   1. Initial program 3.6

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

   if -4.810575736323358e-275 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 0.0

   1. Initial program 35.6

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

   2. Simplified 0.8

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

   4. Applied div-inv 0.8

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

   if 4.1812963521871686e+304 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))

   1. Initial program 63.4

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

   2. Simplified 27.4

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

   3. Taylor expanded around 0 31.1

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

   5. Applied associate-/l* 25.9

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

   7. Applied times-frac 12.3

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

   8. Applied associate-*r* 12.3

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

   10. Applied add-cube-cbrt 12.5

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

   11. Applied associate-*r* 12.5

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

4. Final simplification 3.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} = -\infty:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \frac{t}{\frac{c}{a}}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -4.8105757363233579 \cdot 10^{-275}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 0.0:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y + b\right) \cdot \frac{1}{z} - \left(a \cdot 4\right) \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 4.18129635218716863 \cdot 10^{304}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + \left(\left(9 \cdot \frac{x}{z}\right) \cdot \left(\sqrt[3]{\frac{y}{c}} \cdot \sqrt[3]{\frac{y}{c}}\right)\right) \cdot \sqrt[3]{\frac{y}{c}}\right) - 4 \cdot \frac{t}{\frac{c}{a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 
(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)))