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

Average Error: 20.6 → 4.4

Time: 19.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\frac{z \cdot c}{y}}, 9, \frac{b}{z \cdot c}\right) - 4 \cdot \left(t \cdot \frac{a}{c}\right)\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -1.375150000214844609946251739328002992128 \cdot 10^{-252}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 0.0:\\ \;\;\;\;\left(\sqrt[3]{\frac{\mathsf{fma}\left(y, x \cdot 9, b\right)}{z} - \left(a \cdot 4\right) \cdot t} \cdot \sqrt[3]{\frac{\mathsf{fma}\left(y, x \cdot 9, b\right)}{z} - \left(a \cdot 4\right) \cdot t}\right) \cdot \frac{\sqrt[3]{\frac{\mathsf{fma}\left(y, x \cdot 9, b\right)}{z} - \left(a \cdot 4\right) \cdot t}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 2.230748167825543970616207741182955919334 \cdot 10^{274}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\frac{z \cdot c}{y}}, 9, \frac{b}{z \cdot c}\right) - 4 \cdot \left(t \cdot \frac{a}{c}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r930480 = x;
        double r930481 = 9.0;
        double r930482 = r930480 * r930481;
        double r930483 = y;
        double r930484 = r930482 * r930483;
        double r930485 = z;
        double r930486 = 4.0;
        double r930487 = r930485 * r930486;
        double r930488 = t;
        double r930489 = r930487 * r930488;
        double r930490 = a;
        double r930491 = r930489 * r930490;
        double r930492 = r930484 - r930491;
        double r930493 = b;
        double r930494 = r930492 + r930493;
        double r930495 = c;
        double r930496 = r930485 * r930495;
        double r930497 = r930494 / r930496;
        return r930497;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r930498 = x;
        double r930499 = 9.0;
        double r930500 = r930498 * r930499;
        double r930501 = y;
        double r930502 = r930500 * r930501;
        double r930503 = z;
        double r930504 = 4.0;
        double r930505 = r930503 * r930504;
        double r930506 = t;
        double r930507 = r930505 * r930506;
        double r930508 = a;
        double r930509 = r930507 * r930508;
        double r930510 = r930502 - r930509;
        double r930511 = b;
        double r930512 = r930510 + r930511;
        double r930513 = c;
        double r930514 = r930503 * r930513;
        double r930515 = r930512 / r930514;
        double r930516 = -inf.0;
        bool r930517 = r930515 <= r930516;
        double r930518 = r930514 / r930501;
        double r930519 = r930498 / r930518;
        double r930520 = r930511 / r930514;
        double r930521 = fma(r930519, r930499, r930520);
        double r930522 = r930508 / r930513;
        double r930523 = r930506 * r930522;
        double r930524 = r930504 * r930523;
        double r930525 = r930521 - r930524;
        double r930526 = -1.3751500002148446e-252;
        bool r930527 = r930515 <= r930526;
        double r930528 = 0.0;
        bool r930529 = r930515 <= r930528;
        double r930530 = fma(r930501, r930500, r930511);
        double r930531 = r930530 / r930503;
        double r930532 = r930508 * r930504;
        double r930533 = r930532 * r930506;
        double r930534 = r930531 - r930533;
        double r930535 = cbrt(r930534);
        double r930536 = r930535 * r930535;
        double r930537 = r930535 / r930513;
        double r930538 = r930536 * r930537;
        double r930539 = 2.230748167825544e+274;
        bool r930540 = r930515 <= r930539;
        double r930541 = r930540 ? r930515 : r930525;
        double r930542 = r930529 ? r930538 : r930541;
        double r930543 = r930527 ? r930515 : r930542;
        double r930544 = r930517 ? r930525 : r930543;
        return r930544;
}

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	20.6
Target	14.6
Herbie	4.4

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

Derivation

1. Split input into 3 regimes

2. if (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -inf.0 or 2.230748167825544e+274 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))

   1. Initial program 59.0

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

   2. Simplified 26.1

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

   3. Taylor expanded around 0 28.7

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

   4. Simplified 28.7

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

   6. Applied associate-/l* 19.4

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

   8. Applied *-un-lft-identity 19.4

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

   9. Applied times-frac 14.0

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

   10. Simplified 14.0

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

   if -inf.0 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -1.3751500002148446e-252 or 0.0 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 2.230748167825544e+274

   1. Initial program 3.8

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

   if -1.3751500002148446e-252 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 0.0

   1. Initial program 34.3

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

   2. Simplified 0.5

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

   4. Applied *-un-lft-identity 0.5

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

   5. Applied add-cube-cbrt 1.2

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

   6. Applied times-frac 1.2

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

   7. Simplified 1.2

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

4. Final simplification 4.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\frac{z \cdot c}{y}}, 9, \frac{b}{z \cdot c}\right) - 4 \cdot \left(t \cdot \frac{a}{c}\right)\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -1.375150000214844609946251739328002992128 \cdot 10^{-252}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 0.0:\\ \;\;\;\;\left(\sqrt[3]{\frac{\mathsf{fma}\left(y, x \cdot 9, b\right)}{z} - \left(a \cdot 4\right) \cdot t} \cdot \sqrt[3]{\frac{\mathsf{fma}\left(y, x \cdot 9, b\right)}{z} - \left(a \cdot 4\right) \cdot t}\right) \cdot \frac{\sqrt[3]{\frac{\mathsf{fma}\left(y, x \cdot 9, b\right)}{z} - \left(a \cdot 4\right) \cdot t}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 2.230748167825543970616207741182955919334 \cdot 10^{274}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\frac{z \cdot c}{y}}, 9, \frac{b}{z \cdot c}\right) - 4 \cdot \left(t \cdot \frac{a}{c}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 +o rules:numerics
(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)))