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

Average Error: 20.2 → 3.4

Time: 18.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\frac{z \cdot c}{y}}, 9, \frac{b}{z \cdot c}\right) - \frac{\left(a \cdot 4\right) \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -3.806721957017680858881585199397795276301 \cdot 10^{-283}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -0.0:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(y, x \cdot 9, b\right)}{z} - \left(a \cdot 4\right) \cdot t\right) \cdot \frac{1}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 1.814053920823523978195989056964182842962 \cdot 10^{303}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z} \cdot \frac{y}{c}, 9, \frac{b}{z \cdot c}\right) - \frac{a \cdot 4}{\frac{c}{t}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r816198 = x;
        double r816199 = 9.0;
        double r816200 = r816198 * r816199;
        double r816201 = y;
        double r816202 = r816200 * r816201;
        double r816203 = z;
        double r816204 = 4.0;
        double r816205 = r816203 * r816204;
        double r816206 = t;
        double r816207 = r816205 * r816206;
        double r816208 = a;
        double r816209 = r816207 * r816208;
        double r816210 = r816202 - r816209;
        double r816211 = b;
        double r816212 = r816210 + r816211;
        double r816213 = c;
        double r816214 = r816203 * r816213;
        double r816215 = r816212 / r816214;
        return r816215;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r816216 = x;
        double r816217 = 9.0;
        double r816218 = r816216 * r816217;
        double r816219 = y;
        double r816220 = r816218 * r816219;
        double r816221 = z;
        double r816222 = 4.0;
        double r816223 = r816221 * r816222;
        double r816224 = t;
        double r816225 = r816223 * r816224;
        double r816226 = a;
        double r816227 = r816225 * r816226;
        double r816228 = r816220 - r816227;
        double r816229 = b;
        double r816230 = r816228 + r816229;
        double r816231 = c;
        double r816232 = r816221 * r816231;
        double r816233 = r816230 / r816232;
        double r816234 = -inf.0;
        bool r816235 = r816233 <= r816234;
        double r816236 = r816232 / r816219;
        double r816237 = r816216 / r816236;
        double r816238 = r816229 / r816232;
        double r816239 = fma(r816237, r816217, r816238);
        double r816240 = r816226 * r816222;
        double r816241 = r816240 * r816224;
        double r816242 = r816241 / r816231;
        double r816243 = r816239 - r816242;
        double r816244 = -3.806721957017681e-283;
        bool r816245 = r816233 <= r816244;
        double r816246 = -0.0;
        bool r816247 = r816233 <= r816246;
        double r816248 = fma(r816219, r816218, r816229);
        double r816249 = r816248 / r816221;
        double r816250 = r816249 - r816241;
        double r816251 = 1.0;
        double r816252 = r816251 / r816231;
        double r816253 = r816250 * r816252;
        double r816254 = 1.814053920823524e+303;
        bool r816255 = r816233 <= r816254;
        double r816256 = r816216 / r816221;
        double r816257 = r816219 / r816231;
        double r816258 = r816256 * r816257;
        double r816259 = fma(r816258, r816217, r816238);
        double r816260 = r816231 / r816224;
        double r816261 = r816240 / r816260;
        double r816262 = r816259 - r816261;
        double r816263 = r816255 ? r816233 : r816262;
        double r816264 = r816247 ? r816253 : r816263;
        double r816265 = r816245 ? r816233 : r816264;
        double r816266 = r816235 ? r816243 : r816265;
        return r816266;
}

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	20.2
Target	14.6
Herbie	3.4

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

Derivation

1. Split input into 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 27.3

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

   3. Taylor expanded around 0 32.8

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

   4. Simplified 32.8

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

   6. Applied associate-/l* 15.3

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

   if -inf.0 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -3.806721957017681e-283 or -0.0 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 1.814053920823524e+303

   1. Initial program 3.7

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

   if -3.806721957017681e-283 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -0.0

   1. Initial program 33.4

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

   2. Simplified 0.6

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

   4. Applied div-inv 0.7

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

   if 1.814053920823524e+303 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))

   1. Initial program 62.9

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

   2. Simplified 28.5

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

   3. Taylor expanded around 0 32.4

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

   4. Simplified 32.5

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

   6. Applied associate-/l* 26.3

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

   8. Applied times-frac 10.3

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

4. Final simplification 3.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\frac{z \cdot c}{y}}, 9, \frac{b}{z \cdot c}\right) - \frac{\left(a \cdot 4\right) \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -3.806721957017680858881585199397795276301 \cdot 10^{-283}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -0.0:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(y, x \cdot 9, b\right)}{z} - \left(a \cdot 4\right) \cdot t\right) \cdot \frac{1}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 1.814053920823523978195989056964182842962 \cdot 10^{303}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z} \cdot \frac{y}{c}, 9, \frac{b}{z \cdot c}\right) - \frac{a \cdot 4}{\frac{c}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 +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)))