Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Average Error: 7.0 → 0.3

Time: 15.1s

Precision: 64

Math

\[\left(x \cdot y - z \cdot y\right) \cdot t\]

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot y \le -8.274186156879764240774351835854573918832 \cdot 10^{226}:\\ \;\;\;\;\left(t \cdot y\right) \cdot \left(x - z\right)\\ \mathbf{elif}\;x \cdot y - z \cdot y \le -4.819622087448949409690444355976373823692 \cdot 10^{-247}:\\ \;\;\;\;\left(x \cdot y - z \cdot y\right) \cdot t\\ \mathbf{elif}\;x \cdot y - z \cdot y \le 1.232937047876378353025506941095613328966 \cdot 10^{-242}:\\ \;\;\;\;\left(t \cdot y\right) \cdot \left(x - z\right)\\ \mathbf{elif}\;x \cdot y - z \cdot y \le 1.77758114352811529361559523864976797551 \cdot 10^{299}:\\ \;\;\;\;\left(x \cdot y - z \cdot y\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r329914 = x;
        double r329915 = y;
        double r329916 = r329914 * r329915;
        double r329917 = z;
        double r329918 = r329917 * r329915;
        double r329919 = r329916 - r329918;
        double r329920 = t;
        double r329921 = r329919 * r329920;
        return r329921;
}

↓

double f(double x, double y, double z, double t) {
        double r329922 = x;
        double r329923 = y;
        double r329924 = r329922 * r329923;
        double r329925 = z;
        double r329926 = r329925 * r329923;
        double r329927 = r329924 - r329926;
        double r329928 = -8.274186156879764e+226;
        bool r329929 = r329927 <= r329928;
        double r329930 = t;
        double r329931 = r329930 * r329923;
        double r329932 = r329922 - r329925;
        double r329933 = r329931 * r329932;
        double r329934 = -4.8196220874489494e-247;
        bool r329935 = r329927 <= r329934;
        double r329936 = r329927 * r329930;
        double r329937 = 1.2329370478763784e-242;
        bool r329938 = r329927 <= r329937;
        double r329939 = 1.7775811435281153e+299;
        bool r329940 = r329927 <= r329939;
        double r329941 = r329930 * r329932;
        double r329942 = r329923 * r329941;
        double r329943 = r329940 ? r329936 : r329942;
        double r329944 = r329938 ? r329933 : r329943;
        double r329945 = r329935 ? r329936 : r329944;
        double r329946 = r329929 ? r329933 : r329945;
        return r329946;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.0
Target	3.1
Herbie	0.3

\[\begin{array}{l} \mathbf{if}\;t \lt -9.231879582886776938073886590448747944753 \cdot 10^{-80}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(x - z\right)\\ \mathbf{elif}\;t \lt 2.543067051564877116200336808272775217995 \cdot 10^{83}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(x - z\right)\right) \cdot t\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (- (* x y) (* z y)) < -8.274186156879764e+226 or -4.8196220874489494e-247 < (- (* x y) (* z y)) < 1.2329370478763784e-242

   1. Initial program 21.6

      \[\left(x \cdot y - z \cdot y\right) \cdot t\]
   2. Using strategy rm

   3. Applied distribute-rgt-out-- 21.6

      \[\leadsto \color{blue}{\left(y \cdot \left(x - z\right)\right)} \cdot t\]

   4. Applied associate-*l* 0.6

      \[\leadsto \color{blue}{y \cdot \left(\left(x - z\right) \cdot t\right)}\]

   5. Simplified 0.6

      \[\leadsto y \cdot \color{blue}{\left(t \cdot \left(x - z\right)\right)}\]
   6. Using strategy rm

   7. Applied associate-*r* 0.5

      \[\leadsto \color{blue}{\left(y \cdot t\right) \cdot \left(x - z\right)}\]

   8. Simplified 0.5

      \[\leadsto \color{blue}{\left(t \cdot y\right)} \cdot \left(x - z\right)\]

   if -8.274186156879764e+226 < (- (* x y) (* z y)) < -4.8196220874489494e-247 or 1.2329370478763784e-242 < (- (* x y) (* z y)) < 1.7775811435281153e+299

   1. Initial program 0.2

      \[\left(x \cdot y - z \cdot y\right) \cdot t\]

   if 1.7775811435281153e+299 < (- (* x y) (* z y))

   1. Initial program 58.5

      \[\left(x \cdot y - z \cdot y\right) \cdot t\]
   2. Using strategy rm

   3. Applied distribute-rgt-out-- 58.5

      \[\leadsto \color{blue}{\left(y \cdot \left(x - z\right)\right)} \cdot t\]

   4. Applied associate-*l* 0.3

      \[\leadsto \color{blue}{y \cdot \left(\left(x - z\right) \cdot t\right)}\]

   5. Simplified 0.3

      \[\leadsto y \cdot \color{blue}{\left(t \cdot \left(x - z\right)\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot y \le -8.274186156879764240774351835854573918832 \cdot 10^{226}:\\ \;\;\;\;\left(t \cdot y\right) \cdot \left(x - z\right)\\ \mathbf{elif}\;x \cdot y - z \cdot y \le -4.819622087448949409690444355976373823692 \cdot 10^{-247}:\\ \;\;\;\;\left(x \cdot y - z \cdot y\right) \cdot t\\ \mathbf{elif}\;x \cdot y - z \cdot y \le 1.232937047876378353025506941095613328966 \cdot 10^{-242}:\\ \;\;\;\;\left(t \cdot y\right) \cdot \left(x - z\right)\\ \mathbf{elif}\;x \cdot y - z \cdot y \le 1.77758114352811529361559523864976797551 \cdot 10^{299}:\\ \;\;\;\;\left(x \cdot y - z \cdot y\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (x y z t)
  :name "Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< t -9.231879582886777e-80) (* (* y t) (- x z)) (if (< t 2.543067051564877e+83) (* y (* t (- x z))) (* (* y (- x z)) t)))

  (* (- (* x y) (* z y)) t))