Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Average Error: 7.1 → 0.4

Time: 18.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot y \le -5.992720354667754811931564301530949164485 \cdot 10^{220}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;x \cdot y - z \cdot y \le -2.396235895919902913371156822147246977413 \cdot 10^{-261}:\\ \;\;\;\;t \cdot \left(x \cdot y - z \cdot y\right)\\ \mathbf{elif}\;x \cdot y - z \cdot y \le 2.234588314730933664591068695313985424095 \cdot 10^{-234}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;x \cdot y - z \cdot y \le 4.319761620680668531396215097486412798471 \cdot 10^{234}:\\ \;\;\;\;t \cdot \left(x \cdot y - z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \left(x - z\right)\right) \cdot y\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r26291114 = x;
        double r26291115 = y;
        double r26291116 = r26291114 * r26291115;
        double r26291117 = z;
        double r26291118 = r26291117 * r26291115;
        double r26291119 = r26291116 - r26291118;
        double r26291120 = t;
        double r26291121 = r26291119 * r26291120;
        return r26291121;
}

↓

double f(double x, double y, double z, double t) {
        double r26291122 = x;
        double r26291123 = y;
        double r26291124 = r26291122 * r26291123;
        double r26291125 = z;
        double r26291126 = r26291125 * r26291123;
        double r26291127 = r26291124 - r26291126;
        double r26291128 = -5.992720354667755e+220;
        bool r26291129 = r26291127 <= r26291128;
        double r26291130 = r26291122 - r26291125;
        double r26291131 = t;
        double r26291132 = r26291123 * r26291131;
        double r26291133 = r26291130 * r26291132;
        double r26291134 = -2.396235895919903e-261;
        bool r26291135 = r26291127 <= r26291134;
        double r26291136 = r26291131 * r26291127;
        double r26291137 = 2.2345883147309337e-234;
        bool r26291138 = r26291127 <= r26291137;
        double r26291139 = 4.3197616206806685e+234;
        bool r26291140 = r26291127 <= r26291139;
        double r26291141 = r26291131 * r26291130;
        double r26291142 = r26291141 * r26291123;
        double r26291143 = r26291140 ? r26291136 : r26291142;
        double r26291144 = r26291138 ? r26291133 : r26291143;
        double r26291145 = r26291135 ? r26291136 : r26291144;
        double r26291146 = r26291129 ? r26291133 : r26291145;
        return r26291146;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.1
Target	2.9
Herbie	0.4

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

Derivation

1. Split input into 3 regimes

2. if (- (* x y) (* z y)) < -5.992720354667755e+220 or -2.396235895919903e-261 < (- (* x y) (* z y)) < 2.2345883147309337e-234

   1. Initial program 21.4

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

   2. Simplified 0.6

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

   if -5.992720354667755e+220 < (- (* x y) (* z y)) < -2.396235895919903e-261 or 2.2345883147309337e-234 < (- (* x y) (* z y)) < 4.3197616206806685e+234

   1. Initial program 0.2

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

   if 4.3197616206806685e+234 < (- (* x y) (* z y))

   1. Initial program 38.5

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

   2. Simplified 0.5

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

   4. Applied add-cube-cbrt 1.6

      \[\leadsto \color{blue}{\left(\left(\sqrt[3]{x - z} \cdot \sqrt[3]{x - z}\right) \cdot \sqrt[3]{x - z}\right)} \cdot \left(t \cdot y\right)\]

   5. Applied associate-*l* 1.6

      \[\leadsto \color{blue}{\left(\sqrt[3]{x - z} \cdot \sqrt[3]{x - z}\right) \cdot \left(\sqrt[3]{x - z} \cdot \left(t \cdot y\right)\right)}\]

   6. Taylor expanded around inf 38.5

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

   7. Simplified 0.9

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot y \le -5.992720354667754811931564301530949164485 \cdot 10^{220}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;x \cdot y - z \cdot y \le -2.396235895919902913371156822147246977413 \cdot 10^{-261}:\\ \;\;\;\;t \cdot \left(x \cdot y - z \cdot y\right)\\ \mathbf{elif}\;x \cdot y - z \cdot y \le 2.234588314730933664591068695313985424095 \cdot 10^{-234}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;x \cdot y - z \cdot y \le 4.319761620680668531396215097486412798471 \cdot 10^{234}:\\ \;\;\;\;t \cdot \left(x \cdot y - z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \left(x - z\right)\right) \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x y z t)
  :name "Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3"

  :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))