Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Error: 6.8 → 2.1

Time: 10.5s

Precision: 64

Math

\[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -49355.0736941491268:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y - t}{2}}\\ \mathbf{elif}\;z \le 1.74137454924600484 \cdot 10^{79}:\\ \;\;\;\;1 \cdot \frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x}}{\frac{z}{\sqrt[3]{x}}} \cdot \frac{\sqrt[3]{x}}{\frac{y - t}{2}}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r594117 = x;
        double r594118 = 2.0;
        double r594119 = r594117 * r594118;
        double r594120 = y;
        double r594121 = z;
        double r594122 = r594120 * r594121;
        double r594123 = t;
        double r594124 = r594123 * r594121;
        double r594125 = r594122 - r594124;
        double r594126 = r594119 / r594125;
        return r594126;
}

↓

double f(double x, double y, double z, double t) {
        double r594127 = z;
        double r594128 = -49355.07369414913;
        bool r594129 = r594127 <= r594128;
        double r594130 = x;
        double r594131 = r594130 / r594127;
        double r594132 = y;
        double r594133 = t;
        double r594134 = r594132 - r594133;
        double r594135 = 2.0;
        double r594136 = r594134 / r594135;
        double r594137 = r594131 / r594136;
        double r594138 = 1.741374549246005e+79;
        bool r594139 = r594127 <= r594138;
        double r594140 = 1.0;
        double r594141 = r594127 * r594134;
        double r594142 = r594141 / r594135;
        double r594143 = r594130 / r594142;
        double r594144 = r594140 * r594143;
        double r594145 = cbrt(r594130);
        double r594146 = r594127 / r594145;
        double r594147 = r594145 / r594146;
        double r594148 = r594145 / r594136;
        double r594149 = r594147 * r594148;
        double r594150 = r594139 ? r594144 : r594149;
        double r594151 = r594129 ? r594137 : r594150;
        return r594151;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.8
Target	2.2
Herbie	2.1

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt 1.04502782733012586 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if z < -49355.07369414913

   1. Initial program 11.2

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]

   2. Simplified 9.2

      \[\leadsto \color{blue}{\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 9.2

      \[\leadsto \frac{x}{\frac{z \cdot \left(y - t\right)}{\color{blue}{1 \cdot 2}}}\]

   5. Applied times-frac 9.2

      \[\leadsto \frac{x}{\color{blue}{\frac{z}{1} \cdot \frac{y - t}{2}}}\]

   6. Applied associate-/r* 1.7

      \[\leadsto \color{blue}{\frac{\frac{x}{\frac{z}{1}}}{\frac{y - t}{2}}}\]

   7. Simplified 1.7

      \[\leadsto \frac{\color{blue}{\frac{x}{z}}}{\frac{y - t}{2}}\]

   if -49355.07369414913 < z < 1.741374549246005e+79

   1. Initial program 2.4

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]

   2. Simplified 2.4

      \[\leadsto \color{blue}{\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 2.4

      \[\leadsto \frac{x}{\color{blue}{1 \cdot \frac{z \cdot \left(y - t\right)}{2}}}\]

   5. Applied *-un-lft-identity 2.4

      \[\leadsto \frac{\color{blue}{1 \cdot x}}{1 \cdot \frac{z \cdot \left(y - t\right)}{2}}\]

   6. Applied times-frac 2.4

      \[\leadsto \color{blue}{\frac{1}{1} \cdot \frac{x}{\frac{z \cdot \left(y - t\right)}{2}}}\]

   7. Simplified 2.4

      \[\leadsto \color{blue}{1} \cdot \frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\]

   if 1.741374549246005e+79 < z

   1. Initial program 12.7

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]

   2. Simplified 10.3

      \[\leadsto \color{blue}{\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 10.3

      \[\leadsto \frac{x}{\frac{z \cdot \left(y - t\right)}{\color{blue}{1 \cdot 2}}}\]

   5. Applied times-frac 10.2

      \[\leadsto \frac{x}{\color{blue}{\frac{z}{1} \cdot \frac{y - t}{2}}}\]

   6. Applied add-cube-cbrt 10.6

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\frac{z}{1} \cdot \frac{y - t}{2}}\]

   7. Applied times-frac 1.9

      \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{z}{1}} \cdot \frac{\sqrt[3]{x}}{\frac{y - t}{2}}}\]

   8. Simplified 1.9

      \[\leadsto \color{blue}{\frac{\sqrt[3]{x}}{\frac{z}{\sqrt[3]{x}}}} \cdot \frac{\sqrt[3]{x}}{\frac{y - t}{2}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -49355.0736941491268:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y - t}{2}}\\ \mathbf{elif}\;z \le 1.74137454924600484 \cdot 10^{79}:\\ \;\;\;\;1 \cdot \frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x}}{\frac{z}{\sqrt[3]{x}}} \cdot \frac{\sqrt[3]{x}}{\frac{y - t}{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (if (< (/ (* x 2) (- (* y z) (* t z))) -2.559141628295061e-13) (* (/ x (* (- y t) z)) 2) (if (< (/ (* x 2) (- (* y z) (* t z))) 1.0450278273301259e-269) (/ (* (/ x z) 2) (- y t)) (* (/ x (* (- y t) z)) 2)))

  (/ (* x 2) (- (* y z) (* t z))))