Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Average Error: 7.3 → 6.7

Time: 3.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -3.09483419664136275 \cdot 10^{-215}:\\ \;\;\;\;\left(t \cdot y\right) \cdot \left(x - z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot x + t \cdot \left(-z\right)\right) \cdot y + \left(y \cdot \mathsf{fma}\left(-z, 1, z\right)\right) \cdot t\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r498074 = x;
        double r498075 = y;
        double r498076 = r498074 * r498075;
        double r498077 = z;
        double r498078 = r498077 * r498075;
        double r498079 = r498076 - r498078;
        double r498080 = t;
        double r498081 = r498079 * r498080;
        return r498081;
}

↓

double f(double x, double y, double z, double t) {
        double r498082 = z;
        double r498083 = -3.094834196641363e-215;
        bool r498084 = r498082 <= r498083;
        double r498085 = t;
        double r498086 = y;
        double r498087 = r498085 * r498086;
        double r498088 = x;
        double r498089 = r498088 - r498082;
        double r498090 = r498087 * r498089;
        double r498091 = r498085 * r498088;
        double r498092 = -r498082;
        double r498093 = r498085 * r498092;
        double r498094 = r498091 + r498093;
        double r498095 = r498094 * r498086;
        double r498096 = 1.0;
        double r498097 = fma(r498092, r498096, r498082);
        double r498098 = r498086 * r498097;
        double r498099 = r498098 * r498085;
        double r498100 = r498095 + r498099;
        double r498101 = r498084 ? r498090 : r498100;
        return r498101;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.3
Target	3.0
Herbie	6.7

\[\begin{array}{l} \mathbf{if}\;t \lt -9.2318795828867769 \cdot 10^{-80}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(x - z\right)\\ \mathbf{elif}\;t \lt 2.5430670515648771 \cdot 10^{83}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(x - z\right)\right) \cdot t\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < -3.094834196641363e-215

   1. Initial program 7.7

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

   2. Simplified 7.7

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

   4. Applied associate-*r* 6.6

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

   if -3.094834196641363e-215 < z

   1. Initial program 7.0

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

   2. Simplified 7.0

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

   4. Applied add-cube-cbrt 7.3

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

   5. Applied add-cube-cbrt 7.8

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

   6. Applied prod-diff 7.8

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

   7. Applied distribute-lft-in 7.8

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

   8. Applied distribute-lft-in 7.8

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

   9. Simplified 9.8

      \[\leadsto \color{blue}{\left(t \cdot \left(1 \cdot {\left(\sqrt[3]{x}\right)}^{3} + \left(-z\right)\right)\right) \cdot y} + t \cdot \left(y \cdot \mathsf{fma}\left(-\sqrt[3]{z}, \sqrt[3]{z} \cdot \sqrt[3]{z}, \sqrt[3]{z} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right)\right)\]

   10. Simplified 7.2

       \[\leadsto \left(t \cdot \left(1 \cdot {\left(\sqrt[3]{x}\right)}^{3} + \left(-z\right)\right)\right) \cdot y + \color{blue}{\left(y \cdot \mathsf{fma}\left(-z, 1, z\right)\right) \cdot t}\]
   11. Using strategy rm

   12. Applied distribute-lft-in 7.2

       \[\leadsto \color{blue}{\left(t \cdot \left(1 \cdot {\left(\sqrt[3]{x}\right)}^{3}\right) + t \cdot \left(-z\right)\right)} \cdot y + \left(y \cdot \mathsf{fma}\left(-z, 1, z\right)\right) \cdot t\]

   13. Simplified 6.8

       \[\leadsto \left(\color{blue}{t \cdot x} + t \cdot \left(-z\right)\right) \cdot y + \left(y \cdot \mathsf{fma}\left(-z, 1, z\right)\right) \cdot t\]
3. Recombined 2 regimes into one program.

4. Final simplification 6.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.09483419664136275 \cdot 10^{-215}:\\ \;\;\;\;\left(t \cdot y\right) \cdot \left(x - z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot x + t \cdot \left(-z\right)\right) \cdot y + \left(y \cdot \mathsf{fma}\left(-z, 1, z\right)\right) \cdot t\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 +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))