Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Average Error: 7.1 → 4.0

Time: 3.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -3.63358414705648937 \cdot 10^{163} \lor \neg \left(y \le 9.32128107294202841 \cdot 10^{99}\right):\\ \;\;\;\;\left(t \cdot \left(1 \cdot {\left(\sqrt[3]{x}\right)}^{3} + \left(-z\right)\right)\right) \cdot y + \left(y \cdot \mathsf{fma}\left(-z, 1, z\right)\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(x \cdot y\right) + t \cdot \left(\left(-z\right) \cdot y\right)\\ \end{array}\]

C

double code(double x, double y, double z, double t) {
	return (((x * y) - (z * y)) * t);
}

↓

double code(double x, double y, double z, double t) {
	double temp;
	if (((y <= -3.6335841470564894e+163) || !(y <= 9.321281072942028e+99))) {
		temp = (((t * ((1.0 * pow(cbrt(x), 3.0)) + -z)) * y) + ((y * fma(-z, 1.0, z)) * t));
	} else {
		temp = ((t * (x * y)) + (t * (-z * y)));
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.1
Target	3.0
Herbie	4.0

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

Derivation

1. Split input into 2 regimes

2. if y < -3.6335841470564894e+163 or 9.321281072942028e+99 < y

   1. Initial program 22.4

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

   2. Simplified 22.4

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

   4. Applied add-cube-cbrt 22.8

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

   5. Applied add-cube-cbrt 23.2

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

   6. Applied prod-diff 23.2

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

   7. Applied distribute-lft-in 23.2

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

   8. Applied distribute-lft-in 23.2

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

   9. Simplified 16.6

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

   10. Simplified 5.3

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

   if -3.6335841470564894e+163 < y < 9.321281072942028e+99

   1. Initial program 3.7

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

   2. Simplified 3.7

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

   4. Applied sub-neg 3.7

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

   5. Applied distribute-lft-in 3.7

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

   6. Applied distribute-lft-in 3.7

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

   7. Simplified 3.7

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

   8. Simplified 3.7

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

4. Final simplification 4.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -3.63358414705648937 \cdot 10^{163} \lor \neg \left(y \le 9.32128107294202841 \cdot 10^{99}\right):\\ \;\;\;\;\left(t \cdot \left(1 \cdot {\left(\sqrt[3]{x}\right)}^{3} + \left(-z\right)\right)\right) \cdot y + \left(y \cdot \mathsf{fma}\left(-z, 1, z\right)\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(x \cdot y\right) + t \cdot \left(\left(-z\right) \cdot y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 +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))