Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Average Error: 7.0 → 1.4

Time: 7.8s

Precision: binary64

Cost: 1609

\[ \begin{array}{c}[y, t] = \mathsf{sort}([y, t])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot y - y \cdot z\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+253} \lor \neg \left(t_1 \leq 2 \cdot 10^{+192}\right):\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t\\ \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* (- (* x y) (* z y)) t))

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x y) (* y z))))
   (if (or (<= t_1 -1e+253) (not (<= t_1 2e+192)))
     (* (- x z) (* y t))
     (* t_1 t))))

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 t_1 = (x * y) - (y * z);
	double tmp;
	if ((t_1 <= -1e+253) || !(t_1 <= 2e+192)) {
		tmp = (x - z) * (y * t);
	} else {
		tmp = t_1 * t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x * y) - (z * y)) * t
end function

↓

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * y) - (y * z)
    if ((t_1 <= (-1d+253)) .or. (.not. (t_1 <= 2d+192))) then
        tmp = (x - z) * (y * t)
    else
        tmp = t_1 * t
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * y) - (y * z);
	double tmp;
	if ((t_1 <= -1e+253) || !(t_1 <= 2e+192)) {
		tmp = (x - z) * (y * t);
	} else {
		tmp = t_1 * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	return ((x * y) - (z * y)) * t

↓

def code(x, y, z, t):
	t_1 = (x * y) - (y * z)
	tmp = 0
	if (t_1 <= -1e+253) or not (t_1 <= 2e+192):
		tmp = (x - z) * (y * t)
	else:
		tmp = t_1 * t
	return tmp

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x * y) - Float64(z * y)) * t)
end

↓

function code(x, y, z, t)
	t_1 = Float64(Float64(x * y) - Float64(y * z))
	tmp = 0.0
	if ((t_1 <= -1e+253) || !(t_1 <= 2e+192))
		tmp = Float64(Float64(x - z) * Float64(y * t));
	else
		tmp = Float64(t_1 * t);
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x * y) - (z * y)) * t;
end

↓

function tmp_2 = code(x, y, z, t)
	t_1 = (x * y) - (y * z);
	tmp = 0.0;
	if ((t_1 <= -1e+253) || ~((t_1 <= 2e+192)))
		tmp = (x - z) * (y * t);
	else
		tmp = t_1 * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * y), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]

↓

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e+253], N[Not[LessEqual[t$95$1, 2e+192]], $MachinePrecision]], N[(N[(x - z), $MachinePrecision] * N[(y * t), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * t), $MachinePrecision]]]

Target

Original	7.0
Target	3.2
Herbie	1.4

\[\begin{array}{l} \mathbf{if}\;t < -9.231879582886777 \cdot 10^{-80}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(x - z\right)\\ \mathbf{elif}\;t < 2.543067051564877 \cdot 10^{+83}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(x - z\right)\right) \cdot t\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z y)) < -9.9999999999999994e252 or 2.00000000000000008e192 < (-.f64 (*.f64 x y) (*.f64 z y))

   1. Initial program 32.5

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

   2. Simplified 1.2

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

      [Start] 32.5	\[ \left(x \cdot y - z \cdot y\right) \cdot t \]
      distribute-rgt-out-- [=>] 32.5	\[ \color{blue}{\left(y \cdot \left(x - z\right)\right)} \cdot t \]
      associate-*l* [=>] 1.2	\[ \color{blue}{y \cdot \left(\left(x - z\right) \cdot t\right)} \]
      *-commutative [=>] 1.2	\[ y \cdot \color{blue}{\left(t \cdot \left(x - z\right)\right)} \]

   3. Taylor expanded in x around 0 1.2

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

   4. Simplified 0.8

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

      [Start] 1.2	\[ y \cdot \left(t \cdot x\right) + -1 \cdot \left(y \cdot \left(t \cdot z\right)\right) \]
      mul-1-neg [=>] 1.2	\[ y \cdot \left(t \cdot x\right) + \color{blue}{\left(-y \cdot \left(t \cdot z\right)\right)} \]
      associate-*r* [=>] 1.3	\[ \color{blue}{\left(y \cdot t\right) \cdot x} + \left(-y \cdot \left(t \cdot z\right)\right) \]
      associate-*r* [=>] 0.8	\[ \left(y \cdot t\right) \cdot x + \left(-\color{blue}{\left(y \cdot t\right) \cdot z}\right) \]
      distribute-rgt-neg-out [<=] 0.8	\[ \left(y \cdot t\right) \cdot x + \color{blue}{\left(y \cdot t\right) \cdot \left(-z\right)} \]
      distribute-lft-in [<=] 0.8	\[ \color{blue}{\left(y \cdot t\right) \cdot \left(x + \left(-z\right)\right)} \]
      sub-neg [<=] 0.8	\[ \left(y \cdot t\right) \cdot \color{blue}{\left(x - z\right)} \]
      *-commutative [=>] 0.8	\[ \color{blue}{\left(x - z\right) \cdot \left(y \cdot t\right)} \]

   if -9.9999999999999994e252 < (-.f64 (*.f64 x y) (*.f64 z y)) < 2.00000000000000008e192

   1. Initial program 1.6

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - y \cdot z \leq -1 \cdot 10^{+253} \lor \neg \left(x \cdot y - y \cdot z \leq 2 \cdot 10^{+192}\right):\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y - y \cdot z\right) \cdot t\\ \end{array} \]

Alternatives

Alternative 1
Error	21.8
Cost	913

\[\begin{array}{l} t_1 := \left(-y\right) \cdot \left(z \cdot t\right)\\ \mathbf{if}\;z \leq -3 \cdot 10^{-100}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+46}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+92} \lor \neg \left(z \leq 8.5 \cdot 10^{+179}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternative 2
Error	21.7
Cost	913

\[\begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-92}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+46}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{+92} \lor \neg \left(z \leq 8.5 \cdot 10^{+179}\right):\\ \;\;\;\;\left(-y\right) \cdot \left(z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternative 3
Error	21.9
Cost	913

\[\begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-98}:\\ \;\;\;\;\left(y \cdot z\right) \cdot \left(-t\right)\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+46}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+92} \lor \neg \left(z \leq 8.5 \cdot 10^{+179}\right):\\ \;\;\;\;\left(-y\right) \cdot \left(z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternative 4
Error	7.8
Cost	580

\[\begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+151}:\\ \;\;\;\;\left(y \cdot z\right) \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \end{array} \]

Alternative 5
Error	2.4
Cost	580

\[\begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{-13}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \]

Alternative 6
Error	29.2
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq 20000000000:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \]

Alternative 7
Error	31.5
Cost	320

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

Reproduce

herbie shell --seed 2023018 
(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))