Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Average Error: 7.3 → 1.4

Time: 8.3s

Precision: binary64

Cost: 2513

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

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot y - y \cdot z\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{-152} \lor \neg \left(t_1 \leq 2 \cdot 10^{-45}\right) \land t_1 \leq 10^{+198}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \end{array} \]

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 (<= t_1 (- INFINITY))
     (* (- x z) (* y t))
     (if (or (<= t_1 -1e-152) (and (not (<= t_1 2e-45)) (<= t_1 1e+198)))
       (* t (* y (- x z)))
       (* y (* (- x z) 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 <= -((double) INFINITY)) {
		tmp = (x - z) * (y * t);
	} else if ((t_1 <= -1e-152) || (!(t_1 <= 2e-45) && (t_1 <= 1e+198))) {
		tmp = t * (y * (x - z));
	} else {
		tmp = y * ((x - z) * t);
	}
	return tmp;
}

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 <= -Double.POSITIVE_INFINITY) {
		tmp = (x - z) * (y * t);
	} else if ((t_1 <= -1e-152) || (!(t_1 <= 2e-45) && (t_1 <= 1e+198))) {
		tmp = t * (y * (x - z));
	} else {
		tmp = y * ((x - z) * 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 <= -math.inf:
		tmp = (x - z) * (y * t)
	elif (t_1 <= -1e-152) or (not (t_1 <= 2e-45) and (t_1 <= 1e+198)):
		tmp = t * (y * (x - z))
	else:
		tmp = y * ((x - z) * 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 <= Float64(-Inf))
		tmp = Float64(Float64(x - z) * Float64(y * t));
	elseif ((t_1 <= -1e-152) || (!(t_1 <= 2e-45) && (t_1 <= 1e+198)))
		tmp = Float64(t * Float64(y * Float64(x - z)));
	else
		tmp = Float64(y * Float64(Float64(x - z) * 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 <= -Inf)
		tmp = (x - z) * (y * t);
	elseif ((t_1 <= -1e-152) || (~((t_1 <= 2e-45)) && (t_1 <= 1e+198)))
		tmp = t * (y * (x - z));
	else
		tmp = y * ((x - z) * 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[LessEqual[t$95$1, (-Infinity)], N[(N[(x - z), $MachinePrecision] * N[(y * t), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$1, -1e-152], And[N[Not[LessEqual[t$95$1, 2e-45]], $MachinePrecision], LessEqual[t$95$1, 1e+198]]], N[(t * N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(x - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	7.3
Target	3.5
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 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z y)) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 0.3

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

      [Start] 64.0	\[ \left(x \cdot y - z \cdot y\right) \cdot t \]
      distribute-rgt-out-- [=>] 64.0	\[ \color{blue}{\left(y \cdot \left(x - z\right)\right)} \cdot t \]
      associate-*l* [=>] 0.3	\[ \color{blue}{y \cdot \left(\left(x - z\right) \cdot t\right)} \]
      *-commutative [=>] 0.3	\[ y \cdot \color{blue}{\left(t \cdot \left(x - z\right)\right)} \]

   3. Taylor expanded in x around 0 0.3

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

   4. Simplified 0.3

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

      [Start] 0.3	\[ y \cdot \left(t \cdot x\right) + -1 \cdot \left(y \cdot \left(t \cdot z\right)\right) \]
      mul-1-neg [=>] 0.3	\[ y \cdot \left(t \cdot x\right) + \color{blue}{\left(-y \cdot \left(t \cdot z\right)\right)} \]
      associate-*r* [=>] 0.3	\[ \color{blue}{\left(y \cdot t\right) \cdot x} + \left(-y \cdot \left(t \cdot z\right)\right) \]
      associate-*r* [=>] 0.3	\[ \left(y \cdot t\right) \cdot x + \left(-\color{blue}{\left(y \cdot t\right) \cdot z}\right) \]
      distribute-rgt-neg-out [<=] 0.3	\[ \left(y \cdot t\right) \cdot x + \color{blue}{\left(y \cdot t\right) \cdot \left(-z\right)} \]
      distribute-lft-in [<=] 0.3	\[ \color{blue}{\left(y \cdot t\right) \cdot \left(x + \left(-z\right)\right)} \]
      sub-neg [<=] 0.3	\[ \left(y \cdot t\right) \cdot \color{blue}{\left(x - z\right)} \]
      *-commutative [=>] 0.3	\[ \color{blue}{\left(x - z\right) \cdot \left(y \cdot t\right)} \]

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z y)) < -1.00000000000000007e-152 or 1.99999999999999997e-45 < (-.f64 (*.f64 x y) (*.f64 z y)) < 1.00000000000000002e198

   1. Initial program 0.2

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

   2. Simplified 0.2

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

      [Start] 0.2	\[ \left(x \cdot y - z \cdot y\right) \cdot t \]
      distribute-rgt-out-- [=>] 0.2	\[ \color{blue}{\left(y \cdot \left(x - z\right)\right)} \cdot t \]

   if -1.00000000000000007e-152 < (-.f64 (*.f64 x y) (*.f64 z y)) < 1.99999999999999997e-45 or 1.00000000000000002e198 < (-.f64 (*.f64 x y) (*.f64 z y))

   1. Initial program 11.8

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

   2. Simplified 3.2

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

      [Start] 11.8	\[ \left(x \cdot y - z \cdot y\right) \cdot t \]
      distribute-rgt-out-- [=>] 11.8	\[ \color{blue}{\left(y \cdot \left(x - z\right)\right)} \cdot t \]
      associate-*l* [=>] 3.2	\[ \color{blue}{y \cdot \left(\left(x - z\right) \cdot t\right)} \]
      *-commutative [=>] 3.2	\[ y \cdot \color{blue}{\left(t \cdot \left(x - z\right)\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - y \cdot z \leq -\infty:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;x \cdot y - y \cdot z \leq -1 \cdot 10^{-152} \lor \neg \left(x \cdot y - y \cdot z \leq 2 \cdot 10^{-45}\right) \land x \cdot y - y \cdot z \leq 10^{+198}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	20.3
Cost	913

\[\begin{array}{l} t_1 := y \cdot \left(t \cdot \left(-z\right)\right)\\ \mathbf{if}\;z \leq -6.5 \cdot 10^{+37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -0.00088:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{-131} \lor \neg \left(z \leq 1.75 \cdot 10^{-17}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternative 2
Error	8.2
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq 3.1 \cdot 10^{+91} \lor \neg \left(t \leq 6.2 \cdot 10^{+140}\right):\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(-z\right)\\ \end{array} \]

Alternative 3
Error	3.2
Cost	580

\[\begin{array}{l} \mathbf{if}\;t \leq 3.6 \cdot 10^{+84}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \]

Alternative 4
Error	30.0
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq 3.3 \cdot 10^{+148}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \]

Alternative 5
Error	31.1
Cost	320

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

Reproduce

herbie shell --seed 2022364 
(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))