Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Average Error: 7.3 → 1.4

Time: 13.6s

Precision: binary64

Cost: 1608

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot y - z \cdot y\\ t_2 := y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+202}:\\ \;\;\;\;t_1 \cdot t\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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) (* z y))) (t_2 (* y (* (- x z) t))))
   (if (<= t_1 (- INFINITY)) t_2 (if (<= t_1 4e+202) (* t_1 t) t_2))))

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) - (z * y);
	double t_2 = y * ((x - z) * t);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= 4e+202) {
		tmp = t_1 * t;
	} else {
		tmp = t_2;
	}
	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) - (z * y);
	double t_2 = y * ((x - z) * t);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_1 <= 4e+202) {
		tmp = t_1 * t;
	} else {
		tmp = t_2;
	}
	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) - (z * y)
	t_2 = y * ((x - z) * t)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = t_2
	elif t_1 <= 4e+202:
		tmp = t_1 * t
	else:
		tmp = t_2
	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(z * y))
	t_2 = Float64(y * Float64(Float64(x - z) * t))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= 4e+202)
		tmp = Float64(t_1 * t);
	else
		tmp = t_2;
	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) - (z * y);
	t_2 = y * ((x - z) * t);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = t_2;
	elseif (t_1 <= 4e+202)
		tmp = t_1 * t;
	else
		tmp = t_2;
	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[(z * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(N[(x - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, 4e+202], N[(t$95$1 * t), $MachinePrecision], t$95$2]]]]

Target

Original	7.3
Target	3.1
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)) < -inf.0 or 3.9999999999999996e202 < (-.f64 (*.f64 x y) (*.f64 z y))

   1. Initial program 40.6

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

   2. Simplified 1.2

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

      [Start] 40.6	\[ \left(x \cdot y - z \cdot y\right) \cdot t \]
      rational.json-simplify-2 [=>] 40.6	\[ \color{blue}{t \cdot \left(x \cdot y - z \cdot y\right)} \]
      rational.json-simplify-2 [=>] 40.6	\[ t \cdot \left(\color{blue}{y \cdot x} - z \cdot y\right) \]
      rational.json-simplify-52 [=>] 40.6	\[ t \cdot \color{blue}{\left(y \cdot \left(x - z\right)\right)} \]
      rational.json-simplify-43 [=>] 1.2	\[ \color{blue}{y \cdot \left(\left(x - z\right) \cdot t\right)} \]

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z y)) < 3.9999999999999996e202

   1. Initial program 1.4

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

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot y \leq -\infty:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{elif}\;x \cdot y - z \cdot y \leq 4 \cdot 10^{+202}:\\ \;\;\;\;\left(x \cdot y - z \cdot y\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	20.5
Cost	1176

\[\begin{array}{l} t_1 := x \cdot \left(y \cdot t\right)\\ t_2 := y \cdot \left(t \cdot \left(-z\right)\right)\\ \mathbf{if}\;x \leq -1.06 \cdot 10^{+36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{+24}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -7.4 \cdot 10^{-81}:\\ \;\;\;\;\left(y \cdot x\right) \cdot t\\ \mathbf{elif}\;x \leq -7.8 \cdot 10^{-115}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-137}:\\ \;\;\;\;y \cdot \left(t \cdot x\right)\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+28}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	20.5
Cost	1176

\[\begin{array}{l} t_1 := x \cdot \left(y \cdot t\right)\\ t_2 := y \cdot \left(t \cdot \left(-z\right)\right)\\ \mathbf{if}\;x \leq -4.4 \cdot 10^{+35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.32 \cdot 10^{+25}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(-z\right)\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-81}:\\ \;\;\;\;\left(y \cdot x\right) \cdot t\\ \mathbf{elif}\;x \leq -7.8 \cdot 10^{-115}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-137}:\\ \;\;\;\;y \cdot \left(t \cdot x\right)\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+29}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	20.2
Cost	1176

\[\begin{array}{l} t_1 := x \cdot \left(y \cdot t\right)\\ \mathbf{if}\;x \leq -1.55 \cdot 10^{+35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -8 \cdot 10^{+23}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(-z\right)\\ \mathbf{elif}\;x \leq -7.4 \cdot 10^{-81}:\\ \;\;\;\;\left(y \cdot x\right) \cdot t\\ \mathbf{elif}\;x \leq -7.8 \cdot 10^{-115}:\\ \;\;\;\;y \cdot \left(t \cdot \left(-z\right)\right)\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-137}:\\ \;\;\;\;y \cdot \left(t \cdot x\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+68}:\\ \;\;\;\;\left(y \cdot \left(-z\right)\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	7.8
Cost	712

\[\begin{array}{l} t_1 := y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{if}\;x \leq 3.8 \cdot 10^{+213}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+245}:\\ \;\;\;\;\left(y \cdot x\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	4.9
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq -60000000:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(x - z\right)\right) \cdot t\\ \end{array} \]

Alternative 6
Error	5.2
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq -49000000:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(x - z\right)\right) \cdot t\\ \end{array} \]

Alternative 7
Error	30.6
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{-76}:\\ \;\;\;\;y \cdot \left(t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \]

Alternative 8
Error	31.0
Cost	452

\[\begin{array}{l} \mathbf{if}\;y \leq -50000000:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot t\\ \end{array} \]

Alternative 9
Error	31.7
Cost	320

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

Reproduce

herbie shell --seed 2023075 
(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))