Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Percentage Accurate: 90.6% → 97.0%

Time: 12.4s

Precision: binary64

Cost: 708

• Unsound rule application detected in e-graph. Results may not be sound.

• 24.4% of points produce a very large (infinite) output. You may want to add a precondition.

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+66}:\\ \;\;\;\;\frac{y \cdot t}{\frac{1}{x - z}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= y -2.8e+66) (/ (* y t) (/ 1.0 (- x z))) (* t (* y (- x z)))))

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 tmp;
	if (y <= -2.8e+66) {
		tmp = (y * t) / (1.0 / (x - z));
	} else {
		tmp = t * (y * (x - z));
	}
	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) :: tmp
    if (y <= (-2.8d+66)) then
        tmp = (y * t) / (1.0d0 / (x - z))
    else
        tmp = t * (y * (x - z))
    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 tmp;
	if (y <= -2.8e+66) {
		tmp = (y * t) / (1.0 / (x - z));
	} else {
		tmp = t * (y * (x - z));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t):
	tmp = 0
	if y <= -2.8e+66:
		tmp = (y * t) / (1.0 / (x - z))
	else:
		tmp = t * (y * (x - z))
	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)
	tmp = 0.0
	if (y <= -2.8e+66)
		tmp = Float64(Float64(y * t) / Float64(1.0 / Float64(x - z)));
	else
		tmp = Float64(t * Float64(y * Float64(x - z)));
	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)
	tmp = 0.0;
	if (y <= -2.8e+66)
		tmp = (y * t) / (1.0 / (x - z));
	else
		tmp = t * (y * (x - z));
	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_] := If[LessEqual[y, -2.8e+66], N[(N[(y * t), $MachinePrecision] / N[(1.0 / N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	90.6%
Target	96.1%
Herbie	97.0%

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

Derivation

1. Split input into 2 regimes

2. if y < -2.8000000000000001e66

   1. Initial program 80.9%

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

   2. Simplified 92.9%

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

      [Start] 80.9%	\[ \left(x \cdot y - z \cdot y\right) \cdot t \]
      distribute-rgt-out-- [=>] 88.4%	\[ \color{blue}{\left(y \cdot \left(x - z\right)\right)} \cdot t \]
      associate-*l* [=>] 92.9%	\[ \color{blue}{y \cdot \left(\left(x - z\right) \cdot t\right)} \]

   3. Applied egg-rr 63.2%

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

      [Start] 92.9%	\[ y \cdot \left(\left(x - z\right) \cdot t\right) \]
      add-cbrt-cube [=>] 63.2%	\[ \color{blue}{\sqrt[3]{\left(\left(y \cdot \left(\left(x - z\right) \cdot t\right)\right) \cdot \left(y \cdot \left(\left(x - z\right) \cdot t\right)\right)\right) \cdot \left(y \cdot \left(\left(x - z\right) \cdot t\right)\right)}} \]
      pow3 [=>] 63.2%	\[ \sqrt[3]{\color{blue}{{\left(y \cdot \left(\left(x - z\right) \cdot t\right)\right)}^{3}}} \]

   4. Applied egg-rr 99.7%

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

      [Start] 63.2%	\[ \sqrt[3]{{\left(y \cdot \left(\left(x - z\right) \cdot t\right)\right)}^{3}} \]
      rem-cbrt-cube [=>] 92.9%	\[ \color{blue}{y \cdot \left(\left(x - z\right) \cdot t\right)} \]
      associate-*r* [=>] 88.4%	\[ \color{blue}{\left(y \cdot \left(x - z\right)\right) \cdot t} \]
      *-commutative [=>] 88.4%	\[ \color{blue}{\left(\left(x - z\right) \cdot y\right)} \cdot t \]
      associate-*l* [=>] 99.7%	\[ \color{blue}{\left(x - z\right) \cdot \left(y \cdot t\right)} \]

   5. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{y \cdot t}{\frac{1}{x - z}}} \]
      Step-by-step derivation

      [Start] 99.7%	\[ \left(x - z\right) \cdot \left(y \cdot t\right) \]
      *-commutative [=>] 99.7%	\[ \color{blue}{\left(y \cdot t\right) \cdot \left(x - z\right)} \]
      flip-- [=>] 83.3%	\[ \left(y \cdot t\right) \cdot \color{blue}{\frac{x \cdot x - z \cdot z}{x + z}} \]
      +-commutative [<=] 83.3%	\[ \left(y \cdot t\right) \cdot \frac{x \cdot x - z \cdot z}{\color{blue}{z + x}} \]
      clear-num [=>] 83.2%	\[ \left(y \cdot t\right) \cdot \color{blue}{\frac{1}{\frac{z + x}{x \cdot x - z \cdot z}}} \]
      div-inv [<=] 83.3%	\[ \color{blue}{\frac{y \cdot t}{\frac{z + x}{x \cdot x - z \cdot z}}} \]
      clear-num [=>] 83.3%	\[ \frac{y \cdot t}{\color{blue}{\frac{1}{\frac{x \cdot x - z \cdot z}{z + x}}}} \]
      +-commutative [=>] 83.3%	\[ \frac{y \cdot t}{\frac{1}{\frac{x \cdot x - z \cdot z}{\color{blue}{x + z}}}} \]
      flip-- [<=] 99.7%	\[ \frac{y \cdot t}{\frac{1}{\color{blue}{x - z}}} \]

   if -2.8000000000000001e66 < y

   1. Initial program 92.7%

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

   2. Simplified 94.6%

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

      [Start] 92.7%	\[ \left(x \cdot y - z \cdot y\right) \cdot t \]
      distribute-rgt-out-- [=>] 94.6%	\[ \color{blue}{\left(y \cdot \left(x - z\right)\right)} \cdot t \]

3. Recombined 2 regimes into one program.

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+66}:\\ \;\;\;\;\frac{y \cdot t}{\frac{1}{x - z}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.0%
Cost	708

\[\begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+66}:\\ \;\;\;\;\frac{y \cdot t}{\frac{1}{x - z}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	73.4%
Cost	649

\[\begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{-18} \lor \neg \left(z \leq 1.15 \cdot 10^{-63}\right):\\ \;\;\;\;y \cdot \left(z \cdot \left(-t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 3
Accuracy	73.5%
Cost	649

\[\begin{array}{l} \mathbf{if}\;z \leq -1.18 \cdot 10^{-18} \lor \neg \left(z \leq 6.9 \cdot 10^{-62}\right):\\ \;\;\;\;t \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 4
Accuracy	91.7%
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{+246}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	96.8%
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+67}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	97.0%
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+66}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(x - z\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \]

Alternative 7
Accuracy	55.0%
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{-101}:\\ \;\;\;\;\left(y \cdot t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot x\right)\\ \end{array} \]

Alternative 8
Accuracy	55.2%
Cost	320

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

Reproduce

herbie shell --seed 2023263 
(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))