Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Percentage Accurate: 90.2% → 96.4%

Time: 9.2s

Precision: binary64

Cost: 1737

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

• 24.6% 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

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

↓

\[\begin{array}{l} t_1 := \left(x \cdot y - y \cdot z\right) \cdot t\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+59} \lor \neg \left(t_1 \leq 2 \cdot 10^{-114}\right):\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- (* x y) (* y z)) t)))
   (if (or (<= t_1 -5e+59) (not (<= t_1 2e-114)))
     (* (- x z) (* y t))
     (* y (* t (- 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 t_1 = ((x * y) - (y * z)) * t;
	double tmp;
	if ((t_1 <= -5e+59) || !(t_1 <= 2e-114)) {
		tmp = (x - z) * (y * t);
	} else {
		tmp = y * (t * (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) :: t_1
    real(8) :: tmp
    t_1 = ((x * y) - (y * z)) * t
    if ((t_1 <= (-5d+59)) .or. (.not. (t_1 <= 2d-114))) then
        tmp = (x - z) * (y * t)
    else
        tmp = y * (t * (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 t_1 = ((x * y) - (y * z)) * t;
	double tmp;
	if ((t_1 <= -5e+59) || !(t_1 <= 2e-114)) {
		tmp = (x - z) * (y * t);
	} else {
		tmp = y * (t * (x - z));
	}
	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)) * t
	tmp = 0
	if (t_1 <= -5e+59) or not (t_1 <= 2e-114):
		tmp = (x - z) * (y * t)
	else:
		tmp = y * (t * (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)
	t_1 = Float64(Float64(Float64(x * y) - Float64(y * z)) * t)
	tmp = 0.0
	if ((t_1 <= -5e+59) || !(t_1 <= 2e-114))
		tmp = Float64(Float64(x - z) * Float64(y * t));
	else
		tmp = Float64(y * Float64(t * 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)
	t_1 = ((x * y) - (y * z)) * t;
	tmp = 0.0;
	if ((t_1 <= -5e+59) || ~((t_1 <= 2e-114)))
		tmp = (x - z) * (y * t);
	else
		tmp = y * (t * (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_] := Block[{t$95$1 = N[(N[(N[(x * y), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e+59], N[Not[LessEqual[t$95$1, 2e-114]], $MachinePrecision]], N[(N[(x - z), $MachinePrecision] * N[(y * t), $MachinePrecision]), $MachinePrecision], N[(y * N[(t * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	90.2%
Target	96.0%
Herbie	96.4%

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (*.f64 x y) (*.f64 z y)) t) < -4.9999999999999997e59 or 2.0000000000000001e-114 < (*.f64 (-.f64 (*.f64 x y) (*.f64 z y)) t)

   1. Initial program 91.4%

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

   2. Taylor expanded in x around 0 83.2%

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

   3. Simplified 99.1%

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

      [Start] 83.2	\[ y \cdot \left(t \cdot x\right) + -1 \cdot \left(y \cdot \left(t \cdot z\right)\right) \]
      associate-*r* [=>] 88.8	\[ \color{blue}{\left(y \cdot t\right) \cdot x} + -1 \cdot \left(y \cdot \left(t \cdot z\right)\right) \]
      *-commutative [<=] 88.8	\[ \color{blue}{\left(t \cdot y\right)} \cdot x + -1 \cdot \left(y \cdot \left(t \cdot z\right)\right) \]
      mul-1-neg [=>] 88.8	\[ \left(t \cdot y\right) \cdot x + \color{blue}{\left(-y \cdot \left(t \cdot z\right)\right)} \]
      associate-*r* [=>] 90.1	\[ \left(t \cdot y\right) \cdot x + \left(-\color{blue}{\left(y \cdot t\right) \cdot z}\right) \]
      *-commutative [<=] 90.1	\[ \left(t \cdot y\right) \cdot x + \left(-\color{blue}{\left(t \cdot y\right)} \cdot z\right) \]
      distribute-rgt-neg-in [=>] 90.1	\[ \left(t \cdot y\right) \cdot x + \color{blue}{\left(t \cdot y\right) \cdot \left(-z\right)} \]
      distribute-lft-in [<=] 99.1	\[ \color{blue}{\left(t \cdot y\right) \cdot \left(x + \left(-z\right)\right)} \]
      sub-neg [<=] 99.1	\[ \left(t \cdot y\right) \cdot \color{blue}{\left(x - z\right)} \]
      *-commutative [=>] 99.1	\[ \color{blue}{\left(x - z\right) \cdot \left(t \cdot y\right)} \]
      *-commutative [=>] 99.1	\[ \left(x - z\right) \cdot \color{blue}{\left(y \cdot t\right)} \]

   if -4.9999999999999997e59 < (*.f64 (-.f64 (*.f64 x y) (*.f64 z y)) t) < 2.0000000000000001e-114

   1. Initial program 95.4%

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

   2. Simplified 94.7%

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

      [Start] 95.4	\[ \left(x \cdot y - z \cdot y\right) \cdot t \]
      distribute-rgt-out-- [=>] 95.4	\[ \color{blue}{\left(y \cdot \left(x - z\right)\right)} \cdot t \]
      associate-*l* [=>] 94.7	\[ \color{blue}{y \cdot \left(\left(x - z\right) \cdot t\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot y - y \cdot z\right) \cdot t \leq -5 \cdot 10^{+59} \lor \neg \left(\left(x \cdot y - y \cdot z\right) \cdot t \leq 2 \cdot 10^{-114}\right):\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	92.1%
Cost	845

\[\begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+247}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{elif}\;z \leq -1.04 \cdot 10^{-258} \lor \neg \left(z \leq 3.8 \cdot 10^{-256}\right):\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \end{array} \]

Alternative 2
Accuracy	73.3%
Cost	649

\[\begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+35} \lor \neg \left(z \leq 1.75 \cdot 10^{-57}\right):\\ \;\;\;\;z \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \end{array} \]

Alternative 3
Accuracy	73.2%
Cost	648

\[\begin{array}{l} \mathbf{if}\;z \leq -5.1 \cdot 10^{+39}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{-58}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-t\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	73.0%
Cost	648

\[\begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+45}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-57}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot z\right) \cdot \left(-t\right)\\ \end{array} \]

Alternative 5
Accuracy	96.9%
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-54}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	56.1%
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq 10^{+21}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \]

Alternative 7
Accuracy	52.7%
Cost	320

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

Reproduce

herbie shell --seed 2023160 
(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))