Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Average Error: 10.89% → 2.38%

Time: 9.0s

Precision: binary64

Cost: 1736

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

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot y - y \cdot z\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+188}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+199}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t - \left(y \cdot z\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \end{array} \]

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 -5e+188)
     (* (- x z) (* y t))
     (if (<= t_1 5e+199)
       (- (* (* x y) t) (* (* y z) t))
       (* 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 <= -5e+188) {
		tmp = (x - z) * (y * t);
	} else if (t_1 <= 5e+199) {
		tmp = ((x * y) * t) - ((y * z) * t);
	} else {
		tmp = y * ((x - z) * 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 <= (-5d+188)) then
        tmp = (x - z) * (y * t)
    else if (t_1 <= 5d+199) then
        tmp = ((x * y) * t) - ((y * z) * t)
    else
        tmp = y * ((x - z) * 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 <= -5e+188) {
		tmp = (x - z) * (y * t);
	} else if (t_1 <= 5e+199) {
		tmp = ((x * y) * t) - ((y * z) * t);
	} 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 <= -5e+188:
		tmp = (x - z) * (y * t)
	elif t_1 <= 5e+199:
		tmp = ((x * y) * t) - ((y * z) * t)
	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 <= -5e+188)
		tmp = Float64(Float64(x - z) * Float64(y * t));
	elseif (t_1 <= 5e+199)
		tmp = Float64(Float64(Float64(x * y) * t) - Float64(Float64(y * z) * t));
	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 <= -5e+188)
		tmp = (x - z) * (y * t);
	elseif (t_1 <= 5e+199)
		tmp = ((x * y) * t) - ((y * z) * t);
	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, -5e+188], N[(N[(x - z), $MachinePrecision] * N[(y * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+199], N[(N[(N[(x * y), $MachinePrecision] * t), $MachinePrecision] - N[(N[(y * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(x - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	10.89%
Target	5.36%
Herbie	2.38%

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z y)) < -5.0000000000000001e188

   1. Initial program 38.04

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

   2. Simplified 2.18

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

      [Start] 38.04	\[ \left(x \cdot y - z \cdot y\right) \cdot t \]
      distribute-rgt-out-- [=>] 38.03	\[ \color{blue}{\left(y \cdot \left(x - z\right)\right)} \cdot t \]
      associate-*l* [=>] 2.18	\[ \color{blue}{y \cdot \left(\left(x - z\right) \cdot t\right)} \]
      *-commutative [=>] 2.18	\[ y \cdot \color{blue}{\left(t \cdot \left(x - z\right)\right)} \]

   3. Taylor expanded in x around 0 2.19

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

   4. Simplified 1.77

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

      [Start] 2.19	\[ y \cdot \left(t \cdot x\right) + -1 \cdot \left(y \cdot \left(t \cdot z\right)\right) \]
      mul-1-neg [=>] 2.19	\[ y \cdot \left(t \cdot x\right) + \color{blue}{\left(-y \cdot \left(t \cdot z\right)\right)} \]
      associate-*r* [=>] 2.53	\[ \color{blue}{\left(y \cdot t\right) \cdot x} + \left(-y \cdot \left(t \cdot z\right)\right) \]
      associate-*r* [=>] 1.78	\[ \left(y \cdot t\right) \cdot x + \left(-\color{blue}{\left(y \cdot t\right) \cdot z}\right) \]
      distribute-rgt-neg-out [<=] 1.78	\[ \left(y \cdot t\right) \cdot x + \color{blue}{\left(y \cdot t\right) \cdot \left(-z\right)} \]
      distribute-lft-in [<=] 1.77	\[ \color{blue}{\left(y \cdot t\right) \cdot \left(x + \left(-z\right)\right)} \]
      sub-neg [<=] 1.77	\[ \left(y \cdot t\right) \cdot \color{blue}{\left(x - z\right)} \]
      *-commutative [=>] 1.77	\[ \color{blue}{\left(x - z\right) \cdot \left(y \cdot t\right)} \]

   if -5.0000000000000001e188 < (-.f64 (*.f64 x y) (*.f64 z y)) < 4.9999999999999998e199

   1. Initial program 2.51

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

   2. Applied egg-rr 2.51

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

   if 4.9999999999999998e199 < (-.f64 (*.f64 x y) (*.f64 z y))

   1. Initial program 45.27

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

   2. Simplified 2.01

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

      [Start] 45.27	\[ \left(x \cdot y - z \cdot y\right) \cdot t \]
      distribute-rgt-out-- [=>] 45.27	\[ \color{blue}{\left(y \cdot \left(x - z\right)\right)} \cdot t \]
      associate-*l* [=>] 2.01	\[ \color{blue}{y \cdot \left(\left(x - z\right) \cdot t\right)} \]
      *-commutative [=>] 2.01	\[ y \cdot \color{blue}{\left(t \cdot \left(x - z\right)\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 2.38

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - y \cdot z \leq -5 \cdot 10^{+188}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;x \cdot y - y \cdot z \leq 5 \cdot 10^{+199}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t - \left(y \cdot z\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	2.38%
Cost	1480

\[\begin{array}{l} t_1 := x \cdot y - y \cdot z\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+188}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+199}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \end{array} \]

Alternative 2
Error	31.65%
Cost	913

\[\begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{-90}:\\ \;\;\;\;\left(-y\right) \cdot \left(z \cdot t\right)\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-18}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+21} \lor \neg \left(z \leq 7 \cdot 10^{+46}\right):\\ \;\;\;\;z \cdot \left(t \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \]

Alternative 3
Error	32.45%
Cost	912

\[\begin{array}{l} t_1 := \left(y \cdot z\right) \cdot \left(-t\right)\\ \mathbf{if}\;z \leq -2 \cdot 10^{-118}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-18}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+21}:\\ \;\;\;\;z \cdot \left(t \cdot \left(-y\right)\right)\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{+46}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	11.59%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-236} \lor \neg \left(x \leq 4.6 \cdot 10^{-234}\right):\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(t \cdot \left(-y\right)\right)\\ \end{array} \]

Alternative 5
Error	31.09%
Cost	648

\[\begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+44}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-22}:\\ \;\;\;\;\left(-y\right) \cdot \left(z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \end{array} \]

Alternative 6
Error	4.08%
Cost	580

\[\begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-21}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \]

Alternative 7
Error	45.78%
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq 6600000000000:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \]

Alternative 8
Error	46.48%
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq 10^{-149}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \end{array} \]

Alternative 9
Error	49.58%
Cost	320

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

Reproduce

herbie shell --seed 2023090 
(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))