Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Average Error: 7.1 → 1.6

Time: 10.3s

Precision: binary64

Cost: 1608

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

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot y - y \cdot z\\ \mathbf{if}\;t_1 \leq -4 \cdot 10^{+228}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{elif}\;t_1 \leq 10^{+219}:\\ \;\;\;\;t_1 \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \end{array} \]

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 -4e+228)
     (* y (* t (- x z)))
     (if (<= t_1 1e+219) (* t_1 t) (* (- x z) (* y 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 <= -4e+228) {
		tmp = y * (t * (x - z));
	} else if (t_1 <= 1e+219) {
		tmp = t_1 * t;
	} else {
		tmp = (x - z) * (y * 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 <= (-4d+228)) then
        tmp = y * (t * (x - z))
    else if (t_1 <= 1d+219) then
        tmp = t_1 * t
    else
        tmp = (x - z) * (y * 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 <= -4e+228) {
		tmp = y * (t * (x - z));
	} else if (t_1 <= 1e+219) {
		tmp = t_1 * t;
	} else {
		tmp = (x - z) * (y * 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 <= -4e+228:
		tmp = y * (t * (x - z))
	elif t_1 <= 1e+219:
		tmp = t_1 * t
	else:
		tmp = (x - z) * (y * 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 <= -4e+228)
		tmp = Float64(y * Float64(t * Float64(x - z)));
	elseif (t_1 <= 1e+219)
		tmp = Float64(t_1 * t);
	else
		tmp = Float64(Float64(x - z) * Float64(y * 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 <= -4e+228)
		tmp = y * (t * (x - z));
	elseif (t_1 <= 1e+219)
		tmp = t_1 * t;
	else
		tmp = (x - z) * (y * 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, -4e+228], N[(y * N[(t * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+219], N[(t$95$1 * t), $MachinePrecision], N[(N[(x - z), $MachinePrecision] * N[(y * t), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	7.1
Target	3.7
Herbie	1.6

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z y)) < -3.9999999999999997e228

   1. Initial program 33.8

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

   2. Simplified 0.9

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

      [Start] 33.8	\[ \left(x \cdot y - z \cdot y\right) \cdot t \]
      distribute-rgt-out-- [=>] 33.8	\[ \color{blue}{\left(y \cdot \left(x - z\right)\right)} \cdot t \]
      associate-*l* [=>] 0.9	\[ \color{blue}{y \cdot \left(\left(x - z\right) \cdot t\right)} \]
      *-commutative [=>] 0.9	\[ y \cdot \color{blue}{\left(t \cdot \left(x - z\right)\right)} \]

   if -3.9999999999999997e228 < (-.f64 (*.f64 x y) (*.f64 z y)) < 9.99999999999999965e218

   1. Initial program 1.8

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

   if 9.99999999999999965e218 < (-.f64 (*.f64 x y) (*.f64 z y))

   1. Initial program 32.7

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

   2. Simplified 0.5

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

      [Start] 32.7	\[ \left(x \cdot y - z \cdot y\right) \cdot t \]
      distribute-rgt-out-- [=>] 32.7	\[ \color{blue}{\left(y \cdot \left(x - z\right)\right)} \cdot t \]
      associate-*l* [=>] 0.5	\[ \color{blue}{y \cdot \left(\left(x - z\right) \cdot t\right)} \]
      *-commutative [=>] 0.5	\[ y \cdot \color{blue}{\left(t \cdot \left(x - z\right)\right)} \]

   3. Taylor expanded in x around 0 0.5

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

   4. Simplified 0.9

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

      [Start] 0.5	\[ y \cdot \left(t \cdot x\right) + -1 \cdot \left(y \cdot \left(t \cdot z\right)\right) \]
      mul-1-neg [=>] 0.5	\[ y \cdot \left(t \cdot x\right) + \color{blue}{\left(-y \cdot \left(t \cdot z\right)\right)} \]
      associate-*r* [=>] 0.8	\[ \color{blue}{\left(y \cdot t\right) \cdot x} + \left(-y \cdot \left(t \cdot z\right)\right) \]
      associate-*r* [=>] 0.9	\[ \left(y \cdot t\right) \cdot x + \left(-\color{blue}{\left(y \cdot t\right) \cdot z}\right) \]
      distribute-rgt-neg-out [<=] 0.9	\[ \left(y \cdot t\right) \cdot x + \color{blue}{\left(y \cdot t\right) \cdot \left(-z\right)} \]
      distribute-lft-in [<=] 0.9	\[ \color{blue}{\left(y \cdot t\right) \cdot \left(x + \left(-z\right)\right)} \]
      sub-neg [<=] 0.9	\[ \left(y \cdot t\right) \cdot \color{blue}{\left(x - z\right)} \]
      *-commutative [=>] 0.9	\[ \color{blue}{\left(x - z\right) \cdot \left(y \cdot t\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - y \cdot z \leq -4 \cdot 10^{+228}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{elif}\;x \cdot y - y \cdot z \leq 10^{+219}:\\ \;\;\;\;\left(x \cdot y - y \cdot z\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	21.1
Cost	1178

\[\begin{array}{l} t_1 := y \cdot \left(z \cdot \left(-t\right)\right)\\ \mathbf{if}\;z \leq -4.8 \cdot 10^{+106}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -5.5 \cdot 10^{+70}:\\ \;\;\;\;t \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \leq -1200 \lor \neg \left(z \leq 5.9 \cdot 10^{-30} \lor \neg \left(z \leq 9 \cdot 10^{+110}\right) \land z \leq 4.8 \cdot 10^{+161}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \end{array} \]

Alternative 2
Error	20.8
Cost	1176

\[\begin{array}{l} t_1 := z \cdot \left(y \cdot \left(-t\right)\right)\\ t_2 := y \cdot \left(x \cdot t\right)\\ \mathbf{if}\;z \leq -4.8 \cdot 10^{+106}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.2 \cdot 10^{+71}:\\ \;\;\;\;t \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \leq -4.4 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-29}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+110}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+161}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-t\right)\right)\\ \end{array} \]

Alternative 3
Error	20.0
Cost	912

\[\begin{array}{l} t_1 := \left(y \cdot z\right) \cdot \left(-t\right)\\ \mathbf{if}\;z \leq -4.8 \cdot 10^{+106}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.2 \cdot 10^{+71}:\\ \;\;\;\;t \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-14}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{elif}\;z \leq 7.4 \cdot 10^{-30}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	3.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+20}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \]

Alternative 5
Error	2.5
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-33}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \]

Alternative 6
Error	29.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-8}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternative 7
Error	7.4
Cost	448

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

Alternative 8
Error	32.0
Cost	320

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

Reproduce

herbie shell --seed 2023055 
(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))