Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Average Error: 6.9 → 0.5

Time: 9.8s

Precision: binary64

Cost: 2512

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

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot y - y \cdot z\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+229}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;t_1 \leq -5 \cdot 10^{-232}:\\ \;\;\;\;t_1 \cdot t\\ \mathbf{elif}\;t_1 \leq 10^{-180}:\\ \;\;\;\;y \cdot \left(x \cdot t - z \cdot t\right)\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+177}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \end{array} \]

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 -2e+229)
     (* (- x z) (* y t))
     (if (<= t_1 -5e-232)
       (* t_1 t)
       (if (<= t_1 1e-180)
         (* y (- (* x t) (* z t)))
         (if (<= t_1 4e+177) (* t (* y (- x z))) (* 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 <= -2e+229) {
		tmp = (x - z) * (y * t);
	} else if (t_1 <= -5e-232) {
		tmp = t_1 * t;
	} else if (t_1 <= 1e-180) {
		tmp = y * ((x * t) - (z * t));
	} else if (t_1 <= 4e+177) {
		tmp = t * (y * (x - z));
	} 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 <= (-2d+229)) then
        tmp = (x - z) * (y * t)
    else if (t_1 <= (-5d-232)) then
        tmp = t_1 * t
    else if (t_1 <= 1d-180) then
        tmp = y * ((x * t) - (z * t))
    else if (t_1 <= 4d+177) then
        tmp = t * (y * (x - z))
    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 <= -2e+229) {
		tmp = (x - z) * (y * t);
	} else if (t_1 <= -5e-232) {
		tmp = t_1 * t;
	} else if (t_1 <= 1e-180) {
		tmp = y * ((x * t) - (z * t));
	} else if (t_1 <= 4e+177) {
		tmp = t * (y * (x - z));
	} 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 <= -2e+229:
		tmp = (x - z) * (y * t)
	elif t_1 <= -5e-232:
		tmp = t_1 * t
	elif t_1 <= 1e-180:
		tmp = y * ((x * t) - (z * t))
	elif t_1 <= 4e+177:
		tmp = t * (y * (x - z))
	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 <= -2e+229)
		tmp = Float64(Float64(x - z) * Float64(y * t));
	elseif (t_1 <= -5e-232)
		tmp = Float64(t_1 * t);
	elseif (t_1 <= 1e-180)
		tmp = Float64(y * Float64(Float64(x * t) - Float64(z * t)));
	elseif (t_1 <= 4e+177)
		tmp = Float64(t * Float64(y * Float64(x - z)));
	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 <= -2e+229)
		tmp = (x - z) * (y * t);
	elseif (t_1 <= -5e-232)
		tmp = t_1 * t;
	elseif (t_1 <= 1e-180)
		tmp = y * ((x * t) - (z * t));
	elseif (t_1 <= 4e+177)
		tmp = t * (y * (x - z));
	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, -2e+229], N[(N[(x - z), $MachinePrecision] * N[(y * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e-232], N[(t$95$1 * t), $MachinePrecision], If[LessEqual[t$95$1, 1e-180], N[(y * N[(N[(x * t), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+177], N[(t * N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(x - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	6.9
Target	3.4
Herbie	0.5

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

Derivation

1. Split input into 5 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z y)) < -2e229

   1. Initial program 32.4

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

   2. Simplified 0.6

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

      [Start] 32.4	\[ \left(x \cdot y - z \cdot y\right) \cdot t \]
      distribute-rgt-out-- [=>] 32.4	\[ \color{blue}{\left(y \cdot \left(x - z\right)\right)} \cdot t \]
      associate-*l* [=>] 0.6	\[ \color{blue}{y \cdot \left(\left(x - z\right) \cdot t\right)} \]
      *-commutative [=>] 0.6	\[ y \cdot \color{blue}{\left(t \cdot \left(x - z\right)\right)} \]

   3. Taylor expanded in x around 0 0.6

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

   4. Simplified 0.9

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

      [Start] 0.6	\[ y \cdot \left(t \cdot x\right) + -1 \cdot \left(y \cdot \left(t \cdot z\right)\right) \]
      mul-1-neg [=>] 0.6	\[ y \cdot \left(t \cdot x\right) + \color{blue}{\left(-y \cdot \left(t \cdot z\right)\right)} \]
      associate-*r* [=>] 1.2	\[ \color{blue}{\left(y \cdot t\right) \cdot x} + \left(-y \cdot \left(t \cdot z\right)\right) \]
      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	\[ \color{blue}{\left(x - z\right) \cdot \left(y \cdot t\right)} \]

   if -2e229 < (-.f64 (*.f64 x y) (*.f64 z y)) < -4.9999999999999999e-232

   1. Initial program 0.2

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

   if -4.9999999999999999e-232 < (-.f64 (*.f64 x y) (*.f64 z y)) < 1e-180

   1. Initial program 8.2

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

   2. Simplified 0.8

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

      [Start] 8.2	\[ \left(x \cdot y - z \cdot y\right) \cdot t \]
      distribute-rgt-out-- [=>] 8.2	\[ \color{blue}{\left(y \cdot \left(x - z\right)\right)} \cdot t \]
      associate-*l* [=>] 0.8	\[ \color{blue}{y \cdot \left(\left(x - z\right) \cdot t\right)} \]
      *-commutative [=>] 0.8	\[ y \cdot \color{blue}{\left(t \cdot \left(x - z\right)\right)} \]

   3. Applied egg-rr 0.8

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

   4. Applied egg-rr 0.8

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

   if 1e-180 < (-.f64 (*.f64 x y) (*.f64 z y)) < 4e177

   1. Initial program 0.2

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

   2. Simplified 0.2

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

      [Start] 0.2	\[ \left(x \cdot y - z \cdot y\right) \cdot t \]
      distribute-rgt-out-- [=>] 0.2	\[ \color{blue}{\left(y \cdot \left(x - z\right)\right)} \cdot t \]

   if 4e177 < (-.f64 (*.f64 x y) (*.f64 z y))

   1. Initial program 25.9

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

   2. Simplified 1.6

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

      [Start] 25.9	\[ \left(x \cdot y - z \cdot y\right) \cdot t \]
      distribute-rgt-out-- [=>] 25.9	\[ \color{blue}{\left(y \cdot \left(x - z\right)\right)} \cdot t \]
      associate-*l* [=>] 1.6	\[ \color{blue}{y \cdot \left(\left(x - z\right) \cdot t\right)} \]
      *-commutative [=>] 1.6	\[ y \cdot \color{blue}{\left(t \cdot \left(x - z\right)\right)} \]

3. Recombined 5 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - y \cdot z \leq -2 \cdot 10^{+229}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;x \cdot y - y \cdot z \leq -5 \cdot 10^{-232}:\\ \;\;\;\;\left(x \cdot y - y \cdot z\right) \cdot t\\ \mathbf{elif}\;x \cdot y - y \cdot z \leq 10^{-180}:\\ \;\;\;\;y \cdot \left(x \cdot t - z \cdot t\right)\\ \mathbf{elif}\;x \cdot y - y \cdot z \leq 4 \cdot 10^{+177}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	7.6
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -3.55 \cdot 10^{-242} \lor \neg \left(x \leq 4.8 \cdot 10^{-300}\right):\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-t\right)\right)\\ \end{array} \]

Alternative 2
Error	2.6
Cost	708

\[\begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{+19}:\\ \;\;\;\;y \cdot \left(x \cdot t - z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \end{array} \]

Alternative 3
Error	20.2
Cost	648

\[\begin{array}{l} \mathbf{if}\;x \leq -1.08 \cdot 10^{-44}:\\ \;\;\;\;t \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \leq 10^{-109}:\\ \;\;\;\;y \cdot \left(t \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \end{array} \]

Alternative 4
Error	20.2
Cost	648

\[\begin{array}{l} \mathbf{if}\;x \leq -0.5:\\ \;\;\;\;t \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-119}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \end{array} \]

Alternative 5
Error	2.7
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{+36}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \]

Alternative 6
Error	2.6
Cost	580

\[\begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{+19}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \end{array} \]

Alternative 7
Error	30.0
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq 2 \cdot 10^{-55}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \]

Alternative 8
Error	32.3
Cost	320

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

Reproduce

herbie shell --seed 2023023 
(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))