Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Average Error: 6.8 → 2.1

Time: 6.1s

Precision: binary64

\[[y, t] = \mathsf{sort}([y, t]) \\]

Math

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

↓

\[\begin{array}{l} t_1 := \left(x \cdot y - y \cdot z\right) \cdot t\\ \mathbf{if}\;t_1 \leq -1.6308848601928538 \cdot 10^{+161}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;t_1 \leq 1.1804557241858011 \cdot 10^{+305}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \end{array} \]

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 (<= t_1 -1.6308848601928538e+161)
     (* (- x z) (* y t))
     (if (<= t_1 1.1804557241858011e+305)
       (* t (* y (- x z)))
       (* 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 <= -1.6308848601928538e+161) {
		tmp = (x - z) * (y * t);
	} else if (t_1 <= 1.1804557241858011e+305) {
		tmp = t * (y * (x - z));
	} 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 <= (-1.6308848601928538d+161)) then
        tmp = (x - z) * (y * t)
    else if (t_1 <= 1.1804557241858011d+305) then
        tmp = t * (y * (x - z))
    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 <= -1.6308848601928538e+161) {
		tmp = (x - z) * (y * t);
	} else if (t_1 <= 1.1804557241858011e+305) {
		tmp = t * (y * (x - z));
	} 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 <= -1.6308848601928538e+161:
		tmp = (x - z) * (y * t)
	elif t_1 <= 1.1804557241858011e+305:
		tmp = t * (y * (x - z))
	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 <= -1.6308848601928538e+161)
		tmp = Float64(Float64(x - z) * Float64(y * t));
	elseif (t_1 <= 1.1804557241858011e+305)
		tmp = Float64(t * Float64(y * Float64(x - z)));
	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 <= -1.6308848601928538e+161)
		tmp = (x - z) * (y * t);
	elseif (t_1 <= 1.1804557241858011e+305)
		tmp = t * (y * (x - z));
	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[LessEqual[t$95$1, -1.6308848601928538e+161], N[(N[(x - z), $MachinePrecision] * N[(y * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.1804557241858011e+305], N[(t * N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(t * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.8
Target	3.7
Herbie	2.1

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 (*.f64 x y) (*.f64 z y)) t) < -1.63088486019285377e161

   1. Initial program 18.9

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

   2. Taylor expanded in y around inf 5.2

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

   if -1.63088486019285377e161 < (*.f64 (-.f64 (*.f64 x y) (*.f64 z y)) t) < 1.18045572418580112e305

   1. Initial program 1.7

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

   2. Taylor expanded in y around 0 1.7

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

   if 1.18045572418580112e305 < (*.f64 (-.f64 (*.f64 x y) (*.f64 z y)) t)

   1. Initial program 62.8

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

   2. Simplified 0.4

      \[\leadsto \color{blue}{y \cdot \left(t \cdot \left(x - z\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot y - y \cdot z\right) \cdot t \leq -1.6308848601928538 \cdot 10^{+161}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;\left(x \cdot y - y \cdot z\right) \cdot t \leq 1.1804557241858011 \cdot 10^{+305}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022130 
(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))