Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Average Error: 7.6 → 1.3

Time: 8.4s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_1 := \left(x \cdot y - y \cdot z\right) \cdot t\\ t_2 := y \cdot \left(x \cdot t - z \cdot t\right)\\ \mathbf{if}\;t_1 \leq -9.059539056078665 \cdot 10^{+292}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq -1.5810525527869248 \cdot 10^{-77}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 4.214196659200732 \cdot 10^{-279}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 1.874049262200502 \cdot 10^{+287}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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)) (t_2 (* y (- (* x t) (* z t)))))
   (if (<= t_1 -9.059539056078665e+292)
     t_2
     (if (<= t_1 -1.5810525527869248e-77)
       t_1
       (if (<= t_1 4.214196659200732e-279)
         t_2
         (if (<= t_1 1.874049262200502e+287) (* t (* y (- x z))) t_2))))))

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 t_2 = y * ((x * t) - (z * t));
	double tmp;
	if (t_1 <= -9.059539056078665e+292) {
		tmp = t_2;
	} else if (t_1 <= -1.5810525527869248e-77) {
		tmp = t_1;
	} else if (t_1 <= 4.214196659200732e-279) {
		tmp = t_2;
	} else if (t_1 <= 1.874049262200502e+287) {
		tmp = t * (y * (x - z));
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = ((x * y) - (y * z)) * t
    t_2 = y * ((x * t) - (z * t))
    if (t_1 <= (-9.059539056078665d+292)) then
        tmp = t_2
    else if (t_1 <= (-1.5810525527869248d-77)) then
        tmp = t_1
    else if (t_1 <= 4.214196659200732d-279) then
        tmp = t_2
    else if (t_1 <= 1.874049262200502d+287) then
        tmp = t * (y * (x - z))
    else
        tmp = t_2
    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 t_2 = y * ((x * t) - (z * t));
	double tmp;
	if (t_1 <= -9.059539056078665e+292) {
		tmp = t_2;
	} else if (t_1 <= -1.5810525527869248e-77) {
		tmp = t_1;
	} else if (t_1 <= 4.214196659200732e-279) {
		tmp = t_2;
	} else if (t_1 <= 1.874049262200502e+287) {
		tmp = t * (y * (x - z));
	} else {
		tmp = t_2;
	}
	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
	t_2 = y * ((x * t) - (z * t))
	tmp = 0
	if t_1 <= -9.059539056078665e+292:
		tmp = t_2
	elif t_1 <= -1.5810525527869248e-77:
		tmp = t_1
	elif t_1 <= 4.214196659200732e-279:
		tmp = t_2
	elif t_1 <= 1.874049262200502e+287:
		tmp = t * (y * (x - z))
	else:
		tmp = t_2
	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)
	t_2 = Float64(y * Float64(Float64(x * t) - Float64(z * t)))
	tmp = 0.0
	if (t_1 <= -9.059539056078665e+292)
		tmp = t_2;
	elseif (t_1 <= -1.5810525527869248e-77)
		tmp = t_1;
	elseif (t_1 <= 4.214196659200732e-279)
		tmp = t_2;
	elseif (t_1 <= 1.874049262200502e+287)
		tmp = Float64(t * Float64(y * Float64(x - z)));
	else
		tmp = t_2;
	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;
	t_2 = y * ((x * t) - (z * t));
	tmp = 0.0;
	if (t_1 <= -9.059539056078665e+292)
		tmp = t_2;
	elseif (t_1 <= -1.5810525527869248e-77)
		tmp = t_1;
	elseif (t_1 <= 4.214196659200732e-279)
		tmp = t_2;
	elseif (t_1 <= 1.874049262200502e+287)
		tmp = t * (y * (x - z));
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(y * N[(N[(x * t), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -9.059539056078665e+292], t$95$2, If[LessEqual[t$95$1, -1.5810525527869248e-77], t$95$1, If[LessEqual[t$95$1, 4.214196659200732e-279], t$95$2, If[LessEqual[t$95$1, 1.874049262200502e+287], N[(t * N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.6
Target	3.4
Herbie	1.3

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 (*.f64 x y) (*.f64 z y)) t) < -9.0595390560786649e292 or -1.58105255278692478e-77 < (*.f64 (-.f64 (*.f64 x y) (*.f64 z y)) t) < 4.21419665920073161e-279 or 1.8740492622005021e287 < (*.f64 (-.f64 (*.f64 x y) (*.f64 z y)) t)

   1. Initial program 19.7

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

   2. Taylor expanded in x around 0 2.8

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

   3. Applied distribute-lft-out--_binary64 2.8

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

   if -9.0595390560786649e292 < (*.f64 (-.f64 (*.f64 x y) (*.f64 z y)) t) < -1.58105255278692478e-77

   1. Initial program 0.3

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

   if 4.21419665920073161e-279 < (*.f64 (-.f64 (*.f64 x y) (*.f64 z y)) t) < 1.8740492622005021e287

   1. Initial program 0.4

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

   2. Taylor expanded in y around 0 0.4

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

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot y - y \cdot z\right) \cdot t \leq -9.059539056078665 \cdot 10^{+292}:\\ \;\;\;\;y \cdot \left(x \cdot t - z \cdot t\right)\\ \mathbf{elif}\;\left(x \cdot y - y \cdot z\right) \cdot t \leq -1.5810525527869248 \cdot 10^{-77}:\\ \;\;\;\;\left(x \cdot y - y \cdot z\right) \cdot t\\ \mathbf{elif}\;\left(x \cdot y - y \cdot z\right) \cdot t \leq 4.214196659200732 \cdot 10^{-279}:\\ \;\;\;\;y \cdot \left(x \cdot t - z \cdot t\right)\\ \mathbf{elif}\;\left(x \cdot y - y \cdot z\right) \cdot t \leq 1.874049262200502 \cdot 10^{+287}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot t - z \cdot t\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(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))