Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Error: 6.8 → 1.3

Time: 4.7s

Precision: binary64

Math

\[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]

↓

\[\begin{array}{l} t_1 := y \cdot z - z \cdot t\\ t_2 := 2 \cdot \frac{\frac{x}{y - t}}{z}\\ \mathbf{if}\;t_1 \leq -3.913792715055396 \cdot 10^{+151}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq -1.2691724707153681 \cdot 10^{-107}:\\ \;\;\;\;\frac{2 \cdot x}{t_1}\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \mathbf{elif}\;t_1 \leq 1.4516736965324895 \cdot 10^{+90}:\\ \;\;\;\;\frac{2 \cdot x}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ (* x 2.0) (- (* y z) (* t z))))

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* y z) (* z t))) (t_2 (* 2.0 (/ (/ x (- y t)) z))))
   (if (<= t_1 -3.913792715055396e+151)
     t_2
     (if (<= t_1 -1.2691724707153681e-107)
       (/ (* 2.0 x) t_1)
       (if (<= t_1 0.0)
         (* 2.0 (/ (/ x z) (- y t)))
         (if (<= t_1 1.4516736965324895e+90)
           (/ (* 2.0 x) (* z (- y t)))
           t_2))))))

C

double code(double x, double y, double z, double t) {
	return (x * 2.0) / ((y * z) - (t * z));
}

↓

double code(double x, double y, double z, double t) {
	double t_1 = (y * z) - (z * t);
	double t_2 = 2.0 * ((x / (y - t)) / z);
	double tmp;
	if (t_1 <= -3.913792715055396e+151) {
		tmp = t_2;
	} else if (t_1 <= -1.2691724707153681e-107) {
		tmp = (2.0 * x) / t_1;
	} else if (t_1 <= 0.0) {
		tmp = 2.0 * ((x / z) / (y - t));
	} else if (t_1 <= 1.4516736965324895e+90) {
		tmp = (2.0 * x) / (z * (y - t));
	} 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 * 2.0d0) / ((y * z) - (t * z))
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 = (y * z) - (z * t)
    t_2 = 2.0d0 * ((x / (y - t)) / z)
    if (t_1 <= (-3.913792715055396d+151)) then
        tmp = t_2
    else if (t_1 <= (-1.2691724707153681d-107)) then
        tmp = (2.0d0 * x) / t_1
    else if (t_1 <= 0.0d0) then
        tmp = 2.0d0 * ((x / z) / (y - t))
    else if (t_1 <= 1.4516736965324895d+90) then
        tmp = (2.0d0 * x) / (z * (y - t))
    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 * 2.0) / ((y * z) - (t * z));
}

↓

public static double code(double x, double y, double z, double t) {
	double t_1 = (y * z) - (z * t);
	double t_2 = 2.0 * ((x / (y - t)) / z);
	double tmp;
	if (t_1 <= -3.913792715055396e+151) {
		tmp = t_2;
	} else if (t_1 <= -1.2691724707153681e-107) {
		tmp = (2.0 * x) / t_1;
	} else if (t_1 <= 0.0) {
		tmp = 2.0 * ((x / z) / (y - t));
	} else if (t_1 <= 1.4516736965324895e+90) {
		tmp = (2.0 * x) / (z * (y - t));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	return (x * 2.0) / ((y * z) - (t * z))

↓

def code(x, y, z, t):
	t_1 = (y * z) - (z * t)
	t_2 = 2.0 * ((x / (y - t)) / z)
	tmp = 0
	if t_1 <= -3.913792715055396e+151:
		tmp = t_2
	elif t_1 <= -1.2691724707153681e-107:
		tmp = (2.0 * x) / t_1
	elif t_1 <= 0.0:
		tmp = 2.0 * ((x / z) / (y - t))
	elif t_1 <= 1.4516736965324895e+90:
		tmp = (2.0 * x) / (z * (y - t))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	return Float64(Float64(x * 2.0) / Float64(Float64(y * z) - Float64(t * z)))
end

↓

function code(x, y, z, t)
	t_1 = Float64(Float64(y * z) - Float64(z * t))
	t_2 = Float64(2.0 * Float64(Float64(x / Float64(y - t)) / z))
	tmp = 0.0
	if (t_1 <= -3.913792715055396e+151)
		tmp = t_2;
	elseif (t_1 <= -1.2691724707153681e-107)
		tmp = Float64(Float64(2.0 * x) / t_1);
	elseif (t_1 <= 0.0)
		tmp = Float64(2.0 * Float64(Float64(x / z) / Float64(y - t)));
	elseif (t_1 <= 1.4516736965324895e+90)
		tmp = Float64(Float64(2.0 * x) / Float64(z * Float64(y - t)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * 2.0) / ((y * z) - (t * z));
end

↓

function tmp_2 = code(x, y, z, t)
	t_1 = (y * z) - (z * t);
	t_2 = 2.0 * ((x / (y - t)) / z);
	tmp = 0.0;
	if (t_1 <= -3.913792715055396e+151)
		tmp = t_2;
	elseif (t_1 <= -1.2691724707153681e-107)
		tmp = (2.0 * x) / t_1;
	elseif (t_1 <= 0.0)
		tmp = 2.0 * ((x / z) / (y - t));
	elseif (t_1 <= 1.4516736965324895e+90)
		tmp = (2.0 * x) / (z * (y - t));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * 2.0), $MachinePrecision] / N[(N[(y * z), $MachinePrecision] - N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[(x / N[(y - t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -3.913792715055396e+151], t$95$2, If[LessEqual[t$95$1, -1.2691724707153681e-107], N[(N[(2.0 * x), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(2.0 * N[(N[(x / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.4516736965324895e+90], N[(N[(2.0 * x), $MachinePrecision] / N[(z * N[(y - t), $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	6.8
Target	2.1
Herbie	1.3

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} < -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} < 1.045027827330126 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array} \]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < -3.9137927150553962e151 or 1.4516736965324895e90 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 11.2

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]

   2. Simplified 1.9

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{x}{z}}{y - t}} \]

   3. Applied egg-rr 1.9

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{x}{z} \cdot \frac{1}{y - t}\right)} \]

   4. Applied egg-rr 1.6

      \[\leadsto 2 \cdot \color{blue}{\frac{\frac{x}{y - t}}{z}} \]

   if -3.9137927150553962e151 < (-.f64 (*.f64 y z) (*.f64 t z)) < -1.26917247071536813e-107

   1. Initial program 0.3

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]

   if -1.26917247071536813e-107 < (-.f64 (*.f64 y z) (*.f64 t z)) < -0.0

   1. Initial program 12.6

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]

   2. Simplified 4.2

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{x}{z}}{y - t}} \]

   3. Applied egg-rr 4.3

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{x}{z} \cdot \frac{1}{y - t}\right)} \]

   4. Applied egg-rr 4.2

      \[\leadsto 2 \cdot \color{blue}{\frac{\frac{x}{y - t}}{z}} \]

   5. Taylor expanded in x around 0 12.6

      \[\leadsto \color{blue}{2 \cdot \frac{x}{\left(y - t\right) \cdot z}} \]

   6. Simplified 4.2

      \[\leadsto \color{blue}{\frac{\frac{x}{z}}{y - t} \cdot 2} \]

   if -0.0 < (-.f64 (*.f64 y z) (*.f64 t z)) < 1.4516736965324895e90

   1. Initial program 0.5

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]

   2. Simplified 10.4

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{x}{z}}{y - t}} \]

   3. Applied egg-rr 0.5

      \[\leadsto \color{blue}{\frac{x \cdot 2}{z \cdot \left(y - t\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z - z \cdot t \leq -3.913792715055396 \cdot 10^{+151}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y - t}}{z}\\ \mathbf{elif}\;y \cdot z - z \cdot t \leq -1.2691724707153681 \cdot 10^{-107}:\\ \;\;\;\;\frac{2 \cdot x}{y \cdot z - z \cdot t}\\ \mathbf{elif}\;y \cdot z - z \cdot t \leq 0:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \mathbf{elif}\;y \cdot z - z \cdot t \leq 1.4516736965324895 \cdot 10^{+90}:\\ \;\;\;\;\frac{2 \cdot x}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y - t}}{z}\\ \end{array} \]

Reproduce

herbie shell --seed 2022150 
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (if (< (/ (* x 2.0) (- (* y z) (* t z))) -2.559141628295061e-13) (* (/ x (* (- y t) z)) 2.0) (if (< (/ (* x 2.0) (- (* y z) (* t z))) 1.045027827330126e-269) (/ (* (/ x z) 2.0) (- y t)) (* (/ x (* (- y t) z)) 2.0)))

  (/ (* x 2.0) (- (* y z) (* t z))))