Linear.Projection:perspective from linear-1.19.1.3, B

Average Error: 15.0 → 0.1

Time: 5.5s

Precision: binary64

Cost: 840

Math

\[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]

↓

\[\begin{array}{l} t_0 := y \cdot \frac{x + x}{x - y}\\ \mathbf{if}\;x \leq -21025673474.39561:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.359056827737975 \cdot 10^{-38}:\\ \;\;\;\;\frac{x + x}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (/ (+ x x) (- x y)))))
   (if (<= x -21025673474.39561)
     t_0
     (if (<= x 2.359056827737975e-38) (/ (+ x x) (/ (- x y) y)) t_0))))

C

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

↓

double code(double x, double y) {
	double t_0 = y * ((x + x) / (x - y));
	double tmp;
	if (x <= -21025673474.39561) {
		tmp = t_0;
	} else if (x <= 2.359056827737975e-38) {
		tmp = (x + x) / ((x - y) / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * 2.0d0) * y) / (x - y)
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * ((x + x) / (x - y))
    if (x <= (-21025673474.39561d0)) then
        tmp = t_0
    else if (x <= 2.359056827737975d-38) then
        tmp = (x + x) / ((x - y) / y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	return ((x * 2.0) * y) / (x - y);
}

↓

public static double code(double x, double y) {
	double t_0 = y * ((x + x) / (x - y));
	double tmp;
	if (x <= -21025673474.39561) {
		tmp = t_0;
	} else if (x <= 2.359056827737975e-38) {
		tmp = (x + x) / ((x - y) / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

↓

def code(x, y):
	t_0 = y * ((x + x) / (x - y))
	tmp = 0
	if x <= -21025673474.39561:
		tmp = t_0
	elif x <= 2.359056827737975e-38:
		tmp = (x + x) / ((x - y) / y)
	else:
		tmp = t_0
	return tmp

Julia

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

↓

function code(x, y)
	t_0 = Float64(y * Float64(Float64(x + x) / Float64(x - y)))
	tmp = 0.0
	if (x <= -21025673474.39561)
		tmp = t_0;
	elseif (x <= 2.359056827737975e-38)
		tmp = Float64(Float64(x + x) / Float64(Float64(x - y) / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y)
	t_0 = y * ((x + x) / (x - y));
	tmp = 0.0;
	if (x <= -21025673474.39561)
		tmp = t_0;
	elseif (x <= 2.359056827737975e-38)
		tmp = (x + x) / ((x - y) / y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_] := Block[{t$95$0 = N[(y * N[(N[(x + x), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -21025673474.39561], t$95$0, If[LessEqual[x, 2.359056827737975e-38], N[(N[(x + x), $MachinePrecision] / N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Target

Original	15.0
Target	0.4
Herbie	0.1

\[\begin{array}{l} \mathbf{if}\;x < -1.7210442634149447 \cdot 10^{+81}:\\ \;\;\;\;\frac{2 \cdot x}{x - y} \cdot y\\ \mathbf{elif}\;x < 83645045635564430:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot x}{x - y} \cdot y\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if x < -21025673474.3956108 or 2.359056827737975e-38 < x

   1. Initial program 14.9

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]

   2. Applied egg-rr 0.1

      \[\leadsto \color{blue}{\frac{x + x}{x - y} \cdot y} \]

   if -21025673474.3956108 < x < 2.359056827737975e-38

   1. Initial program 15.1

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]

   2. Applied egg-rr 14.9

      \[\leadsto \color{blue}{\frac{x + x}{x - y} \cdot y} \]

   3. Applied egg-rr 0.0

      \[\leadsto \color{blue}{\frac{x + x}{\frac{x - y}{y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -21025673474.39561:\\ \;\;\;\;y \cdot \frac{x + x}{x - y}\\ \mathbf{elif}\;x \leq 2.359056827737975 \cdot 10^{-38}:\\ \;\;\;\;\frac{x + x}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x + x}{x - y}\\ \end{array} \]

Alternatives

Alternative 1
Error	15.9
Cost	972

\[\begin{array}{l} t_0 := -2 \cdot \left(x + x \cdot \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -11693451672896.02:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 6.428657201951684 \cdot 10^{-53}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;y \leq 4.498192215726338 \cdot 10^{-40}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.0024052127302475993:\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-0.5}\\ \end{array} \]

Alternative 2
Error	4.9
Cost	840

\[\begin{array}{l} t_0 := \frac{x + x}{\frac{x - y}{y}}\\ \mathbf{if}\;y \leq -3.844228748840974 \cdot 10^{-232}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.048255675901521 \cdot 10^{-80}:\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	4.9
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -3.844228748840974 \cdot 10^{-232}:\\ \;\;\;\;\frac{x + x}{\frac{x - y}{y}}\\ \mathbf{elif}\;y \leq 2.048255675901521 \cdot 10^{-80}:\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{0.5 \cdot \frac{x}{y} + -0.5}\\ \end{array} \]

Alternative 4
Error	16.2
Cost	720

\[\begin{array}{l} \mathbf{if}\;y \leq -11693451672896.02:\\ \;\;\;\;\frac{x}{-0.5}\\ \mathbf{elif}\;y \leq 4.1597244160640194 \cdot 10^{-72}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;y \leq 4.498192215726338 \cdot 10^{-40}:\\ \;\;\;\;\frac{x}{-0.5}\\ \mathbf{elif}\;y \leq 0.0024052127302475993:\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-0.5}\\ \end{array} \]

Alternative 5
Error	31.7
Cost	192

\[\frac{x}{-0.5} \]

Reproduce

herbie shell --seed 2022308 
(FPCore (x y)
  :name "Linear.Projection:perspective from linear-1.19.1.3, B"
  :precision binary64

  :herbie-target
  (if (< x -1.7210442634149447e+81) (* (/ (* 2.0 x) (- x y)) y) (if (< x 83645045635564430.0) (/ (* x 2.0) (/ (- x y) y)) (* (/ (* 2.0 x) (- x y)) y)))

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