Linear.Projection:perspective from linear-1.19.1.3, B

Average Error: 15.2 → 0.1

Time: 5.0s

Precision: binary64

Cost: 840

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x + x}{\frac{x - y}{y}}\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{+21}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-21}:\\ \;\;\;\;\left(2 \cdot y\right) \cdot \frac{x}{x - y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x x) (/ (- x y) y))))
   (if (<= y -2.3e+21)
     t_0
     (if (<= y 3.6e-21) (* (* 2.0 y) (/ x (- x 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 = (x + x) / ((x - y) / y);
	double tmp;
	if (y <= -2.3e+21) {
		tmp = t_0;
	} else if (y <= 3.6e-21) {
		tmp = (2.0 * y) * (x / (x - 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 = (x + x) / ((x - y) / y)
    if (y <= (-2.3d+21)) then
        tmp = t_0
    else if (y <= 3.6d-21) then
        tmp = (2.0d0 * y) * (x / (x - 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 = (x + x) / ((x - y) / y);
	double tmp;
	if (y <= -2.3e+21) {
		tmp = t_0;
	} else if (y <= 3.6e-21) {
		tmp = (2.0 * y) * (x / (x - 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 = (x + x) / ((x - y) / y)
	tmp = 0
	if y <= -2.3e+21:
		tmp = t_0
	elif y <= 3.6e-21:
		tmp = (2.0 * y) * (x / (x - 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(Float64(x + x) / Float64(Float64(x - y) / y))
	tmp = 0.0
	if (y <= -2.3e+21)
		tmp = t_0;
	elseif (y <= 3.6e-21)
		tmp = Float64(Float64(2.0 * y) * Float64(x / Float64(x - 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 = (x + x) / ((x - y) / y);
	tmp = 0.0;
	if (y <= -2.3e+21)
		tmp = t_0;
	elseif (y <= 3.6e-21)
		tmp = (2.0 * y) * (x / (x - 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[(N[(x + x), $MachinePrecision] / N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.3e+21], t$95$0, If[LessEqual[y, 3.6e-21], N[(N[(2.0 * y), $MachinePrecision] * N[(x / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Target

Original	15.2
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 y < -2.3e21 or 3.59999999999999989e-21 < y

   1. Initial program 16.5

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

   2. Simplified 13.9

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

      [Start] 16.5	\[ \frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
      rational.json-simplify-2 [=>] 16.5	\[ \frac{\color{blue}{y \cdot \left(x \cdot 2\right)}}{x - y} \]
      rational.json-simplify-43 [=>] 16.6	\[ \frac{\color{blue}{x \cdot \left(2 \cdot y\right)}}{x - y} \]
      rational.json-simplify-49 [=>] 13.9	\[ \color{blue}{\left(2 \cdot y\right) \cdot \frac{x}{x - y}} \]

   3. Applied egg-rr 0.2

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

   if -2.3e21 < y < 3.59999999999999989e-21

   1. Initial program 13.8

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

   2. Simplified 0.1

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

      [Start] 13.8	\[ \frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
      rational.json-simplify-2 [=>] 13.8	\[ \frac{\color{blue}{y \cdot \left(x \cdot 2\right)}}{x - y} \]
      rational.json-simplify-43 [=>] 13.8	\[ \frac{\color{blue}{x \cdot \left(2 \cdot y\right)}}{x - y} \]
      rational.json-simplify-49 [=>] 0.1	\[ \color{blue}{\left(2 \cdot y\right) \cdot \frac{x}{x - y}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+21}:\\ \;\;\;\;\frac{x + x}{\frac{x - y}{y}}\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-21}:\\ \;\;\;\;\left(2 \cdot y\right) \cdot \frac{x}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + x}{\frac{x - y}{y}}\\ \end{array} \]

Alternatives

Alternative 1
Error	3.8
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+191}:\\ \;\;\;\;x \cdot -2\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+116}:\\ \;\;\;\;\left(2 \cdot y\right) \cdot \frac{x}{x - y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2\\ \end{array} \]

Alternative 2
Error	0.1
Cost	840

\[\begin{array}{l} t_0 := \left(x \cdot 2\right) \cdot \frac{y}{x - y}\\ \mathbf{if}\;y \leq -4 \cdot 10^{+20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\left(2 \cdot y\right) \cdot \frac{x}{x - y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	17.7
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-79}:\\ \;\;\;\;y + y\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{-83}:\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;y + y\\ \end{array} \]

Alternative 4
Error	31.8
Cost	192

\[x \cdot -2 \]

Reproduce

herbie shell --seed 2023064 
(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)))