Linear.Projection:perspective from linear-1.19.1.3, A

Average Error: 0.04% → 0.05%

Time: 3.7s

Precision: binary64

Cost: 576

Math

\[\frac{x + y}{x - y} \]

↓

\[\frac{1}{\frac{x - y}{x + y}} \]

FPCore

(FPCore (x y) :precision binary64 (/ (+ x y) (- x y)))

↓

(FPCore (x y) :precision binary64 (/ 1.0 (/ (- x y) (+ x y))))

C

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

↓

double code(double x, double y) {
	return 1.0 / ((x - y) / (x + y));
}

Fortran

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

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / ((x - y) / (x + y))
end function

Java

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

↓

public static double code(double x, double y) {
	return 1.0 / ((x - y) / (x + y));
}

Python

def code(x, y):
	return (x + y) / (x - y)

↓

def code(x, y):
	return 1.0 / ((x - y) / (x + y))

Julia

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

↓

function code(x, y)
	return Float64(1.0 / Float64(Float64(x - y) / Float64(x + y)))
end

MATLAB

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

↓

function tmp = code(x, y)
	tmp = 1.0 / ((x - y) / (x + y));
end

Wolfram

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

↓

code[x_, y_] := N[(1.0 / N[(N[(x - y), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	0.04%
Target	0.04%
Herbie	0.05%

\[\frac{1}{\frac{x}{x + y} - \frac{y}{x + y}} \]

Derivation

1. Initial program 0.04

   \[\frac{x + y}{x - y} \]

2. Applied egg-rr 48.3

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

3. Applied egg-rr 0.05

   \[\leadsto \color{blue}{\frac{1}{\frac{x - y}{x + y}}} \]

4. Final simplification 0.05

   \[\leadsto \frac{1}{\frac{x - y}{x + y}} \]

Alternatives

Alternative 1
Error	24.83%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{-17} \lor \neg \left(x \leq 7.2 \cdot 10^{-53}\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 2
Error	24.38%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{-17} \lor \neg \left(x \leq 7.8 \cdot 10^{-53}\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \end{array} \]

Alternative 3
Error	0.04%
Cost	448

\[\frac{x + y}{x - y} \]

Alternative 4
Error	25.47%
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-15}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{-53}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	51.03%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023121 
(FPCore (x y)
  :name "Linear.Projection:perspective from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (/ 1.0 (- (/ x (+ x y)) (/ y (+ x y))))

  (/ (+ x y) (- x y)))