Linear.Projection:perspective from linear-1.19.1.3, A

Average Error: 0.04% → 0.04%

Time: 2.8s

Precision: binary64

Cost: 448

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Python

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Target

Original	0.04%
Target	0.04%
Herbie	0.04%

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

Derivation

1. Initial program 0.04

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

2. Final simplification 0.04

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

Alternatives

Alternative 1
Error	26.29%
Cost	976

\[\begin{array}{l} t_0 := 1 + 2 \cdot \frac{y}{x}\\ \mathbf{if}\;x \leq -4 \cdot 10^{-14}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-153}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-134}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-41}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	25.74%
Cost	976

\[\begin{array}{l} t_0 := 1 + 2 \cdot \frac{y}{x}\\ t_1 := -2 \cdot \frac{x}{y} + -1\\ \mathbf{if}\;x \leq -3 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-134}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-43}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	26.77%
Cost	592

\[\begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-153}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-133}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-55}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Error	50.99%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023090 
(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)))