Linear.Projection:perspective from linear-1.19.1.3, A

Average Error: 0.05% → 0.05%

Time: 5.8s

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	0.05%
Target	0.04%
Herbie	0.05%

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

Derivation

1. Initial program 0.05

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

2. Final simplification 0.05

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

Alternatives

Alternative 1
Error	25.84%
Cost	978

\[\begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{+82} \lor \neg \left(y \leq -1150 \lor \neg \left(y \leq -3.6 \cdot 10^{-51}\right) \land y \leq 4.1 \cdot 10^{-9}\right):\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \end{array} \]

Alternative 2
Error	26.34%
Cost	977

\[\begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+80}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -1150 \lor \neg \left(y \leq -4 \cdot 10^{-51}\right) \land y \leq 7.2 \cdot 10^{-9}:\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 3
Error	27.8%
Cost	592

\[\begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+92}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -200000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.5 \cdot 10^{-83}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 4
Error	49.93%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023088 
(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)))