Linear.Projection:perspective from linear-1.19.1.3, A

Percentage Accurate: 99.9% → 99.9%

Time: 6.7s

Precision: binary64

Cost: 576

• Unsound rule application detected in e-graph. Results may not be sound.

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	99.9%
Target	100.0%
Herbie	99.9%

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

Derivation

1. Initial program 99.9%

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

2. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{{\left(\frac{x - y}{x + y}\right)}^{-1}} \]
   Step-by-step derivation

   [Start] 99.9%	\[ \frac{x + y}{x - y} \]
   clear-num [=>] 99.9%	\[ \color{blue}{\frac{1}{\frac{x - y}{x + y}}} \]
   inv-pow [=>] 99.9%	\[ \color{blue}{{\left(\frac{x - y}{x + y}\right)}^{-1}} \]

3. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\frac{1}{\frac{x - y}{y + x}}} \]
   Step-by-step derivation

   [Start] 99.9%	\[ {\left(\frac{x - y}{x + y}\right)}^{-1} \]
   unpow-1 [=>] 99.9%	\[ \color{blue}{\frac{1}{\frac{x - y}{x + y}}} \]
   +-commutative [=>] 99.9%	\[ \frac{1}{\frac{x - y}{\color{blue}{y + x}}} \]

4. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.9%
Cost	576

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

Alternative 2
Accuracy	73.9%
Cost	1242

\[\begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+129}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -2.85 \cdot 10^{+100} \lor \neg \left(y \leq -2.95 \cdot 10^{-32}\right) \land \left(y \leq 1.15 \cdot 10^{-100} \lor \neg \left(y \leq 1.15 \cdot 10^{-92}\right) \land y \leq 2600000000000\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 3
Accuracy	75.2%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-14} \lor \neg \left(x \leq 3.2 \cdot 10^{+36}\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \end{array} \]

Alternative 4
Accuracy	99.9%
Cost	448

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

Alternative 5
Accuracy	74.0%
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{+36}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Accuracy	50.3%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023269 
(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)))