Linear.Projection:perspective from linear-1.19.1.3, A

Average Accuracy: 100.0% → 100.0%

Time: 10.0s

Precision: binary64

Cost: 704

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Target

Original	100.0%
Target	100.0%
Herbie	100.0%

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

Derivation

1. Initial program 100.0%

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

2. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\log \left(e^{\frac{x + y}{x - y}}\right)} \]
   Proof

   [Start] 100.0	\[ \frac{x + y}{x - y} \]
   add-log-exp [=>] 99.9	\[ \color{blue}{\log \left(e^{\frac{x + y}{x - y}}\right)} \]

3. Applied egg-rr 34.1%

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

   [Start] 99.9	\[ \log \left(e^{\frac{x + y}{x - y}}\right) \]
   add-log-exp [<=] 100.0	\[ \color{blue}{\frac{x + y}{x - y}} \]
   flip3-+ [=>] 33.8	\[ \frac{\color{blue}{\frac{{x}^{3} + {y}^{3}}{x \cdot x + \left(y \cdot y - x \cdot y\right)}}}{x - y} \]
   associate-/r* [<=] 33.8	\[ \color{blue}{\frac{{x}^{3} + {y}^{3}}{\left(x \cdot x + \left(y \cdot y - x \cdot y\right)\right) \cdot \left(x - y\right)}} \]
   sum-cubes [=>] 33.9	\[ \frac{\color{blue}{\left(x \cdot x + \left(y \cdot y - x \cdot y\right)\right) \cdot \left(x + y\right)}}{\left(x \cdot x + \left(y \cdot y - x \cdot y\right)\right) \cdot \left(x - y\right)} \]
   associate-/l* [=>] 34.1	\[ \color{blue}{\frac{x \cdot x + \left(y \cdot y - x \cdot y\right)}{\frac{\left(x \cdot x + \left(y \cdot y - x \cdot y\right)\right) \cdot \left(x - y\right)}{x + y}}} \]
   fma-def [=>] 34.1	\[ \frac{\color{blue}{\mathsf{fma}\left(x, x, y \cdot y - x \cdot y\right)}}{\frac{\left(x \cdot x + \left(y \cdot y - x \cdot y\right)\right) \cdot \left(x - y\right)}{x + y}} \]
   distribute-rgt-out-- [=>] 34.1	\[ \frac{\mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(y - x\right)}\right)}{\frac{\left(x \cdot x + \left(y \cdot y - x \cdot y\right)\right) \cdot \left(x - y\right)}{x + y}} \]
   *-commutative [=>] 34.1	\[ \frac{\mathsf{fma}\left(x, x, y \cdot \left(y - x\right)\right)}{\frac{\color{blue}{\left(x - y\right) \cdot \left(x \cdot x + \left(y \cdot y - x \cdot y\right)\right)}}{x + y}} \]
   fma-def [=>] 34.1	\[ \frac{\mathsf{fma}\left(x, x, y \cdot \left(y - x\right)\right)}{\frac{\left(x - y\right) \cdot \color{blue}{\mathsf{fma}\left(x, x, y \cdot y - x \cdot y\right)}}{x + y}} \]
   distribute-rgt-out-- [=>] 34.1	\[ \frac{\mathsf{fma}\left(x, x, y \cdot \left(y - x\right)\right)}{\frac{\left(x - y\right) \cdot \mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(y - x\right)}\right)}{x + y}} \]

4. Simplified 34.0%

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

   [Start] 34.1	\[ \frac{\mathsf{fma}\left(x, x, y \cdot \left(y - x\right)\right)}{\frac{\left(x - y\right) \cdot \mathsf{fma}\left(x, x, y \cdot \left(y - x\right)\right)}{x + y}} \]
   associate-/r/ [=>] 34.0	\[ \color{blue}{\frac{\mathsf{fma}\left(x, x, y \cdot \left(y - x\right)\right)}{\left(x - y\right) \cdot \mathsf{fma}\left(x, x, y \cdot \left(y - x\right)\right)} \cdot \left(x + y\right)} \]

5. Applied egg-rr 100.0%

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

   [Start] 34.0	\[ \frac{\mathsf{fma}\left(x, x, y \cdot \left(y - x\right)\right)}{\left(x - y\right) \cdot \mathsf{fma}\left(x, x, y \cdot \left(y - x\right)\right)} \cdot \left(x + y\right) \]
   +-commutative [=>] 34.0	\[ \frac{\mathsf{fma}\left(x, x, y \cdot \left(y - x\right)\right)}{\left(x - y\right) \cdot \mathsf{fma}\left(x, x, y \cdot \left(y - x\right)\right)} \cdot \color{blue}{\left(y + x\right)} \]
   distribute-rgt-in [=>] 34.0	\[ \color{blue}{y \cdot \frac{\mathsf{fma}\left(x, x, y \cdot \left(y - x\right)\right)}{\left(x - y\right) \cdot \mathsf{fma}\left(x, x, y \cdot \left(y - x\right)\right)} + x \cdot \frac{\mathsf{fma}\left(x, x, y \cdot \left(y - x\right)\right)}{\left(x - y\right) \cdot \mathsf{fma}\left(x, x, y \cdot \left(y - x\right)\right)}} \]

6. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	73.8%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{+70} \lor \neg \left(x \leq 100000000\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 2
Accuracy	74.5%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{+69} \lor \neg \left(x \leq 2600000000000\right):\\ \;\;\;\;1 + 2 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{y} + -1\\ \end{array} \]

Alternative 3
Accuracy	100.0%
Cost	448

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

Alternative 4
Accuracy	72.9%
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+85}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 450000000:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Accuracy	49.3%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023137 
(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)))