Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, B

Average Error: 8.0 → 0.0

Time: 2.4s

Precision: binary64

Cost: 448

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Fortran

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

↓

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

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Target

Original	8.0
Target	0.0
Herbie	0.0

\[\begin{array}{l} \mathbf{if}\;y < -3693.8482788297247:\\ \;\;\;\;\frac{x}{y \cdot y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y < 6799310503.41891:\\ \;\;\;\;\frac{x \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y} - \left(\frac{x}{y} - x\right)\\ \end{array} \]

Derivation

1. Initial program 8.0

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

2. Simplified 0.0

   \[\leadsto \color{blue}{x \cdot \frac{y}{y + 1}} \]
   Proof
   (*.f64 x (/.f64 y (+.f64 y 1))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 x y) (+.f64 y 1))): 32 points increase in error, 5 points decrease in error

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	1.2
Cost	584

\[\begin{array}{l} t_0 := x - \frac{x}{y}\\ \mathbf{if}\;y \leq -218.53919179681262:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.607625927258275 \cdot 10^{-8}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	1.4
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -218.53919179681262:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.607625927258275 \cdot 10^{-8}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	0.1
Cost	448

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

Alternative 4
Error	31.2
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022291 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, B"
  :precision binary64

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ x (* y y)) (- (/ x y) x)) (if (< y 6799310503.41891) (/ (* x y) (+ y 1.0)) (- (/ x (* y y)) (- (/ x y) x))))

  (/ (* x y) (+ y 1.0)))