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

Average Error: 8.3 → 0.1

Time: 1.9s

Precision: binary64

Cost: 448

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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 + 1.0d0) / y)
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 + 1.0) / y);
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Target

Original	8.3
Target	0.0
Herbie	0.1

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

Derivation

1. Initial program 8.3

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

2. Simplified 0.1

   \[\leadsto \color{blue}{\frac{x}{\frac{y + 1}{y}}} \]
   Proof
   (/.f64 x (/.f64 (+.f64 y 1) y)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x y) (+.f64 y 1))): 42 points increase in error, 19 points decrease in error

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	1.1
Cost	584

\[\begin{array}{l} t_0 := x - \frac{x}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.3:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	1.4
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	31.1
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022338 
(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)))