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

Average Accuracy: 87.8% → 99.8%

Time: 3.4s

Precision: binary64

Cost: 448

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

double code(double x, double y) {
	return x / (1.0 + (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 / (1.0d0 + (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 / (1.0 + (1.0 / y));
}

Python

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

↓

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

MATLAB

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

↓

function tmp = code(x, y)
	tmp = x / (1.0 + (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[(1.0 + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	87.8%
Target	100.0%
Herbie	99.8%

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

Derivation

1. Initial program 87.8%

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

2. Simplified 99.8%

   \[\leadsto \color{blue}{\frac{x}{\frac{y + 1}{y}}} \]
   Proof

   [Start] 87.8	\[ \frac{x \cdot y}{y + 1} \]
   associate-/l* [=>] 99.8	\[ \color{blue}{\frac{x}{\frac{y + 1}{y}}} \]

3. Taylor expanded in y around 0 99.8%

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

4. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	98.3%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.35\right):\\ \;\;\;\;x - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 2
Accuracy	97.7%
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
Accuracy	50.5%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023129 
(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)))