Statistics.Sample:$skurtosis from math-functions-0.1.5.2

Average Accuracy: 94.1% → 99.9%

Time: 2.9s

Precision: binary64

Cost: 448

• 25.4% of points produce a very large (infinite) output. You may want to add a precondition.

Math

\[\frac{x}{y \cdot y} - 3 \]

↓

\[\frac{\frac{x}{y}}{y} - 3 \]

FPCore

(FPCore (x y) :precision binary64 (- (/ x (* y y)) 3.0))

↓

(FPCore (x y) :precision binary64 (- (/ (/ x y) y) 3.0))

C

double code(double x, double y) {
	return (x / (y * y)) - 3.0;
}

↓

double code(double x, double y) {
	return ((x / y) / y) - 3.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x / (y * y)) - 3.0d0
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x / y) / y) - 3.0d0
end function

Java

public static double code(double x, double y) {
	return (x / (y * y)) - 3.0;
}

↓

public static double code(double x, double y) {
	return ((x / y) / y) - 3.0;
}

Python

def code(x, y):
	return (x / (y * y)) - 3.0

↓

def code(x, y):
	return ((x / y) / y) - 3.0

Julia

function code(x, y)
	return Float64(Float64(x / Float64(y * y)) - 3.0)
end

↓

function code(x, y)
	return Float64(Float64(Float64(x / y) / y) - 3.0)
end

MATLAB

function tmp = code(x, y)
	tmp = (x / (y * y)) - 3.0;
end

↓

function tmp = code(x, y)
	tmp = ((x / y) / y) - 3.0;
end

Wolfram

code[x_, y_] := N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] - 3.0), $MachinePrecision]

↓

code[x_, y_] := N[(N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision] - 3.0), $MachinePrecision]

Target

Original	94.1%
Target	99.9%
Herbie	99.9%

\[\frac{\frac{x}{y}}{y} - 3 \]

Derivation

1. Initial program 94.2%

   \[\frac{x}{y \cdot y} - 3 \]

2. Taylor expanded in x around 0 94.2%

   \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} - 3 \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} - 3 \]
   Step-by-step derivation

   [Start] 94.2	\[ \frac{x}{{y}^{2}} - 3 \]
   unpow2 [=>] 94.2	\[ \frac{x}{\color{blue}{y \cdot y}} - 3 \]
   associate-/r* [=>] 99.9	\[ \color{blue}{\frac{\frac{x}{y}}{y}} - 3 \]

4. Final simplification 99.9%

   \[\leadsto \frac{\frac{x}{y}}{y} - 3 \]

Alternatives

Alternative 1
Accuracy	94.1%
Cost	448

\[\frac{x}{y \cdot y} - 3 \]

Reproduce

herbie shell --seed 2023159 
(FPCore (x y)
  :name "Statistics.Sample:$skurtosis from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (- (/ (/ x y) y) 3.0)

  (- (/ x (* y y)) 3.0))