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

Average Error: 5.2 → 0.1

Time: 3.3s

Precision: binary64

Cost: 708

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{-310}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y} + -3\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 5e-310) (* (/ x y) (/ 1.0 y)) (+ (/ x (* y y)) -3.0)))

C

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

↓

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e-310) {
		tmp = (x / y) * (1.0 / y);
	} else {
		tmp = (x / (y * y)) + -3.0;
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((y * y) <= 5d-310) then
        tmp = (x / y) * (1.0d0 / y)
    else
        tmp = (x / (y * y)) + (-3.0d0)
    end if
    code = tmp
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) {
	double tmp;
	if ((y * y) <= 5e-310) {
		tmp = (x / y) * (1.0 / y);
	} else {
		tmp = (x / (y * y)) + -3.0;
	}
	return tmp;
}

Python

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

↓

def code(x, y):
	tmp = 0
	if (y * y) <= 5e-310:
		tmp = (x / y) * (1.0 / y)
	else:
		tmp = (x / (y * y)) + -3.0
	return tmp

Julia

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

↓

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 5e-310)
		tmp = Float64(Float64(x / y) * Float64(1.0 / y));
	else
		tmp = Float64(Float64(x / Float64(y * y)) + -3.0);
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 5e-310)
		tmp = (x / y) * (1.0 / y);
	else
		tmp = (x / (y * y)) + -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 5e-310], N[(N[(x / y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] + -3.0), $MachinePrecision]]

Target

Original	5.2
Target	0.1
Herbie	0.1

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 4.999999999999985e-310

   1. Initial program 60.9

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

   2. Applied egg-rr 0.4

      \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{1}{y}} - 3 \]

   3. Applied egg-rr 0.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y}, -3\right)} \]

   4. Taylor expanded in x around inf 60.9

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

   5. Simplified 60.9

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
      Proof
      (/.f64 x (*.f64 y y)): 0 points increase in error, 0 points decrease in error
      (/.f64 x (Rewrite<= unpow2_binary64 (pow.f64 y 2))): 0 points increase in error, 0 points decrease in error

   6. Applied egg-rr 0.4

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

   if 4.999999999999985e-310 < (*.f64 y y)

   1. Initial program 0.1

      \[\frac{x}{y \cdot y} - 3 \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{-310}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y} + -3\\ \end{array} \]

Alternatives

Alternative 1
Error	1.1
Cost	1224

\[\begin{array}{l} t_0 := \frac{x}{y \cdot y}\\ t_1 := \frac{x}{y} \cdot \frac{1}{y}\\ \mathbf{if}\;t_0 \leq -1000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 2:\\ \;\;\;\;-3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	6.2
Cost	1096

\[\begin{array}{l} t_0 := \frac{x}{y \cdot y}\\ \mathbf{if}\;t_0 \leq -3:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 3:\\ \;\;\;\;-3\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.1
Cost	448

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

Alternative 4
Error	20.8
Cost	64

\[-3 \]

Reproduce

herbie shell --seed 2022325 
(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))