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

Average Error: 4.9 → 0.2

Time: 6.0s

Precision: binary64

Cost: 1224

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{y \cdot y}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+178}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+14}:\\ \;\;\;\;t_0 + -3\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y y))))
   (if (<= t_0 -5e+178)
     (* (/ x y) (/ 1.0 y))
     (if (<= t_0 5e+14) (+ t_0 -3.0) (/ (/ x y) y)))))

C

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

↓

double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (t_0 <= -5e+178) {
		tmp = (x / y) * (1.0 / y);
	} else if (t_0 <= 5e+14) {
		tmp = t_0 + -3.0;
	} else {
		tmp = (x / y) / y;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = x / (y * y)
    if (t_0 <= (-5d+178)) then
        tmp = (x / y) * (1.0d0 / y)
    else if (t_0 <= 5d+14) then
        tmp = t_0 + (-3.0d0)
    else
        tmp = (x / y) / y
    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 t_0 = x / (y * y);
	double tmp;
	if (t_0 <= -5e+178) {
		tmp = (x / y) * (1.0 / y);
	} else if (t_0 <= 5e+14) {
		tmp = t_0 + -3.0;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

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

↓

def code(x, y):
	t_0 = x / (y * y)
	tmp = 0
	if t_0 <= -5e+178:
		tmp = (x / y) * (1.0 / y)
	elif t_0 <= 5e+14:
		tmp = t_0 + -3.0
	else:
		tmp = (x / y) / y
	return tmp

Julia

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

↓

function code(x, y)
	t_0 = Float64(x / Float64(y * y))
	tmp = 0.0
	if (t_0 <= -5e+178)
		tmp = Float64(Float64(x / y) * Float64(1.0 / y));
	elseif (t_0 <= 5e+14)
		tmp = Float64(t_0 + -3.0);
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y)
	t_0 = x / (y * y);
	tmp = 0.0;
	if (t_0 <= -5e+178)
		tmp = (x / y) * (1.0 / y);
	elseif (t_0 <= 5e+14)
		tmp = t_0 + -3.0;
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+178], N[(N[(x / y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+14], N[(t$95$0 + -3.0), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]

Target

Original	4.9
Target	0.1
Herbie	0.2

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 x (*.f64 y y)) < -4.9999999999999999e178

   1. Initial program 30.8

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

   2. Taylor expanded in x around 0 30.8

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

   3. Simplified 31.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, {y}^{-2}, -3\right)} \]
      Proof

      [Start] 30.8	\[ \frac{x}{{y}^{2}} - 3 \]
      unpow2 [=>] 30.8	\[ \frac{x}{\color{blue}{y \cdot y}} - 3 \]
      *-lft-identity [<=] 30.8	\[ \frac{\color{blue}{1 \cdot x}}{y \cdot y} - 3 \]
      times-frac [=>] 0.4	\[ \color{blue}{\frac{1}{y} \cdot \frac{x}{y}} - 3 \]
      unpow-1 [<=] 0.4	\[ \color{blue}{{y}^{-1}} \cdot \frac{x}{y} - 3 \]
      metadata-eval [<=] 0.4	\[ {y}^{\color{blue}{\left(\frac{-2}{2}\right)}} \cdot \frac{x}{y} - 3 \]
      *-commutative [<=] 0.4	\[ \color{blue}{\frac{x}{y} \cdot {y}^{\left(\frac{-2}{2}\right)}} - 3 \]
      *-lft-identity [<=] 0.4	\[ \frac{\color{blue}{1 \cdot x}}{y} \cdot {y}^{\left(\frac{-2}{2}\right)} - 3 \]
      associate-*l/ [<=] 0.6	\[ \color{blue}{\left(\frac{1}{y} \cdot x\right)} \cdot {y}^{\left(\frac{-2}{2}\right)} - 3 \]
      *-commutative [=>] 0.6	\[ \color{blue}{\left(x \cdot \frac{1}{y}\right)} \cdot {y}^{\left(\frac{-2}{2}\right)} - 3 \]
      unpow-1 [<=] 0.6	\[ \left(x \cdot \color{blue}{{y}^{-1}}\right) \cdot {y}^{\left(\frac{-2}{2}\right)} - 3 \]
      metadata-eval [<=] 0.6	\[ \left(x \cdot {y}^{\color{blue}{\left(\frac{-2}{2}\right)}}\right) \cdot {y}^{\left(\frac{-2}{2}\right)} - 3 \]
      associate-*r* [<=] 31.7	\[ \color{blue}{x \cdot \left({y}^{\left(\frac{-2}{2}\right)} \cdot {y}^{\left(\frac{-2}{2}\right)}\right)} - 3 \]
      sqr-pow [<=] 31.6	\[ x \cdot \color{blue}{{y}^{-2}} - 3 \]
      fma-neg [=>] 31.6	\[ \color{blue}{\mathsf{fma}\left(x, {y}^{-2}, -3\right)} \]
      metadata-eval [=>] 31.6	\[ \mathsf{fma}\left(x, {y}^{-2}, \color{blue}{-3}\right) \]

   4. Taylor expanded in x around inf 30.8

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

   5. Simplified 30.8

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

      [Start] 30.8	\[ \frac{x}{{y}^{2}} \]
      unpow2 [=>] 30.8	\[ \frac{x}{\color{blue}{y \cdot y}} \]

   6. Applied egg-rr 0.4

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

   if -4.9999999999999999e178 < (/.f64 x (*.f64 y y)) < 5e14

   1. Initial program 0.1

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

   if 5e14 < (/.f64 x (*.f64 y y))

   1. Initial program 15.8

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

   2. Taylor expanded in x around 0 15.8

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

   3. Simplified 16.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, {y}^{-2}, -3\right)} \]
      Proof

      [Start] 15.8	\[ \frac{x}{{y}^{2}} - 3 \]
      unpow2 [=>] 15.8	\[ \frac{x}{\color{blue}{y \cdot y}} - 3 \]
      *-lft-identity [<=] 15.8	\[ \frac{\color{blue}{1 \cdot x}}{y \cdot y} - 3 \]
      times-frac [=>] 0.4	\[ \color{blue}{\frac{1}{y} \cdot \frac{x}{y}} - 3 \]
      unpow-1 [<=] 0.4	\[ \color{blue}{{y}^{-1}} \cdot \frac{x}{y} - 3 \]
      metadata-eval [<=] 0.4	\[ {y}^{\color{blue}{\left(\frac{-2}{2}\right)}} \cdot \frac{x}{y} - 3 \]
      *-commutative [<=] 0.4	\[ \color{blue}{\frac{x}{y} \cdot {y}^{\left(\frac{-2}{2}\right)}} - 3 \]
      *-lft-identity [<=] 0.4	\[ \frac{\color{blue}{1 \cdot x}}{y} \cdot {y}^{\left(\frac{-2}{2}\right)} - 3 \]
      associate-*l/ [<=] 0.5	\[ \color{blue}{\left(\frac{1}{y} \cdot x\right)} \cdot {y}^{\left(\frac{-2}{2}\right)} - 3 \]
      *-commutative [=>] 0.5	\[ \color{blue}{\left(x \cdot \frac{1}{y}\right)} \cdot {y}^{\left(\frac{-2}{2}\right)} - 3 \]
      unpow-1 [<=] 0.5	\[ \left(x \cdot \color{blue}{{y}^{-1}}\right) \cdot {y}^{\left(\frac{-2}{2}\right)} - 3 \]
      metadata-eval [<=] 0.5	\[ \left(x \cdot {y}^{\color{blue}{\left(\frac{-2}{2}\right)}}\right) \cdot {y}^{\left(\frac{-2}{2}\right)} - 3 \]
      associate-*r* [<=] 17.1	\[ \color{blue}{x \cdot \left({y}^{\left(\frac{-2}{2}\right)} \cdot {y}^{\left(\frac{-2}{2}\right)}\right)} - 3 \]
      sqr-pow [<=] 16.9	\[ x \cdot \color{blue}{{y}^{-2}} - 3 \]
      fma-neg [=>] 16.9	\[ \color{blue}{\mathsf{fma}\left(x, {y}^{-2}, -3\right)} \]
      metadata-eval [=>] 16.9	\[ \mathsf{fma}\left(x, {y}^{-2}, \color{blue}{-3}\right) \]

   4. Taylor expanded in x around inf 15.8

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

   5. Simplified 0.3

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

      [Start] 15.8	\[ \frac{x}{{y}^{2}} \]
      unpow2 [=>] 15.8	\[ \frac{x}{\color{blue}{y \cdot y}} \]
      associate-/r* [=>] 0.3	\[ \color{blue}{\frac{\frac{x}{y}}{y}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y \cdot y} \leq -5 \cdot 10^{+178}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \mathbf{elif}\;\frac{x}{y \cdot y} \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{y \cdot y} + -3\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternatives

Alternative 1
Error	5.8
Cost	1097

\[\begin{array}{l} t_0 := \frac{x}{y \cdot y}\\ \mathbf{if}\;t_0 \leq -3 \lor \neg \left(t_0 \leq 3\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-3\\ \end{array} \]

Alternative 2
Error	1.3
Cost	1097

\[\begin{array}{l} t_0 := \frac{x}{y \cdot y}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+16} \lor \neg \left(t_0 \leq 0.2\right):\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;-3\\ \end{array} \]

Alternative 3
Error	0.1
Cost	448

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

Alternative 4
Error	20.9
Cost	64

\[-3 \]

Reproduce

herbie shell --seed 2023018 
(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))