Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F

Average Error: 11.3 → 0.4

Time: 4.7s

Precision: binary64

Math

\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]

↓

\[\begin{array}{l} t_0 := {\left(x \cdot e^{y}\right)}^{-1}\\ \mathbf{if}\;x \leq -85145579.98679124:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.41414108793425003:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (exp (* x (log (/ x (+ x y))))) x))

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (* x (exp y)) -1.0)))
   (if (<= x -85145579.98679124)
     t_0
     (if (<= x 0.41414108793425003) (/ 1.0 x) t_0))))

C

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

↓

double code(double x, double y) {
	double t_0 = pow((x * exp(y)), -1.0);
	double tmp;
	if (x <= -85145579.98679124) {
		tmp = t_0;
	} else if (x <= 0.41414108793425003) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = exp((x * log((x / (x + y))))) / x
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 * exp(y)) ** (-1.0d0)
    if (x <= (-85145579.98679124d0)) then
        tmp = t_0
    else if (x <= 0.41414108793425003d0) then
        tmp = 1.0d0 / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	return Math.exp((x * Math.log((x / (x + y))))) / x;
}

↓

public static double code(double x, double y) {
	double t_0 = Math.pow((x * Math.exp(y)), -1.0);
	double tmp;
	if (x <= -85145579.98679124) {
		tmp = t_0;
	} else if (x <= 0.41414108793425003) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	return math.exp((x * math.log((x / (x + y))))) / x

↓

def code(x, y):
	t_0 = math.pow((x * math.exp(y)), -1.0)
	tmp = 0
	if x <= -85145579.98679124:
		tmp = t_0
	elif x <= 0.41414108793425003:
		tmp = 1.0 / x
	else:
		tmp = t_0
	return tmp

Julia

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

↓

function code(x, y)
	t_0 = Float64(x * exp(y)) ^ -1.0
	tmp = 0.0
	if (x <= -85145579.98679124)
		tmp = t_0;
	elseif (x <= 0.41414108793425003)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y)
	t_0 = (x * exp(y)) ^ -1.0;
	tmp = 0.0;
	if (x <= -85145579.98679124)
		tmp = t_0;
	elseif (x <= 0.41414108793425003)
		tmp = 1.0 / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_] := Block[{t$95$0 = N[Power[N[(x * N[Exp[y], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]}, If[LessEqual[x, -85145579.98679124], t$95$0, If[LessEqual[x, 0.41414108793425003], N[(1.0 / x), $MachinePrecision], t$95$0]]]

Error

Bits error versus x

Bits error versus y

Target

Original	11.3
Target	7.9
Herbie	0.4

\[\begin{array}{l} \mathbf{if}\;y < -3.7311844206647956 \cdot 10^{+94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y < 2.817959242728288 \cdot 10^{+37}:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y < 2.347387415166998 \cdot 10^{+178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if x < -85145579.9867912382 or 0.41414108793425003 < x

   1. Initial program 10.6

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]

   2. Simplified 10.6

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]

   3. Taylor expanded in x around inf 0.1

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

   4. Applied egg-rr 0.1

      \[\leadsto \color{blue}{{\left(x \cdot e^{y}\right)}^{-1}} \]

   if -85145579.9867912382 < x < 0.41414108793425003

   1. Initial program 12.0

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]

   2. Simplified 12.0

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]

   3. Taylor expanded in x around 0 0.7

      \[\leadsto \frac{\color{blue}{1}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -85145579.98679124:\\ \;\;\;\;{\left(x \cdot e^{y}\right)}^{-1}\\ \mathbf{elif}\;x \leq 0.41414108793425003:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot e^{y}\right)}^{-1}\\ \end{array} \]

Reproduce

herbie shell --seed 2022133 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"
  :precision binary64

  :herbie-target
  (if (< y -3.7311844206647956e+94) (/ (exp (/ -1.0 y)) x) (if (< y 2.817959242728288e+37) (/ (pow (/ x (+ y x)) x) x) (if (< y 2.347387415166998e+178) (log (exp (/ (pow (/ x (+ y x)) x) x))) (/ (exp (/ -1.0 y)) x))))

  (/ (exp (* x (log (/ x (+ x y))))) x))