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

Average Error: 10.6 → 0.9

Time: 5.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+32}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-33}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot e^{y}\right)}^{-1}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= x -2e+32)
   (/ (exp (- y)) x)
   (if (<= x 1.8e-33) (/ 1.0 x) (pow (* x (exp y)) -1.0))))

C

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

↓

double code(double x, double y) {
	double tmp;
	if (x <= -2e+32) {
		tmp = exp(-y) / x;
	} else if (x <= 1.8e-33) {
		tmp = 1.0 / x;
	} else {
		tmp = pow((x * exp(y)), -1.0);
	}
	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) :: tmp
    if (x <= (-2d+32)) then
        tmp = exp(-y) / x
    else if (x <= 1.8d-33) then
        tmp = 1.0d0 / x
    else
        tmp = (x * exp(y)) ** (-1.0d0)
    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 tmp;
	if (x <= -2e+32) {
		tmp = Math.exp(-y) / x;
	} else if (x <= 1.8e-33) {
		tmp = 1.0 / x;
	} else {
		tmp = Math.pow((x * Math.exp(y)), -1.0);
	}
	return tmp;
}

Python

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

↓

def code(x, y):
	tmp = 0
	if x <= -2e+32:
		tmp = math.exp(-y) / x
	elif x <= 1.8e-33:
		tmp = 1.0 / x
	else:
		tmp = math.pow((x * math.exp(y)), -1.0)
	return tmp

Julia

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

↓

function code(x, y)
	tmp = 0.0
	if (x <= -2e+32)
		tmp = Float64(exp(Float64(-y)) / x);
	elseif (x <= 1.8e-33)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(x * exp(y)) ^ -1.0;
	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)
	tmp = 0.0;
	if (x <= -2e+32)
		tmp = exp(-y) / x;
	elseif (x <= 1.8e-33)
		tmp = 1.0 / x;
	else
		tmp = (x * exp(y)) ^ -1.0;
	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_] := If[LessEqual[x, -2e+32], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 1.8e-33], N[(1.0 / x), $MachinePrecision], N[Power[N[(x * N[Exp[y], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]]

Error

Bits error versus x

Bits error versus y

Target

Original	10.6
Target	7.7
Herbie	0.9

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

Derivation

1. Split input into 3 regimes

2. if x < -2.00000000000000011e32

   1. Initial program 12.5

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

   2. Simplified 12.5

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

   3. Taylor expanded in x around inf 0.0

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

   if -2.00000000000000011e32 < x < 1.80000000000000017e-33

   1. Initial program 10.8

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

   2. Simplified 10.8

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

   3. Taylor expanded in x around 0 0.5

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

   if 1.80000000000000017e-33 < x

   1. Initial program 9.0

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

   2. Simplified 9.0

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

   3. Taylor expanded in x around inf 2.2

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

   4. Applied egg-rr 2.2

      \[\leadsto \color{blue}{{\left(x \cdot e^{y}\right)}^{-1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+32}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-33}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot e^{y}\right)}^{-1}\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(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))