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

Average Error: 11.5 → 1.0

Time: 5.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{e^{-y}}{x}\\ t_1 := \log \left(\frac{x}{x + y}\right)\\ t_2 := \frac{e^{x \cdot t_1}}{x}\\ t_3 := \sqrt{{\left(e^{x}\right)}^{t_1}}\\ \mathbf{if}\;t_2 \leq -234.66687405323307:\\ \;\;\;\;\frac{t_3 \cdot t_3}{x}\\ \mathbf{elif}\;t_2 \leq -4.399415670006494 \cdot 10^{-261}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\left(-1 \cdot \frac{y}{x}\right)}}{x}\\ \mathbf{elif}\;t_2 \leq 3.046188998270413 \cdot 10^{-149}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{{\left(e^{x}\right)}^{\left(2 \cdot t_1\right)}}}{x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (exp (- y)) x))
        (t_1 (log (/ x (+ x y))))
        (t_2 (/ (exp (* x t_1)) x))
        (t_3 (sqrt (pow (exp x) t_1))))
   (if (<= t_2 -234.66687405323307)
     (/ (* t_3 t_3) x)
     (if (<= t_2 -4.399415670006494e-261)
       t_0
       (if (<= t_2 0.0)
         (/ (pow (exp x) (* -1.0 (/ y x))) x)
         (if (<= t_2 3.046188998270413e-149)
           t_0
           (/ (sqrt (pow (exp x) (* 2.0 t_1))) x)))))))

C

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

↓

double code(double x, double y) {
	double t_0 = exp(-y) / x;
	double t_1 = log((x / (x + y)));
	double t_2 = exp((x * t_1)) / x;
	double t_3 = sqrt(pow(exp(x), t_1));
	double tmp;
	if (t_2 <= -234.66687405323307) {
		tmp = (t_3 * t_3) / x;
	} else if (t_2 <= -4.399415670006494e-261) {
		tmp = t_0;
	} else if (t_2 <= 0.0) {
		tmp = pow(exp(x), (-1.0 * (y / x))) / x;
	} else if (t_2 <= 3.046188998270413e-149) {
		tmp = t_0;
	} else {
		tmp = sqrt(pow(exp(x), (2.0 * t_1))) / x;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = exp(-y) / x
    t_1 = log((x / (x + y)))
    t_2 = exp((x * t_1)) / x
    t_3 = sqrt((exp(x) ** t_1))
    if (t_2 <= (-234.66687405323307d0)) then
        tmp = (t_3 * t_3) / x
    else if (t_2 <= (-4.399415670006494d-261)) then
        tmp = t_0
    else if (t_2 <= 0.0d0) then
        tmp = (exp(x) ** ((-1.0d0) * (y / x))) / x
    else if (t_2 <= 3.046188998270413d-149) then
        tmp = t_0
    else
        tmp = sqrt((exp(x) ** (2.0d0 * t_1))) / x
    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.exp(-y) / x;
	double t_1 = Math.log((x / (x + y)));
	double t_2 = Math.exp((x * t_1)) / x;
	double t_3 = Math.sqrt(Math.pow(Math.exp(x), t_1));
	double tmp;
	if (t_2 <= -234.66687405323307) {
		tmp = (t_3 * t_3) / x;
	} else if (t_2 <= -4.399415670006494e-261) {
		tmp = t_0;
	} else if (t_2 <= 0.0) {
		tmp = Math.pow(Math.exp(x), (-1.0 * (y / x))) / x;
	} else if (t_2 <= 3.046188998270413e-149) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt(Math.pow(Math.exp(x), (2.0 * t_1))) / x;
	}
	return tmp;
}

Python

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

↓

def code(x, y):
	t_0 = math.exp(-y) / x
	t_1 = math.log((x / (x + y)))
	t_2 = math.exp((x * t_1)) / x
	t_3 = math.sqrt(math.pow(math.exp(x), t_1))
	tmp = 0
	if t_2 <= -234.66687405323307:
		tmp = (t_3 * t_3) / x
	elif t_2 <= -4.399415670006494e-261:
		tmp = t_0
	elif t_2 <= 0.0:
		tmp = math.pow(math.exp(x), (-1.0 * (y / x))) / x
	elif t_2 <= 3.046188998270413e-149:
		tmp = t_0
	else:
		tmp = math.sqrt(math.pow(math.exp(x), (2.0 * t_1))) / x
	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(exp(Float64(-y)) / x)
	t_1 = log(Float64(x / Float64(x + y)))
	t_2 = Float64(exp(Float64(x * t_1)) / x)
	t_3 = sqrt((exp(x) ^ t_1))
	tmp = 0.0
	if (t_2 <= -234.66687405323307)
		tmp = Float64(Float64(t_3 * t_3) / x);
	elseif (t_2 <= -4.399415670006494e-261)
		tmp = t_0;
	elseif (t_2 <= 0.0)
		tmp = Float64((exp(x) ^ Float64(-1.0 * Float64(y / x))) / x);
	elseif (t_2 <= 3.046188998270413e-149)
		tmp = t_0;
	else
		tmp = Float64(sqrt((exp(x) ^ Float64(2.0 * t_1))) / x);
	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 = exp(-y) / x;
	t_1 = log((x / (x + y)));
	t_2 = exp((x * t_1)) / x;
	t_3 = sqrt((exp(x) ^ t_1));
	tmp = 0.0;
	if (t_2 <= -234.66687405323307)
		tmp = (t_3 * t_3) / x;
	elseif (t_2 <= -4.399415670006494e-261)
		tmp = t_0;
	elseif (t_2 <= 0.0)
		tmp = (exp(x) ^ (-1.0 * (y / x))) / x;
	elseif (t_2 <= 3.046188998270413e-149)
		tmp = t_0;
	else
		tmp = sqrt((exp(x) ^ (2.0 * t_1))) / x;
	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[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = N[Log[N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Exp[N[(x * t$95$1), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[Power[N[Exp[x], $MachinePrecision], t$95$1], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, -234.66687405323307], N[(N[(t$95$3 * t$95$3), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$2, -4.399415670006494e-261], t$95$0, If[LessEqual[t$95$2, 0.0], N[(N[Power[N[Exp[x], $MachinePrecision], N[(-1.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$2, 3.046188998270413e-149], t$95$0, N[(N[Sqrt[N[Power[N[Exp[x], $MachinePrecision], N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]]]]]]]]]

Error

Bits error versus x

Bits error versus y

Target

Original	11.5
Target	7.9
Herbie	1.0

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (exp.f64 (*.f64 x (log.f64 (/.f64 x (+.f64 x y))))) x) < -234.66687405323307

   1. Initial program 11.0

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

   2. Simplified 0.1

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

   3. Applied egg-rr 0.1

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

   if -234.66687405323307 < (/.f64 (exp.f64 (*.f64 x (log.f64 (/.f64 x (+.f64 x y))))) x) < -4.3994156700064944e-261 or 0.0 < (/.f64 (exp.f64 (*.f64 x (log.f64 (/.f64 x (+.f64 x y))))) x) < 3.04618899827041309e-149

   1. Initial program 13.7

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

   2. Simplified 13.7

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

   3. Taylor expanded in x around inf 25.8

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

   4. Taylor expanded in x around 0 0.1

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

   if -4.3994156700064944e-261 < (/.f64 (exp.f64 (*.f64 x (log.f64 (/.f64 x (+.f64 x y))))) x) < 0.0

   1. Initial program 26.3

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

   2. Simplified 5.4

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

   3. Taylor expanded in x around inf 1.8

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

   if 3.04618899827041309e-149 < (/.f64 (exp.f64 (*.f64 x (log.f64 (/.f64 x (+.f64 x y))))) x)

   1. Initial program 2.3

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

   2. Simplified 2.4

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

   3. Applied egg-rr 2.4

      \[\leadsto \frac{\color{blue}{\sqrt{{\left(e^{x}\right)}^{\left(2 \cdot \log \left(\frac{x}{x + y}\right)\right)}}}}{x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \leq -234.66687405323307:\\ \;\;\;\;\frac{\sqrt{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}} \cdot \sqrt{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}}{x}\\ \mathbf{elif}\;\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \leq -4.399415670006494 \cdot 10^{-261}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{elif}\;\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \leq 0:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\left(-1 \cdot \frac{y}{x}\right)}}{x}\\ \mathbf{elif}\;\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \leq 3.046188998270413 \cdot 10^{-149}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{{\left(e^{x}\right)}^{\left(2 \cdot \log \left(\frac{x}{x + y}\right)\right)}}}{x}\\ \end{array} \]

Reproduce

herbie shell --seed 2022152 
(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))