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

Average Error: 10.9 → 0.6

Time: 19.6s

Precision: binary64

Cost: 60688

Math

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

↓

\[\begin{array}{l} t_0 := \frac{e^{-y}}{x}\\ t_1 := \frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\\ \mathbf{if}\;t_1 \leq -1000000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{-308}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\left(0 - \left(-1 - \frac{1}{x}\right)\right) - 1\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \end{array} \]

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 (/ (exp (* x (log (/ x (+ x y))))) x)))
   (if (<= t_1 -1000000.0)
     (/ 1.0 x)
     (if (<= t_1 -1e-308)
       t_0
       (if (<= t_1 0.0)
         (- (- 0.0 (- -1.0 (/ 1.0 x))) 1.0)
         (if (<= t_1 5e-10) t_0 (/ (pow (/ x (+ y x)) x) 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 = exp((x * log((x / (x + y))))) / x;
	double tmp;
	if (t_1 <= -1000000.0) {
		tmp = 1.0 / x;
	} else if (t_1 <= -1e-308) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = (0.0 - (-1.0 - (1.0 / x))) - 1.0;
	} else if (t_1 <= 5e-10) {
		tmp = t_0;
	} else {
		tmp = pow((x / (y + x)), x) / 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) :: tmp
    t_0 = exp(-y) / x
    t_1 = exp((x * log((x / (x + y))))) / x
    if (t_1 <= (-1000000.0d0)) then
        tmp = 1.0d0 / x
    else if (t_1 <= (-1d-308)) then
        tmp = t_0
    else if (t_1 <= 0.0d0) then
        tmp = (0.0d0 - ((-1.0d0) - (1.0d0 / x))) - 1.0d0
    else if (t_1 <= 5d-10) then
        tmp = t_0
    else
        tmp = ((x / (y + x)) ** x) / 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.exp((x * Math.log((x / (x + y))))) / x;
	double tmp;
	if (t_1 <= -1000000.0) {
		tmp = 1.0 / x;
	} else if (t_1 <= -1e-308) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = (0.0 - (-1.0 - (1.0 / x))) - 1.0;
	} else if (t_1 <= 5e-10) {
		tmp = t_0;
	} else {
		tmp = Math.pow((x / (y + x)), x) / 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.exp((x * math.log((x / (x + y))))) / x
	tmp = 0
	if t_1 <= -1000000.0:
		tmp = 1.0 / x
	elif t_1 <= -1e-308:
		tmp = t_0
	elif t_1 <= 0.0:
		tmp = (0.0 - (-1.0 - (1.0 / x))) - 1.0
	elif t_1 <= 5e-10:
		tmp = t_0
	else:
		tmp = math.pow((x / (y + x)), x) / 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 = Float64(exp(Float64(x * log(Float64(x / Float64(x + y))))) / x)
	tmp = 0.0
	if (t_1 <= -1000000.0)
		tmp = Float64(1.0 / x);
	elseif (t_1 <= -1e-308)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(0.0 - Float64(-1.0 - Float64(1.0 / x))) - 1.0);
	elseif (t_1 <= 5e-10)
		tmp = t_0;
	else
		tmp = Float64((Float64(x / Float64(y + x)) ^ x) / 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 = exp((x * log((x / (x + y))))) / x;
	tmp = 0.0;
	if (t_1 <= -1000000.0)
		tmp = 1.0 / x;
	elseif (t_1 <= -1e-308)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = (0.0 - (-1.0 - (1.0 / x))) - 1.0;
	elseif (t_1 <= 5e-10)
		tmp = t_0;
	else
		tmp = ((x / (y + x)) ^ x) / 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[(N[Exp[N[(x * N[Log[N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$1, -1000000.0], N[(1.0 / x), $MachinePrecision], If[LessEqual[t$95$1, -1e-308], t$95$0, If[LessEqual[t$95$1, 0.0], N[(N[(0.0 - N[(-1.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[t$95$1, 5e-10], t$95$0, N[(N[Power[N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision], x], $MachinePrecision] / x), $MachinePrecision]]]]]]]

Target

Original	10.9
Target	7.7
Herbie	0.6

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (exp.f64 (*.f64 x (log.f64 (/.f64 x (+.f64 x y))))) x) < -1e6

   1. Initial program 11.4

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

   2. Taylor expanded in x around 0 0.5

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

   if -1e6 < (/.f64 (exp.f64 (*.f64 x (log.f64 (/.f64 x (+.f64 x y))))) x) < -9.9999999999999991e-309 or 0.0 < (/.f64 (exp.f64 (*.f64 x (log.f64 (/.f64 x (+.f64 x y))))) x) < 5.00000000000000031e-10

   1. Initial program 12.2

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

   2. Taylor expanded in x around inf 0.3

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

   3. Simplified 0.3

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

      [Start] 0.3	\[ \frac{e^{-1 \cdot y}}{x} \]
      rational.json-simplify-2 [=>] 0.3	\[ \frac{e^{\color{blue}{y \cdot -1}}}{x} \]
      rational.json-simplify-9 [=>] 0.3	\[ \frac{e^{\color{blue}{-y}}}{x} \]

   if -9.9999999999999991e-309 < (/.f64 (exp.f64 (*.f64 x (log.f64 (/.f64 x (+.f64 x y))))) x) < 0.0

   1. Initial program 25.8

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

   2. Taylor expanded in y around 0 62.7

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

   3. Simplified 62.7

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

      [Start] 62.7	\[ \frac{1 + -1 \cdot y}{x} \]
      rational.json-simplify-17 [=>] 62.7	\[ \frac{\color{blue}{-1 \cdot y - -1}}{x} \]
      rational.json-simplify-2 [=>] 62.7	\[ \frac{\color{blue}{y \cdot -1} - -1}{x} \]
      rational.json-simplify-9 [=>] 62.7	\[ \frac{\color{blue}{\left(-y\right)} - -1}{x} \]
      rational.json-simplify-12 [=>] 62.7	\[ \frac{\color{blue}{\left(0 - y\right)} - -1}{x} \]
      rational.json-simplify-42 [=>] 62.7	\[ \frac{\color{blue}{\left(0 - -1\right) - y}}{x} \]
      metadata-eval [=>] 62.7	\[ \frac{\color{blue}{1} - y}{x} \]

   4. Applied egg-rr 62.9

      \[\leadsto \color{blue}{\left(0 - \left(-1 - \frac{1 - y}{x}\right)\right) - 1} \]

   5. Taylor expanded in y around 0 3.8

      \[\leadsto \left(0 - \left(-1 - \color{blue}{\frac{1}{x}}\right)\right) - 1 \]

   if 5.00000000000000031e-10 < (/.f64 (exp.f64 (*.f64 x (log.f64 (/.f64 x (+.f64 x y))))) x)

   1. Initial program 0.0

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

   2. Applied egg-rr 0.0

      \[\leadsto \frac{\color{blue}{{\left(\frac{x}{x + y}\right)}^{x} + 0}}{x} \]

   3. Simplified 0.0

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

      [Start] 0.0	\[ \frac{{\left(\frac{x}{x + y}\right)}^{x} + 0}{x} \]
      rational.json-simplify-4 [=>] 0.0	\[ \frac{\color{blue}{{\left(\frac{x}{x + y}\right)}^{x}}}{x} \]
      rational.json-simplify-1 [=>] 0.0	\[ \frac{{\left(\frac{x}{\color{blue}{y + x}}\right)}^{x}}{x} \]

3. Recombined 4 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \leq -1000000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \leq -1 \cdot 10^{-308}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{elif}\;\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \leq 0:\\ \;\;\;\;\left(0 - \left(-1 - \frac{1}{x}\right)\right) - 1\\ \mathbf{elif}\;\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	6920

\[\begin{array}{l} t_0 := \frac{e^{-y}}{x}\\ \mathbf{if}\;x \leq -9.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.9 \cdot 10^{-5}:\\ \;\;\;\;\left(0 - \left(-1 - \frac{1}{x}\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	1.5
Cost	708

\[\begin{array}{l} \mathbf{if}\;y \leq 18.5:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0 - \left(-1 - \frac{1}{x}\right)\right) - 1\\ \end{array} \]

Alternative 3
Error	7.9
Cost	588

\[\begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{+26}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{+113}:\\ \;\;\;\;0\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+186}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4
Error	9.4
Cost	192

\[\frac{1}{x} \]

Reproduce

herbie shell --seed 2023077 
(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))