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

Percentage Accurate: 78.2% → 99.4%

Time: 8.7s

Precision: binary64

Cost: 19977

• Unsound rule application detected in e-graph. Results may not be sound.

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+58} \lor \neg \left(x \leq 5 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{\frac{1}{x}}{e^{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.9e+58) (not (<= x 5e-8)))
   (/ (/ 1.0 x) (exp y))
   (/ (pow (exp x) (log (/ x (+ x y)))) x)))

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 <= -2.9e+58) || !(x <= 5e-8)) {
		tmp = (1.0 / x) / exp(y);
	} else {
		tmp = pow(exp(x), log((x / (x + y)))) / 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) :: tmp
    if ((x <= (-2.9d+58)) .or. (.not. (x <= 5d-8))) then
        tmp = (1.0d0 / x) / exp(y)
    else
        tmp = (exp(x) ** log((x / (x + y)))) / 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 tmp;
	if ((x <= -2.9e+58) || !(x <= 5e-8)) {
		tmp = (1.0 / x) / Math.exp(y);
	} else {
		tmp = Math.pow(Math.exp(x), Math.log((x / (x + y)))) / x;
	}
	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 <= -2.9e+58) or not (x <= 5e-8):
		tmp = (1.0 / x) / math.exp(y)
	else:
		tmp = math.pow(math.exp(x), math.log((x / (x + y)))) / 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)
	tmp = 0.0
	if ((x <= -2.9e+58) || !(x <= 5e-8))
		tmp = Float64(Float64(1.0 / x) / exp(y));
	else
		tmp = Float64((exp(x) ^ log(Float64(x / Float64(x + y)))) / 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)
	tmp = 0.0;
	if ((x <= -2.9e+58) || ~((x <= 5e-8)))
		tmp = (1.0 / x) / exp(y);
	else
		tmp = (exp(x) ^ log((x / (x + y)))) / 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_] := If[Or[LessEqual[x, -2.9e+58], N[Not[LessEqual[x, 5e-8]], $MachinePrecision]], N[(N[(1.0 / x), $MachinePrecision] / N[Exp[y], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Exp[x], $MachinePrecision], N[Log[N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]]

Target

Original	78.2%
Target	78.1%
Herbie	99.4%

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

Derivation

1. Split input into 2 regimes

2. if x < -2.90000000000000002e58 or 4.9999999999999998e-8 < x

   1. Initial program 70.1%

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

   2. Simplified 70.1%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
      Step-by-step derivation

      [Start] 70.1%	\[ \frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
      *-commutative [=>] 70.1%	\[ \frac{e^{\color{blue}{\log \left(\frac{x}{x + y}\right) \cdot x}}}{x} \]
      exp-to-pow [=>] 70.1%	\[ \frac{\color{blue}{{\left(\frac{x}{x + y}\right)}^{x}}}{x} \]

   3. Taylor expanded in x around inf 100.0%

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

   4. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{e^{-y}}}{x} \]
      Step-by-step derivation

      [Start] 100.0%	\[ \frac{e^{-1 \cdot y}}{x} \]
      mul-1-neg [=>] 100.0%	\[ \frac{e^{\color{blue}{-y}}}{x} \]

   5. Applied egg-rr 100.0%

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

      [Start] 100.0%	\[ \frac{e^{-y}}{x} \]
      clear-num [=>] 100.0%	\[ \color{blue}{\frac{1}{\frac{x}{e^{-y}}}} \]
      inv-pow [=>] 100.0%	\[ \color{blue}{{\left(\frac{x}{e^{-y}}\right)}^{-1}} \]
      exp-neg [=>] 100.0%	\[ {\left(\frac{x}{\color{blue}{\frac{1}{e^{y}}}}\right)}^{-1} \]
      associate-/r/ [=>] 100.0%	\[ {\color{blue}{\left(\frac{x}{1} \cdot e^{y}\right)}}^{-1} \]
      /-rgt-identity [=>] 100.0%	\[ {\left(\color{blue}{x} \cdot e^{y}\right)}^{-1} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\frac{1}{x}}{e^{y}}} \]
      Step-by-step derivation

      [Start] 100.0%	\[ {\left(x \cdot e^{y}\right)}^{-1} \]
      unpow-1 [=>] 100.0%	\[ \color{blue}{\frac{1}{x \cdot e^{y}}} \]
      associate-/r* [=>] 100.0%	\[ \color{blue}{\frac{\frac{1}{x}}{e^{y}}} \]

   if -2.90000000000000002e58 < x < 4.9999999999999998e-8

   1. Initial program 82.3%

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

   2. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}} \]
      Step-by-step derivation

      [Start] 82.3%	\[ \frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
      exp-prod [=>] 99.5%	\[ \frac{\color{blue}{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}}{x} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+58} \lor \neg \left(x \leq 5 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{\frac{1}{x}}{e^{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.4%
Cost	19977

\[\begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+58} \lor \neg \left(x \leq 5 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{\frac{1}{x}}{e^{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \end{array} \]

Alternative 2
Accuracy	99.3%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq -17 \lor \neg \left(x \leq 5 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{\frac{1}{x}}{e^{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 3
Accuracy	99.3%
Cost	6921

\[\begin{array}{l} \mathbf{if}\;x \leq -17 \lor \neg \left(x \leq 5 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 4
Accuracy	82.2%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -17:\\ \;\;\;\;\frac{\left(y + -1\right) \cdot \left(y + -1\right)}{x}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+54}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1}{x \cdot y}\\ \end{array} \]

Alternative 5
Accuracy	82.3%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -17:\\ \;\;\;\;\frac{\frac{x - x \cdot y}{x}}{x}\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{+54}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1}{x \cdot y}\\ \end{array} \]

Alternative 6
Accuracy	78.1%
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{+54}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1}{x \cdot y}\\ \end{array} \]

Alternative 7
Accuracy	4.2%
Cost	192

\[y + y \]

Alternative 8
Accuracy	74.4%
Cost	192

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

Alternative 9
Accuracy	2.9%
Cost	128

\[-y \]

Reproduce

herbie shell --seed 2023165 
(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))