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

Average Accuracy: 82.4% → 99.8%

Time: 10.9s

Precision: binary64

Cost: 19976

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{+46}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{elif}\;x \leq 15:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{e^{y}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= x -4.1e+46)
   (/ (exp (- y)) x)
   (if (<= x 15.0)
     (/ (pow (exp x) (log (/ x (+ x y)))) x)
     (/ (/ 1.0 x) (exp y)))))

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 <= -4.1e+46) {
		tmp = exp(-y) / x;
	} else if (x <= 15.0) {
		tmp = pow(exp(x), log((x / (x + y)))) / x;
	} else {
		tmp = (1.0 / x) / exp(y);
	}
	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 <= (-4.1d+46)) then
        tmp = exp(-y) / x
    else if (x <= 15.0d0) then
        tmp = (exp(x) ** log((x / (x + y)))) / x
    else
        tmp = (1.0d0 / x) / exp(y)
    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 <= -4.1e+46) {
		tmp = Math.exp(-y) / x;
	} else if (x <= 15.0) {
		tmp = Math.pow(Math.exp(x), Math.log((x / (x + y)))) / x;
	} else {
		tmp = (1.0 / x) / Math.exp(y);
	}
	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 <= -4.1e+46:
		tmp = math.exp(-y) / x
	elif x <= 15.0:
		tmp = math.pow(math.exp(x), math.log((x / (x + y)))) / x
	else:
		tmp = (1.0 / x) / math.exp(y)
	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 <= -4.1e+46)
		tmp = Float64(exp(Float64(-y)) / x);
	elseif (x <= 15.0)
		tmp = Float64((exp(x) ^ log(Float64(x / Float64(x + y)))) / x);
	else
		tmp = Float64(Float64(1.0 / x) / exp(y));
	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 <= -4.1e+46)
		tmp = exp(-y) / x;
	elseif (x <= 15.0)
		tmp = (exp(x) ^ log((x / (x + y)))) / x;
	else
		tmp = (1.0 / x) / exp(y);
	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, -4.1e+46], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 15.0], N[(N[Power[N[Exp[x], $MachinePrecision], N[Log[N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / N[Exp[y], $MachinePrecision]), $MachinePrecision]]]

Target

Original	82.4%
Target	87.8%
Herbie	99.8%

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

Derivation

1. Split input into 3 regimes

2. if x < -4.1e46

   1. Initial program 80.1%

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

   2. Simplified 80.1%

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

      [Start] 80.1	\[ \frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
      *-commutative [=>] 80.1	\[ \frac{e^{\color{blue}{\log \left(\frac{x}{x + y}\right) \cdot x}}}{x} \]
      exp-to-pow [=>] 80.1	\[ \frac{\color{blue}{{\left(\frac{x}{x + y}\right)}^{x}}}{x} \]

   3. Taylor expanded in x around inf 100.0%

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

   4. Simplified 100.0%

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

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

   if -4.1e46 < x < 15

   1. Initial program 82.5%

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

   2. Simplified 99.6%

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

      [Start] 82.5	\[ \frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
      exp-prod [=>] 99.6	\[ \frac{\color{blue}{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}}{x} \]

   if 15 < x

   1. Initial program 83.9%

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

   2. Simplified 83.9%

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

      [Start] 83.9	\[ \frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
      *-commutative [=>] 83.9	\[ \frac{e^{\color{blue}{\log \left(\frac{x}{x + y}\right) \cdot x}}}{x} \]
      exp-to-pow [=>] 83.9	\[ \frac{\color{blue}{{\left(\frac{x}{x + y}\right)}^{x}}}{x} \]

   3. Taylor expanded in x around inf 99.9%

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

   4. Simplified 99.9%

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

      [Start] 99.9	\[ \frac{e^{-1 \cdot y}}{x} \]
      mul-1-neg [=>] 99.9	\[ \frac{e^{\color{blue}{-y}}}{x} \]

   5. Applied egg-rr 99.9%

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

      [Start] 99.9	\[ \frac{e^{-y}}{x} \]
      clear-num [=>] 99.9	\[ \color{blue}{\frac{1}{\frac{x}{e^{-y}}}} \]
      inv-pow [=>] 99.9	\[ \color{blue}{{\left(\frac{x}{e^{-y}}\right)}^{-1}} \]
      div-inv [=>] 99.9	\[ {\color{blue}{\left(x \cdot \frac{1}{e^{-y}}\right)}}^{-1} \]
      add-sqr-sqrt [=>] 32.3	\[ {\left(x \cdot \frac{1}{e^{\color{blue}{\sqrt{-y} \cdot \sqrt{-y}}}}\right)}^{-1} \]
      sqrt-unprod [=>] 65.2	\[ {\left(x \cdot \frac{1}{e^{\color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}}}\right)}^{-1} \]
      sqr-neg [=>] 65.2	\[ {\left(x \cdot \frac{1}{e^{\sqrt{\color{blue}{y \cdot y}}}}\right)}^{-1} \]
      sqrt-unprod [<=] 32.9	\[ {\left(x \cdot \frac{1}{e^{\color{blue}{\sqrt{y} \cdot \sqrt{y}}}}\right)}^{-1} \]
      add-sqr-sqrt [<=] 63.9	\[ {\left(x \cdot \frac{1}{e^{\color{blue}{y}}}\right)}^{-1} \]
      exp-neg [<=] 63.9	\[ {\left(x \cdot \color{blue}{e^{-y}}\right)}^{-1} \]
      add-sqr-sqrt [=>] 31.0	\[ {\left(x \cdot e^{\color{blue}{\sqrt{-y} \cdot \sqrt{-y}}}\right)}^{-1} \]
      sqrt-unprod [=>] 98.6	\[ {\left(x \cdot e^{\color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}}\right)}^{-1} \]
      sqr-neg [=>] 98.6	\[ {\left(x \cdot e^{\sqrt{\color{blue}{y \cdot y}}}\right)}^{-1} \]
      sqrt-unprod [<=] 67.6	\[ {\left(x \cdot e^{\color{blue}{\sqrt{y} \cdot \sqrt{y}}}\right)}^{-1} \]
      add-sqr-sqrt [<=] 99.9	\[ {\left(x \cdot e^{\color{blue}{y}}\right)}^{-1} \]

   6. Taylor expanded in x around 0 99.9%

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

   7. Simplified 99.9%

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

      [Start] 99.9	\[ \frac{1}{e^{y} \cdot x} \]
      associate-/l/ [<=] 99.9	\[ \color{blue}{\frac{\frac{1}{x}}{e^{y}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{+46}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{elif}\;x \leq 15:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{e^{y}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.2%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+50}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{elif}\;x \leq 0.58:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{e^{y}}\\ \end{array} \]

Alternative 2
Accuracy	99.3%
Cost	6921

\[\begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{+46} \lor \neg \left(x \leq 0.6\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 3
Accuracy	97.9%
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 38:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{x}\right) + -1\\ \end{array} \]

Alternative 4
Accuracy	84.4%
Cost	452

\[\begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{+59}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot y}\\ \end{array} \]

Alternative 5
Accuracy	85.0%
Cost	192

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

Reproduce

herbie shell --seed 2023151 
(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))