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

Average Error: 11.7 → 0.4

Time: 10.7s

Precision: binary64

Cost: 60688

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{x + y}\\ t_1 := \log t_0\\ t_2 := \frac{e^{x \cdot t_1}}{x}\\ t_3 := \frac{{\left(e^{x}\right)}^{t_1}}{x}\\ \mathbf{if}\;t_2 \leq -20000:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq -1 \cdot 10^{-300}:\\ \;\;\;\;{\left(x \cdot e^{y}\right)}^{-1}\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{-59}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{t_0}^{x}}{x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y)))
        (t_1 (log t_0))
        (t_2 (/ (exp (* x t_1)) x))
        (t_3 (/ (pow (exp x) t_1) x)))
   (if (<= t_2 -20000.0)
     t_3
     (if (<= t_2 -1e-300)
       (pow (* x (exp y)) -1.0)
       (if (<= t_2 0.0)
         t_3
         (if (<= t_2 2e-59) (/ (exp (- y)) x) (/ (pow t_0 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 = x / (x + y);
	double t_1 = log(t_0);
	double t_2 = exp((x * t_1)) / x;
	double t_3 = pow(exp(x), t_1) / x;
	double tmp;
	if (t_2 <= -20000.0) {
		tmp = t_3;
	} else if (t_2 <= -1e-300) {
		tmp = pow((x * exp(y)), -1.0);
	} else if (t_2 <= 0.0) {
		tmp = t_3;
	} else if (t_2 <= 2e-59) {
		tmp = exp(-y) / x;
	} else {
		tmp = pow(t_0, 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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = x / (x + y)
    t_1 = log(t_0)
    t_2 = exp((x * t_1)) / x
    t_3 = (exp(x) ** t_1) / x
    if (t_2 <= (-20000.0d0)) then
        tmp = t_3
    else if (t_2 <= (-1d-300)) then
        tmp = (x * exp(y)) ** (-1.0d0)
    else if (t_2 <= 0.0d0) then
        tmp = t_3
    else if (t_2 <= 2d-59) then
        tmp = exp(-y) / x
    else
        tmp = (t_0 ** 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 = x / (x + y);
	double t_1 = Math.log(t_0);
	double t_2 = Math.exp((x * t_1)) / x;
	double t_3 = Math.pow(Math.exp(x), t_1) / x;
	double tmp;
	if (t_2 <= -20000.0) {
		tmp = t_3;
	} else if (t_2 <= -1e-300) {
		tmp = Math.pow((x * Math.exp(y)), -1.0);
	} else if (t_2 <= 0.0) {
		tmp = t_3;
	} else if (t_2 <= 2e-59) {
		tmp = Math.exp(-y) / x;
	} else {
		tmp = Math.pow(t_0, 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 = x / (x + y)
	t_1 = math.log(t_0)
	t_2 = math.exp((x * t_1)) / x
	t_3 = math.pow(math.exp(x), t_1) / x
	tmp = 0
	if t_2 <= -20000.0:
		tmp = t_3
	elif t_2 <= -1e-300:
		tmp = math.pow((x * math.exp(y)), -1.0)
	elif t_2 <= 0.0:
		tmp = t_3
	elif t_2 <= 2e-59:
		tmp = math.exp(-y) / x
	else:
		tmp = math.pow(t_0, 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(x / Float64(x + y))
	t_1 = log(t_0)
	t_2 = Float64(exp(Float64(x * t_1)) / x)
	t_3 = Float64((exp(x) ^ t_1) / x)
	tmp = 0.0
	if (t_2 <= -20000.0)
		tmp = t_3;
	elseif (t_2 <= -1e-300)
		tmp = Float64(x * exp(y)) ^ -1.0;
	elseif (t_2 <= 0.0)
		tmp = t_3;
	elseif (t_2 <= 2e-59)
		tmp = Float64(exp(Float64(-y)) / x);
	else
		tmp = Float64((t_0 ^ 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 = x / (x + y);
	t_1 = log(t_0);
	t_2 = exp((x * t_1)) / x;
	t_3 = (exp(x) ^ t_1) / x;
	tmp = 0.0;
	if (t_2 <= -20000.0)
		tmp = t_3;
	elseif (t_2 <= -1e-300)
		tmp = (x * exp(y)) ^ -1.0;
	elseif (t_2 <= 0.0)
		tmp = t_3;
	elseif (t_2 <= 2e-59)
		tmp = exp(-y) / x;
	else
		tmp = (t_0 ^ 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[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Log[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Exp[N[(x * t$95$1), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[Exp[x], $MachinePrecision], t$95$1], $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$2, -20000.0], t$95$3, If[LessEqual[t$95$2, -1e-300], N[Power[N[(x * N[Exp[y], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[t$95$2, 0.0], t$95$3, If[LessEqual[t$95$2, 2e-59], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], N[(N[Power[t$95$0, x], $MachinePrecision] / x), $MachinePrecision]]]]]]]]]

Target

Original	11.7
Target	8.1
Herbie	0.4

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (exp.f64 (*.f64 x (log.f64 (/.f64 x (+.f64 x y))))) x) < -2e4 or -1.00000000000000003e-300 < (/.f64 (exp.f64 (*.f64 x (log.f64 (/.f64 x (+.f64 x y))))) x) < -0.0

   1. Initial program 17.3

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

   2. Simplified 0.6

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

      [Start] 17.3	\[ \frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
      exp-prod [=>] 0.6	\[ \frac{\color{blue}{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}}{x} \]

   if -2e4 < (/.f64 (exp.f64 (*.f64 x (log.f64 (/.f64 x (+.f64 x y))))) x) < -1.00000000000000003e-300

   1. Initial program 12.6

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

   2. Simplified 12.6

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

      [Start] 12.6	\[ \frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
      *-commutative [=>] 12.6	\[ \frac{e^{\color{blue}{\log \left(\frac{x}{x + y}\right) \cdot x}}}{x} \]
      exp-to-pow [=>] 12.6	\[ \frac{\color{blue}{{\left(\frac{x}{x + y}\right)}^{x}}}{x} \]

   3. Taylor expanded in x around inf 0.3

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

   4. Simplified 0.3

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

      [Start] 0.3	\[ \frac{e^{-1 \cdot y}}{x} \]
      mul-1-neg [=>] 0.3	\[ \frac{e^{\color{blue}{-y}}}{x} \]

   5. Applied egg-rr 0.3

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

   if -0.0 < (/.f64 (exp.f64 (*.f64 x (log.f64 (/.f64 x (+.f64 x y))))) x) < 2.0000000000000001e-59

   1. Initial program 14.1

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

   2. Simplified 14.1

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

      [Start] 14.1	\[ \frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
      *-commutative [=>] 14.1	\[ \frac{e^{\color{blue}{\log \left(\frac{x}{x + y}\right) \cdot x}}}{x} \]
      exp-to-pow [=>] 14.1	\[ \frac{\color{blue}{{\left(\frac{x}{x + y}\right)}^{x}}}{x} \]

   3. Taylor expanded in x around inf 0.0

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

   4. Simplified 0.0

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

      [Start] 0.0	\[ \frac{e^{-1 \cdot y}}{x} \]
      mul-1-neg [=>] 0.0	\[ \frac{e^{\color{blue}{-y}}}{x} \]

   if 2.0000000000000001e-59 < (/.f64 (exp.f64 (*.f64 x (log.f64 (/.f64 x (+.f64 x y))))) x)

   1. Initial program 0.5

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

   2. Simplified 0.5

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

      [Start] 0.5	\[ \frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
      *-commutative [=>] 0.5	\[ \frac{e^{\color{blue}{\log \left(\frac{x}{x + y}\right) \cdot x}}}{x} \]
      exp-to-pow [=>] 0.5	\[ \frac{\color{blue}{{\left(\frac{x}{x + y}\right)}^{x}}}{x} \]

3. Recombined 4 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \leq -20000:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \mathbf{elif}\;\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \leq -1 \cdot 10^{-300}:\\ \;\;\;\;{\left(x \cdot e^{y}\right)}^{-1}\\ \mathbf{elif}\;\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \leq 0:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \mathbf{elif}\;\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \leq 2 \cdot 10^{-59}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	13188

\[\begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;{\left(x \cdot e^{y}\right)}^{-1}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-16}:\\ \;\;\;\;-1 + \left(1 + \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \end{array} \]

Alternative 2
Error	0.5
Cost	6921

\[\begin{array}{l} \mathbf{if}\;x \leq -0.72 \lor \neg \left(x \leq 1.85 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + \frac{1}{x}\right)\\ \end{array} \]

Alternative 3
Error	1.4
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 40:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + \frac{1}{x}\right)\\ \end{array} \]

Alternative 4
Error	8.3
Cost	452

\[\begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{-32}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x}\\ \end{array} \]

Alternative 5
Error	9.8
Cost	192

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

Reproduce

herbie shell --seed 2023039 
(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))