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

Average Error: 11.5 → 1.2

Time: 9.6s

Precision: binary64

Cost: 60816

Math

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

↓

\[\begin{array}{l} t_0 := \frac{e^{-y}}{x}\\ t_1 := \frac{x}{x + y}\\ t_2 := \frac{e^{x \cdot \log t_1}}{x}\\ \mathbf{if}\;t_2 \leq -1000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;t_2 \leq -1 \cdot 10^{-300}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\left(1 + \frac{1}{x}\right) + -1\\ \mathbf{elif}\;t_2 \leq 10^{-113}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot {t_1}^{x}\\ \end{array} \]

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 (/ x (+ x y)))
        (t_2 (/ (exp (* x (log t_1))) x)))
   (if (<= t_2 -1000.0)
     (/ 1.0 x)
     (if (<= t_2 -1e-300)
       t_0
       (if (<= t_2 0.0)
         (+ (+ 1.0 (/ 1.0 x)) -1.0)
         (if (<= t_2 1e-113) t_0 (* (/ 1.0 x) (pow 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 = x / (x + y);
	double t_2 = exp((x * log(t_1))) / x;
	double tmp;
	if (t_2 <= -1000.0) {
		tmp = 1.0 / x;
	} else if (t_2 <= -1e-300) {
		tmp = t_0;
	} else if (t_2 <= 0.0) {
		tmp = (1.0 + (1.0 / x)) + -1.0;
	} else if (t_2 <= 1e-113) {
		tmp = t_0;
	} else {
		tmp = (1.0 / x) * pow(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) :: tmp
    t_0 = exp(-y) / x
    t_1 = x / (x + y)
    t_2 = exp((x * log(t_1))) / x
    if (t_2 <= (-1000.0d0)) then
        tmp = 1.0d0 / x
    else if (t_2 <= (-1d-300)) then
        tmp = t_0
    else if (t_2 <= 0.0d0) then
        tmp = (1.0d0 + (1.0d0 / x)) + (-1.0d0)
    else if (t_2 <= 1d-113) then
        tmp = t_0
    else
        tmp = (1.0d0 / x) * (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 = x / (x + y);
	double t_2 = Math.exp((x * Math.log(t_1))) / x;
	double tmp;
	if (t_2 <= -1000.0) {
		tmp = 1.0 / x;
	} else if (t_2 <= -1e-300) {
		tmp = t_0;
	} else if (t_2 <= 0.0) {
		tmp = (1.0 + (1.0 / x)) + -1.0;
	} else if (t_2 <= 1e-113) {
		tmp = t_0;
	} else {
		tmp = (1.0 / x) * Math.pow(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 = x / (x + y)
	t_2 = math.exp((x * math.log(t_1))) / x
	tmp = 0
	if t_2 <= -1000.0:
		tmp = 1.0 / x
	elif t_2 <= -1e-300:
		tmp = t_0
	elif t_2 <= 0.0:
		tmp = (1.0 + (1.0 / x)) + -1.0
	elif t_2 <= 1e-113:
		tmp = t_0
	else:
		tmp = (1.0 / x) * math.pow(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 = Float64(x / Float64(x + y))
	t_2 = Float64(exp(Float64(x * log(t_1))) / x)
	tmp = 0.0
	if (t_2 <= -1000.0)
		tmp = Float64(1.0 / x);
	elseif (t_2 <= -1e-300)
		tmp = t_0;
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(1.0 + Float64(1.0 / x)) + -1.0);
	elseif (t_2 <= 1e-113)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / x) * (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 = x / (x + y);
	t_2 = exp((x * log(t_1))) / x;
	tmp = 0.0;
	if (t_2 <= -1000.0)
		tmp = 1.0 / x;
	elseif (t_2 <= -1e-300)
		tmp = t_0;
	elseif (t_2 <= 0.0)
		tmp = (1.0 + (1.0 / x)) + -1.0;
	elseif (t_2 <= 1e-113)
		tmp = t_0;
	else
		tmp = (1.0 / x) * (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[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Exp[N[(x * N[Log[t$95$1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$2, -1000.0], N[(1.0 / x), $MachinePrecision], If[LessEqual[t$95$2, -1e-300], t$95$0, If[LessEqual[t$95$2, 0.0], N[(N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t$95$2, 1e-113], t$95$0, N[(N[(1.0 / x), $MachinePrecision] * N[Power[t$95$1, x], $MachinePrecision]), $MachinePrecision]]]]]]]]

Target

Original	11.5
Target	7.9
Herbie	1.2

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (exp.f64 (*.f64 x (log.f64 (/.f64 x (+.f64 x y))))) x) < -1e3

   1. Initial program 12.8

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

   2. Simplified 12.8

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
      Proof
      (/.f64 (pow.f64 (/.f64 x (+.f64 x y)) x) x): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 (/.f64 x (+.f64 x y))) x))) x): 1 points increase in error, 0 points decrease in error
      (/.f64 (exp.f64 (Rewrite<= *-commutative_binary64 (*.f64 x (log.f64 (/.f64 x (+.f64 x y)))))) x): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in x around 0 0.7

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

   if -1e3 < (/.f64 (exp.f64 (*.f64 x (log.f64 (/.f64 x (+.f64 x y))))) x) < -1.00000000000000003e-300 or 0.0 < (/.f64 (exp.f64 (*.f64 x (log.f64 (/.f64 x (+.f64 x y))))) x) < 9.99999999999999979e-114

   1. Initial program 13.9

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

   2. Simplified 13.9

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
      Proof
      (/.f64 (pow.f64 (/.f64 x (+.f64 x y)) x) x): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 (/.f64 x (+.f64 x y))) x))) x): 1 points increase in error, 0 points decrease in error
      (/.f64 (exp.f64 (Rewrite<= *-commutative_binary64 (*.f64 x (log.f64 (/.f64 x (+.f64 x y)))))) x): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in x around inf 0.1

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

   4. Simplified 0.1

      \[\leadsto \color{blue}{\frac{e^{-y}}{x}} \]
      Proof
      (/.f64 (exp.f64 (neg.f64 y)) x): 0 points increase in error, 0 points decrease in error
      (/.f64 (exp.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 y))) x): 0 points increase in error, 0 points decrease in error

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

   1. Initial program 24.8

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

   2. Simplified 24.8

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
      Proof
      (/.f64 (pow.f64 (/.f64 x (+.f64 x y)) x) x): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 (/.f64 x (+.f64 x y))) x))) x): 1 points increase in error, 0 points decrease in error
      (/.f64 (exp.f64 (Rewrite<= *-commutative_binary64 (*.f64 x (log.f64 (/.f64 x (+.f64 x y)))))) x): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in y around 0 60.0

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

   4. Simplified 60.0

      \[\leadsto \color{blue}{\frac{1}{x} - \frac{y}{x}} \]
      Proof
      (-.f64 (/.f64 1 x) (/.f64 y x)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 1 x) (neg.f64 (/.f64 y x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 x) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 y x)))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 63.0

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

   6. Taylor expanded in y around 0 43.3

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

   7. Applied egg-rr 5.9

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

   if 9.99999999999999979e-114 < (/.f64 (exp.f64 (*.f64 x (log.f64 (/.f64 x (+.f64 x y))))) x)

   1. Initial program 1.3

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

   2. Simplified 1.3

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
      Proof
      (/.f64 (pow.f64 (/.f64 x (+.f64 x y)) x) x): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 (/.f64 x (+.f64 x y))) x))) x): 1 points increase in error, 0 points decrease in error
      (/.f64 (exp.f64 (Rewrite<= *-commutative_binary64 (*.f64 x (log.f64 (/.f64 x (+.f64 x y)))))) x): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 1.3

      \[\leadsto \color{blue}{{\left(\frac{x}{x + y}\right)}^{x} \cdot \frac{1}{x}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \leq -1000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \leq -1 \cdot 10^{-300}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{elif}\;\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \leq 0:\\ \;\;\;\;\left(1 + \frac{1}{x}\right) + -1\\ \mathbf{elif}\;\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \leq 10^{-113}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot {\left(\frac{x}{x + y}\right)}^{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	6920

\[\begin{array}{l} t_0 := \frac{e^{-y}}{x}\\ \mathbf{if}\;x \leq -7.2 \cdot 10^{+45}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 190:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	8.0
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{+33}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+220}:\\ \;\;\;\;\frac{x}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 3
Error	1.5
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 100000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{x}\right) + -1\\ \end{array} \]

Alternative 4
Error	10.0
Cost	192

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

Reproduce

herbie shell --seed 2022300 
(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))