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

Average Error: 10.9 → 1.4

Time: 8.5s

Precision: binary64

Cost: 60688

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{x + y}\\ t_1 := \log t_0\\ t_2 := \frac{e^{x \cdot t_1}}{x}\\ t_3 := e^{-y}\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{+19}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;t_2 \leq -5 \cdot 10^{-290}:\\ \;\;\;\;\frac{t_3 + 0.5 \cdot \frac{t_3}{\frac{x}{y \cdot y}}}{x}\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{t_1}}{x}\\ \mathbf{elif}\;t_2 \leq 10^{-55}:\\ \;\;\;\;\frac{t_3}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{t_0}^{x}}{x}\\ \end{array} \]

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 (exp (- y))))
   (if (<= t_2 -1e+19)
     (/ 1.0 x)
     (if (<= t_2 -5e-290)
       (/ (+ t_3 (* 0.5 (/ t_3 (/ x (* y y))))) x)
       (if (<= t_2 0.0)
         (/ (pow (exp x) t_1) x)
         (if (<= t_2 1e-55) (/ t_3 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 = exp(-y);
	double tmp;
	if (t_2 <= -1e+19) {
		tmp = 1.0 / x;
	} else if (t_2 <= -5e-290) {
		tmp = (t_3 + (0.5 * (t_3 / (x / (y * y))))) / x;
	} else if (t_2 <= 0.0) {
		tmp = pow(exp(x), t_1) / x;
	} else if (t_2 <= 1e-55) {
		tmp = t_3 / 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(-y)
    if (t_2 <= (-1d+19)) then
        tmp = 1.0d0 / x
    else if (t_2 <= (-5d-290)) then
        tmp = (t_3 + (0.5d0 * (t_3 / (x / (y * y))))) / x
    else if (t_2 <= 0.0d0) then
        tmp = (exp(x) ** t_1) / x
    else if (t_2 <= 1d-55) then
        tmp = t_3 / 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.exp(-y);
	double tmp;
	if (t_2 <= -1e+19) {
		tmp = 1.0 / x;
	} else if (t_2 <= -5e-290) {
		tmp = (t_3 + (0.5 * (t_3 / (x / (y * y))))) / x;
	} else if (t_2 <= 0.0) {
		tmp = Math.pow(Math.exp(x), t_1) / x;
	} else if (t_2 <= 1e-55) {
		tmp = t_3 / 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.exp(-y)
	tmp = 0
	if t_2 <= -1e+19:
		tmp = 1.0 / x
	elif t_2 <= -5e-290:
		tmp = (t_3 + (0.5 * (t_3 / (x / (y * y))))) / x
	elif t_2 <= 0.0:
		tmp = math.pow(math.exp(x), t_1) / x
	elif t_2 <= 1e-55:
		tmp = t_3 / 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 = exp(Float64(-y))
	tmp = 0.0
	if (t_2 <= -1e+19)
		tmp = Float64(1.0 / x);
	elseif (t_2 <= -5e-290)
		tmp = Float64(Float64(t_3 + Float64(0.5 * Float64(t_3 / Float64(x / Float64(y * y))))) / x);
	elseif (t_2 <= 0.0)
		tmp = Float64((exp(x) ^ t_1) / x);
	elseif (t_2 <= 1e-55)
		tmp = Float64(t_3 / 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(-y);
	tmp = 0.0;
	if (t_2 <= -1e+19)
		tmp = 1.0 / x;
	elseif (t_2 <= -5e-290)
		tmp = (t_3 + (0.5 * (t_3 / (x / (y * y))))) / x;
	elseif (t_2 <= 0.0)
		tmp = (exp(x) ^ t_1) / x;
	elseif (t_2 <= 1e-55)
		tmp = t_3 / 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[Exp[(-y)], $MachinePrecision]}, If[LessEqual[t$95$2, -1e+19], N[(1.0 / x), $MachinePrecision], If[LessEqual[t$95$2, -5e-290], N[(N[(t$95$3 + N[(0.5 * N[(t$95$3 / N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[Power[N[Exp[x], $MachinePrecision], t$95$1], $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$2, 1e-55], N[(t$95$3 / x), $MachinePrecision], N[(N[Power[t$95$0, x], $MachinePrecision] / x), $MachinePrecision]]]]]]]]]

Target

Original	10.9
Target	8.2
Herbie	1.4

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

Derivation

1. Split input into 5 regimes

2. if (/.f64 (exp.f64 (*.f64 x (log.f64 (/.f64 x (+.f64 x y))))) x) < -1e19

   1. Initial program 12.2

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

   2. Simplified 0.0

      \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}} \]
      Proof
      (/.f64 (pow.f64 (exp.f64 x) (log.f64 (/.f64 x (+.f64 x y)))) x): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= exp-prod_binary64 (exp.f64 (*.f64 x (log.f64 (/.f64 x (+.f64 x y)))))) x): 20 points increase in error, 5 points decrease in error

   3. Taylor expanded in x around 0 0.0

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

   if -1e19 < (/.f64 (exp.f64 (*.f64 x (log.f64 (/.f64 x (+.f64 x y))))) x) < -5.0000000000000001e-290

   1. Initial program 11.4

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

   2. Simplified 11.4

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
      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 4.1

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot y} + 0.5 \cdot \frac{e^{-1 \cdot y} \cdot \left(-1 \cdot {y}^{2} + 2 \cdot {y}^{2}\right)}{x}}}{x} \]

   4. Simplified 4.1

      \[\leadsto \frac{\color{blue}{e^{-y} + 0.5 \cdot \frac{e^{-y}}{\frac{x}{\left(y \cdot y\right) \cdot 1}}}}{x} \]
      Proof
      (+.f64 (exp.f64 (neg.f64 y)) (*.f64 1/2 (/.f64 (exp.f64 (neg.f64 y)) (/.f64 x (*.f64 (*.f64 y y) 1))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (exp.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 y))) (*.f64 1/2 (/.f64 (exp.f64 (neg.f64 y)) (/.f64 x (*.f64 (*.f64 y y) 1))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (exp.f64 (*.f64 -1 y)) (*.f64 1/2 (/.f64 (exp.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 y))) (/.f64 x (*.f64 (*.f64 y y) 1))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (exp.f64 (*.f64 -1 y)) (*.f64 1/2 (/.f64 (exp.f64 (*.f64 -1 y)) (/.f64 x (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 y 2)) 1))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (exp.f64 (*.f64 -1 y)) (*.f64 1/2 (/.f64 (exp.f64 (*.f64 -1 y)) (/.f64 x (*.f64 (pow.f64 y 2) (Rewrite<= metadata-eval (+.f64 -1 2))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (exp.f64 (*.f64 -1 y)) (*.f64 1/2 (/.f64 (exp.f64 (*.f64 -1 y)) (/.f64 x (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 -1 (pow.f64 y 2)) (*.f64 2 (pow.f64 y 2)))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (exp.f64 (*.f64 -1 y)) (*.f64 1/2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (exp.f64 (*.f64 -1 y)) (+.f64 (*.f64 -1 (pow.f64 y 2)) (*.f64 2 (pow.f64 y 2)))) x)))): 2 points increase in error, 5 points decrease in error

   if -5.0000000000000001e-290 < (/.f64 (exp.f64 (*.f64 x (log.f64 (/.f64 x (+.f64 x y))))) x) < 0.0

   1. Initial program 22.8

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

   2. Simplified 2.8

      \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}} \]
      Proof
      (/.f64 (pow.f64 (exp.f64 x) (log.f64 (/.f64 x (+.f64 x y)))) x): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= exp-prod_binary64 (exp.f64 (*.f64 x (log.f64 (/.f64 x (+.f64 x y)))))) x): 20 points increase in error, 5 points decrease in error

   if 0.0 < (/.f64 (exp.f64 (*.f64 x (log.f64 (/.f64 x (+.f64 x y))))) x) < 9.99999999999999995e-56

   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
      (/.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

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

   4. Simplified 0.1

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

   if 9.99999999999999995e-56 < (/.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
      (/.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. Recombined 5 regimes into one program.

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \leq -1 \cdot 10^{+19}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \leq -5 \cdot 10^{-290}:\\ \;\;\;\;\frac{e^{-y} + 0.5 \cdot \frac{e^{-y}}{\frac{x}{y \cdot y}}}{x}\\ \mathbf{elif}\;\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \leq 0:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\log \left(\frac{x}{x + y}\right)}}{x}\\ \mathbf{elif}\;\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \leq 10^{-55}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.0
Cost	60688

\[\begin{array}{l} t_0 := \frac{e^{-y}}{x}\\ t_1 := \frac{x}{x + y}\\ t_2 := \frac{e^{x \cdot \log t_1}}{x}\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{+19}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{-307}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;1 + \left(\frac{1}{x} + -1\right)\\ \mathbf{elif}\;t_2 \leq 10^{-55}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{{t_1}^{x}}{x}\\ \end{array} \]

Alternative 2
Error	1.6
Cost	60688

\[\begin{array}{l} t_0 := \frac{x}{x + y}\\ t_1 := \frac{e^{x \cdot \log t_0}}{x}\\ t_2 := e^{-y}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+19}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-307}:\\ \;\;\;\;\frac{t_2 + 0.5 \cdot \frac{t_2}{\frac{x}{y \cdot y}}}{x}\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;1 + \left(\frac{1}{x} + -1\right)\\ \mathbf{elif}\;t_1 \leq 10^{-55}:\\ \;\;\;\;\frac{t_2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{t_0}^{x}}{x}\\ \end{array} \]

Alternative 3
Error	0.4
Cost	6920

\[\begin{array}{l} t_0 := \frac{e^{-y}}{x}\\ \mathbf{if}\;x \leq -0.52:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;1 + \left(\frac{1}{x} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	1.5
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 102:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{1}{x} + -1\right)\\ \end{array} \]

Alternative 5
Error	10.1
Cost	192

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

Reproduce

herbie shell --seed 2022330 
(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))