Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A

Average Error: 19.9 → 9.5

Time: 9.9s

Precision: binary64

Cost: 1352

\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]

Math

\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{+92}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-146}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))

↓

(FPCore (x y)
 :precision binary64
 (if (<= x -2.45e+92)
   (/ y (* (+ 1.0 x) x))
   (if (<= x -1e-146)
     (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0)))
     (/ x (+ y (* y y))))))

C

double code(double x, double y) {
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
}

↓

double code(double x, double y) {
	double tmp;
	if (x <= -2.45e+92) {
		tmp = y / ((1.0 + x) * x);
	} else if (x <= -1e-146) {
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	} else {
		tmp = x / (y + (y * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0d0))
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.45d+92)) then
        tmp = y / ((1.0d0 + x) * x)
    else if (x <= (-1d-146)) then
        tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0d0))
    else
        tmp = x / (y + (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
}

↓

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.45e+92) {
		tmp = y / ((1.0 + x) * x);
	} else if (x <= -1e-146) {
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	} else {
		tmp = x / (y + (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0))

↓

def code(x, y):
	tmp = 0
	if x <= -2.45e+92:
		tmp = y / ((1.0 + x) * x)
	elif x <= -1e-146:
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0))
	else:
		tmp = x / (y + (y * y))
	return tmp

Julia

function code(x, y)
	return Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(Float64(x + y) + 1.0)))
end

↓

function code(x, y)
	tmp = 0.0
	if (x <= -2.45e+92)
		tmp = Float64(y / Float64(Float64(1.0 + x) * x));
	elseif (x <= -1e-146)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(Float64(x + y) + 1.0)));
	else
		tmp = Float64(x / Float64(y + Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp = code(x, y)
	tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
end

↓

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.45e+92)
		tmp = y / ((1.0 + x) * x);
	elseif (x <= -1e-146)
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	else
		tmp = x / (y + (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := If[LessEqual[x, -2.45e+92], N[(y / N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1e-146], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	19.9
Target	0.1
Herbie	9.5

\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}} \]

Derivation

1. Split input into 3 regimes

2. if x < -2.4500000000000001e92

   1. Initial program 26.0

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around 0 10.5

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

   if -2.4500000000000001e92 < x < -1.00000000000000003e-146

   1. Initial program 9.5

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   if -1.00000000000000003e-146 < x

   1. Initial program 22.3

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in x around 0 8.7

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

   3. Simplified 8.7

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

      [Start] 8.7	\[ \frac{x}{y \cdot \left(1 + y\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-37 [=>] 8.7	\[ \frac{x}{\color{blue}{1 \cdot y + y \cdot y}} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 8.7	\[ \frac{x}{\color{blue}{y \cdot 1} + y \cdot y} \]
      rational_best_oopsla_all_46_json_45_simplify-52 [=>] 8.7	\[ \frac{x}{\color{blue}{y} + y \cdot y} \]

3. Recombined 3 regimes into one program.

4. Final simplification 9.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{+92}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-146}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \end{array} \]

Alternatives

Alternative 1
Error	22.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 2.02 \cdot 10^{-163}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \end{array} \]

Alternative 2
Error	13.8
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{-24}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \end{array} \]

Alternative 3
Error	35.6
Cost	324

\[\begin{array}{l} \mathbf{if}\;y \leq 2.02 \cdot 10^{-163}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4
Error	46.9
Cost	192

\[\frac{x}{y} \]

Alternative 5
Error	61.6
Cost	128

\[-y \]

Alternative 6
Error	61.8
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023090 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))