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

Average Error: 20.1 → 0.1

Time: 14.5s

Precision: binary64

Cost: 1088

Math

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

↓

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

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (* (/ x (+ x y)) (/ (/ y (+ y (+ x 1.0))) (+ x 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) {
	return (x / (x + y)) * ((y / (y + (x + 1.0))) / (x + y));
}

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
    code = (x / (x + y)) * ((y / (y + (x + 1.0d0))) / (x + y))
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) {
	return (x / (x + y)) * ((y / (y + (x + 1.0))) / (x + y));
}

Python

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

↓

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

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)
	return Float64(Float64(x / Float64(x + y)) * Float64(Float64(y / Float64(y + Float64(x + 1.0))) / Float64(x + y)))
end

MATLAB

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

↓

function tmp = code(x, y)
	tmp = (x / (x + y)) * ((y / (y + (x + 1.0))) / (x + y));
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_] := N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(y / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	20.1
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 20.1

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

2. Simplified 16.9

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

   [Start] 20.1	\[ \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   associate-/r* [=>] 16.9	\[ \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]
   associate-+l+ [=>] 16.9	\[ \frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\color{blue}{x + \left(y + 1\right)}} \]

3. Applied egg-rr 36.5

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{x + \left(y + 1\right)}\right)} - 1} \]

4. Simplified 0.1

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

   [Start] 36.5	\[ e^{\mathsf{log1p}\left(\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{x + \left(y + 1\right)}\right)} - 1 \]
   expm1-def [=>] 17.0	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{x + \left(y + 1\right)}\right)\right)} \]
   expm1-log1p [=>] 16.9	\[ \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{x + \left(y + 1\right)}} \]
   times-frac [=>] 0.1	\[ \frac{\color{blue}{\frac{x}{x + y} \cdot \frac{y}{x + y}}}{x + \left(y + 1\right)} \]
   associate-*r/ [<=] 0.1	\[ \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{x + y}}{x + \left(y + 1\right)}} \]
   associate-/l/ [=>] 4.1	\[ \frac{x}{x + y} \cdot \color{blue}{\frac{y}{\left(x + \left(y + 1\right)\right) \cdot \left(x + y\right)}} \]
   associate-/r* [=>] 0.1	\[ \frac{x}{x + y} \cdot \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)}}{x + y}} \]
   associate-+r+ [=>] 0.1	\[ \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]
   +-commutative [=>] 0.1	\[ \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{\left(y + x\right)} + 1}}{x + y} \]
   associate-+l+ [=>] 0.1	\[ \frac{x}{x + y} \cdot \frac{\frac{y}{\color{blue}{y + \left(x + 1\right)}}}{x + y} \]

5. Final simplification 0.1

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

Alternatives

Alternative 1
Error	25.1
Cost	1228

\[\begin{array}{l} t_0 := \frac{\frac{y}{x + y}}{1 + \left(x + y \cdot 2\right)}\\ \mathbf{if}\;x \leq -13500:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -4.9 \cdot 10^{-52}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-135}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 2
Error	25.3
Cost	1100

\[\begin{array}{l} t_0 := \frac{\frac{y}{y + \left(x + 1\right)}}{x + y}\\ \mathbf{if}\;x \leq -29000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-52}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -1.32 \cdot 10^{-136}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 3
Error	11.8
Cost	1092

\[\begin{array}{l} \mathbf{if}\;x \leq -280:\\ \;\;\;\;\frac{\frac{y}{x + y}}{1 + \left(x + y \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{\frac{y}{y + 1}}{x + y}\\ \end{array} \]

Alternative 4
Error	25.4
Cost	972

\[\begin{array}{l} t_0 := \frac{\frac{y}{x}}{x + \left(y + 1\right)}\\ \mathbf{if}\;x \leq -14000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-52}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-135}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 5
Error	25.9
Cost	845

\[\begin{array}{l} \mathbf{if}\;x \leq -3400:\\ \;\;\;\;y \cdot \frac{1}{x \cdot x}\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-52} \lor \neg \left(x \leq -2.8 \cdot 10^{-135}\right):\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x}\\ \end{array} \]

Alternative 6
Error	25.5
Cost	845

\[\begin{array}{l} \mathbf{if}\;x \leq -28000:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-52} \lor \neg \left(x \leq -2.8 \cdot 10^{-135}\right):\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x}\\ \end{array} \]

Alternative 7
Error	28.6
Cost	844

\[\begin{array}{l} t_0 := y \cdot \frac{1}{x \cdot x}\\ \mathbf{if}\;y \leq -2.7 \cdot 10^{-118}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-180}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 2200000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 8
Error	28.6
Cost	844

\[\begin{array}{l} t_0 := y \cdot \frac{1}{x \cdot x}\\ \mathbf{if}\;y \leq -8.2 \cdot 10^{-117}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-180}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 2300000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + y}\\ \end{array} \]

Alternative 9
Error	25.5
Cost	844

\[\begin{array}{l} \mathbf{if}\;x \leq -5200:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-52}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-136}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 10
Error	29.3
Cost	716

\[\begin{array}{l} t_0 := \frac{y}{x \cdot x}\\ \mathbf{if}\;y \leq -1.45 \cdot 10^{-117}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-181}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 2100000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 11
Error	28.6
Cost	716

\[\begin{array}{l} t_0 := \frac{y}{x \cdot x}\\ \mathbf{if}\;y \leq -2.7 \cdot 10^{-118}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-180}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 2300000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 12
Error	36.0
Cost	452

\[\begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{-15}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 13
Error	46.9
Cost	324

\[\begin{array}{l} \mathbf{if}\;y \leq 520000:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y}\\ \end{array} \]

Alternative 14
Error	61.3
Cost	192

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

Reproduce

herbie shell --seed 2023031 
(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))))