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

Average Error: 20.5 → 0.1

Time: 17.6s

Precision: binary64

Cost: 1088

Math

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

↓

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

FPCore

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

↓

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

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

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

Python

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

↓

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

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(Float64(y / Float64(x + Float64(y + 1.0))) * Float64(x / Float64(y + x))) / Float64(y + x))
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 = ((y / (x + (y + 1.0))) * (x / (y + x))) / (y + x);
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[(N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]

Target

Original	20.5
Target	0.2
Herbie	0.1

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

Derivation

1. Initial program 20.5

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

2. Simplified 17.4

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

   [Start] 20.5	\[ \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   associate-/r* [=>] 17.4	\[ \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]
   associate-+l+ [=>] 17.4	\[ \frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\color{blue}{x + \left(y + 1\right)}} \]

3. Applied egg-rr 0.1

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

4. Applied egg-rr 0.1

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

5. Final simplification 0.1

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

Alternatives

Alternative 1
Error	26.5
Cost	1100

\[\begin{array}{l} t_0 := \frac{\frac{y}{y + x}}{y + \left(x + 1\right)}\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+25}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]

Alternative 2
Error	23.5
Cost	1096

\[\begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{-221}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{y + \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 1.28 \cdot 10^{+109}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(\left(y + 1\right) + x \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\frac{y + x}{x}}\\ \end{array} \]

Alternative 3
Error	23.5
Cost	1096

\[\begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{-221}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{y + \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 1.28 \cdot 10^{+109}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(\left(y + 1\right) + x \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\frac{y + x}{x}}\\ \end{array} \]

Alternative 4
Error	12.6
Cost	1092

\[\begin{array}{l} t_0 := \frac{x}{y + x}\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{+25}:\\ \;\;\;\;t_0 \cdot \left(\frac{y}{y + x} \cdot \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 \cdot \frac{y}{y + 1}}{y + x}\\ \end{array} \]

Alternative 5
Error	0.1
Cost	1088

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

Alternative 6
Error	27.1
Cost	976

\[\begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{+24}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-154}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-78}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 7
Error	26.9
Cost	976

\[\begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{+25}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-20}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-154}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-78}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \end{array} \]

Alternative 8
Error	26.6
Cost	972

\[\begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+25}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-154}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]

Alternative 9
Error	26.7
Cost	972

\[\begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+25}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-24}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-154}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]

Alternative 10
Error	31.3
Cost	848

\[\begin{array}{l} t_0 := \frac{x}{y \cdot y}\\ \mathbf{if}\;x \leq -3.3 \cdot 10^{+25}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-174}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	30.8
Cost	848

\[\begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+25}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-174}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-113}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 12
Error	30.3
Cost	848

\[\begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+25}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-24}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq -9.2 \cdot 10^{-176}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-114}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 13
Error	26.7
Cost	844

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+25}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-154}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 14
Error	26.7
Cost	844

\[\begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+25}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-24}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq -2.75 \cdot 10^{-155}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]

Alternative 15
Error	26.7
Cost	844

\[\begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+25}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-19}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-154}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]

Alternative 16
Error	35.6
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-163}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 0.76:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 17
Error	47.3
Cost	324

\[\begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+24}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 18
Error	43.3
Cost	324

\[\begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-174}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 19
Error	61.3
Cost	192

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

Alternative 20
Error	61.8
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023060 
(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))))