Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D

Average Error: 0.2 → 0.2

Time: 8.0s

Precision: binary64

Cost: 7232

Math

\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

↓

\[\left(1 - \frac{\frac{y}{3}}{\sqrt{x}}\right) - \frac{1}{x \cdot 9} \]

FPCore

(FPCore (x y)
 :precision binary64
 (- (- 1.0 (/ 1.0 (* x 9.0))) (/ y (* 3.0 (sqrt x)))))

↓

(FPCore (x y)
 :precision binary64
 (- (- 1.0 (/ (/ y 3.0) (sqrt x))) (/ 1.0 (* x 9.0))))

C

double code(double x, double y) {
	return (1.0 - (1.0 / (x * 9.0))) - (y / (3.0 * sqrt(x)));
}

↓

double code(double x, double y) {
	return (1.0 - ((y / 3.0) / sqrt(x))) - (1.0 / (x * 9.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - (1.0d0 / (x * 9.0d0))) - (y / (3.0d0 * sqrt(x)))
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - ((y / 3.0d0) / sqrt(x))) - (1.0d0 / (x * 9.0d0))
end function

Java

public static double code(double x, double y) {
	return (1.0 - (1.0 / (x * 9.0))) - (y / (3.0 * Math.sqrt(x)));
}

↓

public static double code(double x, double y) {
	return (1.0 - ((y / 3.0) / Math.sqrt(x))) - (1.0 / (x * 9.0));
}

Python

def code(x, y):
	return (1.0 - (1.0 / (x * 9.0))) - (y / (3.0 * math.sqrt(x)))

↓

def code(x, y):
	return (1.0 - ((y / 3.0) / math.sqrt(x))) - (1.0 / (x * 9.0))

Julia

function code(x, y)
	return Float64(Float64(1.0 - Float64(1.0 / Float64(x * 9.0))) - Float64(y / Float64(3.0 * sqrt(x))))
end

↓

function code(x, y)
	return Float64(Float64(1.0 - Float64(Float64(y / 3.0) / sqrt(x))) - Float64(1.0 / Float64(x * 9.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - (1.0 / (x * 9.0))) - (y / (3.0 * sqrt(x)));
end

↓

function tmp = code(x, y)
	tmp = (1.0 - ((y / 3.0) / sqrt(x))) - (1.0 / (x * 9.0));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := N[(N[(1.0 - N[(N[(y / 3.0), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	0.2
Target	0.2
Herbie	0.2

\[\left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

Derivation

1. Initial program 0.2

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

2. Simplified 0.2

   \[\leadsto \color{blue}{\left(1 - \frac{\frac{y}{3}}{\sqrt{x}}\right) - \frac{1}{x \cdot 9}} \]
   Proof

   [Start] 0.2	\[ \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   rational.json-simplify-42 [=>] 0.2	\[ \color{blue}{\left(1 - \frac{y}{3 \cdot \sqrt{x}}\right) - \frac{1}{x \cdot 9}} \]
   rational.json-simplify-46 [=>] 0.2	\[ \left(1 - \color{blue}{\frac{\frac{y}{3}}{\sqrt{x}}}\right) - \frac{1}{x \cdot 9} \]

3. Final simplification 0.2

   \[\leadsto \left(1 - \frac{\frac{y}{3}}{\sqrt{x}}\right) - \frac{1}{x \cdot 9} \]

Alternatives

Alternative 1
Error	0.2
Cost	7232

\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

Alternative 2
Error	5.3
Cost	7112

\[\begin{array}{l} t_0 := y \cdot \left(\sqrt{\frac{1}{x}} \cdot -0.3333333333333333\right)\\ \mathbf{if}\;y \leq -2.9 \cdot 10^{+69}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+42}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.3
Cost	7104

\[\left(1 - \frac{0.1111111111111111}{x}\right) - \frac{0.3333333333333333}{\sqrt{x}} \cdot y \]

Alternative 4
Error	0.2
Cost	7104

\[\left(1 - \frac{0.1111111111111111}{x}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

Alternative 5
Error	0.2
Cost	7104

\[\left(1 - \frac{\frac{y}{3}}{\sqrt{x}}\right) - \frac{0.1111111111111111}{x} \]

Alternative 6
Error	22.2
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;-18 \cdot \frac{0.006172839506172839}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	22.1
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\frac{1.9051973784484073 \cdot 10^{-5}}{x} \cdot -5832\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Error	22.1
Cost	324

\[\begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Error	21.5
Cost	320

\[1 - \frac{0.1111111111111111}{x} \]

Alternative 10
Error	42.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023074 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :herbie-target
  (- (- 1.0 (/ (/ 1.0 x) 9.0)) (/ y (* 3.0 (sqrt x))))

  (- (- 1.0 (/ 1.0 (* x 9.0))) (/ y (* 3.0 (sqrt x)))))