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

Average Accuracy: 99.7% → 99.6%

Time: 10.7s

Precision: binary64

Cost: 7232

Math

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

↓

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

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 (* 0.1111111111111111 (/ -1.0 x))) (/ y (* 3.0 (sqrt x)))))

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 + (0.1111111111111111 * (-1.0 / x))) - (y / (3.0 * sqrt(x)));
}

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 + (0.1111111111111111d0 * ((-1.0d0) / x))) - (y / (3.0d0 * sqrt(x)))
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 + (0.1111111111111111 * (-1.0 / x))) - (y / (3.0 * Math.sqrt(x)));
}

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 + (0.1111111111111111 * (-1.0 / x))) - (y / (3.0 * math.sqrt(x)))

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(0.1111111111111111 * Float64(-1.0 / x))) - Float64(y / Float64(3.0 * sqrt(x))))
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 + (0.1111111111111111 * (-1.0 / x))) - (y / (3.0 * sqrt(x)));
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[(0.1111111111111111 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	99.7%
Target	99.7%
Herbie	99.6%

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

Derivation

1. Initial program 99.7%

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

2. Simplified 99.6%

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

   [Start] 99.7	\[ \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   *-commutative [=>] 99.7	\[ \left(1 - \frac{1}{\color{blue}{9 \cdot x}}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   associate-/r* [=>] 99.6	\[ \left(1 - \color{blue}{\frac{\frac{1}{9}}{x}}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   metadata-eval [=>] 99.6	\[ \left(1 - \frac{\color{blue}{0.1111111111111111}}{x}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

3. Applied egg-rr 99.6%

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

   [Start] 99.6	\[ \left(1 - \frac{0.1111111111111111}{x}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   clear-num [=>] 99.6	\[ \left(1 - \color{blue}{\frac{1}{\frac{x}{0.1111111111111111}}}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   associate-/r/ [=>] 99.6	\[ \left(1 - \color{blue}{\frac{1}{x} \cdot 0.1111111111111111}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

4. Final simplification 99.6%

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

Alternatives

Alternative 1
Accuracy	94.0%
Cost	7240

\[\begin{array}{l} \mathbf{if}\;y \leq -2.65 \cdot 10^{+88}:\\ \;\;\;\;1 - y \cdot \frac{0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+34}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(y \cdot \sqrt{\frac{1}{x}}\right) \cdot -0.3333333333333333\\ \end{array} \]

Alternative 2
Accuracy	94.1%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -2.65 \cdot 10^{+88} \lor \neg \left(y \leq 2.05 \cdot 10^{+34}\right):\\ \;\;\;\;1 - y \cdot \frac{0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \end{array} \]

Alternative 3
Accuracy	99.6%
Cost	7104

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

Alternative 4
Accuracy	99.6%
Cost	7104

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

Alternative 5
Accuracy	91.2%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;y \leq -2.65 \cdot 10^{+88} \lor \neg \left(y \leq 3.8 \cdot 10^{+34}\right):\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \end{array} \]

Alternative 6
Accuracy	91.3%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+88} \lor \neg \left(y \leq 3.8 \cdot 10^{+34}\right):\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \end{array} \]

Alternative 7
Accuracy	91.2%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;y \leq -2.65 \cdot 10^{+88}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+34}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \]

Alternative 8
Accuracy	64.4%
Cost	452

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

Alternative 9
Accuracy	65.4%
Cost	448

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

Alternative 10
Accuracy	64.4%
Cost	324

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

Alternative 11
Accuracy	65.4%
Cost	320

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

Alternative 12
Accuracy	33.4%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023151 
(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)))))