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

Average Error: 0.33% → 0.38%

Time: 10.1s

Precision: binary64

Cost: 7104

Math

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

↓

\[\left(1 + \frac{-0.1111111111111111}{x}\right) - \frac{y}{\sqrt{x \cdot 9}} \]

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 (/ -0.1111111111111111 x)) (/ y (sqrt (* 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 + (-0.1111111111111111 / x)) - (y / sqrt((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 + ((-0.1111111111111111d0) / x)) - (y / sqrt((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 + (-0.1111111111111111 / x)) - (y / Math.sqrt((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 + (-0.1111111111111111 / x)) - (y / math.sqrt((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(-0.1111111111111111 / x)) - Float64(y / sqrt(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 + (-0.1111111111111111 / x)) - (y / sqrt((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[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision] - N[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	0.33%
Target	0.34%
Herbie	0.38%

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

Derivation

1. Initial program 0.33

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

2. Simplified 0.38

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

   [Start] 0.33	\[ \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   sub-neg [=>] 0.33	\[ \color{blue}{\left(1 + \left(-\frac{1}{x \cdot 9}\right)\right)} - \frac{y}{3 \cdot \sqrt{x}} \]
   +-commutative [=>] 0.33	\[ \color{blue}{\left(\left(-\frac{1}{x \cdot 9}\right) + 1\right)} - \frac{y}{3 \cdot \sqrt{x}} \]
   neg-sub0 [=>] 0.33	\[ \left(\color{blue}{\left(0 - \frac{1}{x \cdot 9}\right)} + 1\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   associate-+l- [=>] 0.33	\[ \color{blue}{\left(0 - \left(\frac{1}{x \cdot 9} - 1\right)\right)} - \frac{y}{3 \cdot \sqrt{x}} \]
   associate-+l- [<=] 0.33	\[ \color{blue}{\left(\left(0 - \frac{1}{x \cdot 9}\right) + 1\right)} - \frac{y}{3 \cdot \sqrt{x}} \]
   neg-sub0 [<=] 0.33	\[ \left(\color{blue}{\left(-\frac{1}{x \cdot 9}\right)} + 1\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   +-commutative [<=] 0.33	\[ \color{blue}{\left(1 + \left(-\frac{1}{x \cdot 9}\right)\right)} - \frac{y}{3 \cdot \sqrt{x}} \]
   distribute-neg-frac [=>] 0.33	\[ \left(1 + \color{blue}{\frac{-1}{x \cdot 9}}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   *-commutative [=>] 0.33	\[ \left(1 + \frac{-1}{\color{blue}{9 \cdot x}}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   associate-/r* [=>] 0.38	\[ \left(1 + \color{blue}{\frac{\frac{-1}{9}}{x}}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   metadata-eval [=>] 0.38	\[ \left(1 + \frac{\frac{\color{blue}{-1}}{9}}{x}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   metadata-eval [=>] 0.38	\[ \left(1 + \frac{\color{blue}{-0.1111111111111111}}{x}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

3. Applied egg-rr 0.38

   \[\leadsto \left(1 + \frac{-0.1111111111111111}{x}\right) - \frac{y}{\color{blue}{\sqrt{x \cdot 9}}} \]

4. Final simplification 0.38

   \[\leadsto \left(1 + \frac{-0.1111111111111111}{x}\right) - \frac{y}{\sqrt{x \cdot 9}} \]

Alternatives

Alternative 1
Error	5.68%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+44} \lor \neg \left(y \leq 10^{+31}\right):\\ \;\;\;\;1 + y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]

Alternative 2
Error	5.6%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+41}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+33}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\\ \end{array} \]

Alternative 3
Error	5.68%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+44}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+35}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + {x}^{-0.5} \cdot \left(y \cdot -0.3333333333333333\right)\\ \end{array} \]

Alternative 4
Error	5.66%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+42}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \left(y \cdot \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+35}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + {x}^{-0.5} \cdot \left(y \cdot -0.3333333333333333\right)\\ \end{array} \]

Alternative 5
Error	5.68%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -2.65 \cdot 10^{+44} \lor \neg \left(y \leq 1.35 \cdot 10^{+32}\right):\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]

Alternative 6
Error	8.45%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+46}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+76}:\\ \;\;\;\;1 + -0.1111111111111111 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot \left(y \cdot -0.3333333333333333\right)\\ \end{array} \]

Alternative 7
Error	8.46%
Cost	7112

\[\begin{array}{l} t_0 := \sqrt{\frac{1}{x}}\\ \mathbf{if}\;y \leq -7.8 \cdot 10^{+45}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot t_0\right)\\ \mathbf{elif}\;y \leq 1.32 \cdot 10^{+76}:\\ \;\;\;\;1 + -0.1111111111111111 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(y \cdot -0.3333333333333333\right)\\ \end{array} \]

Alternative 8
Error	8.45%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+46}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+76}:\\ \;\;\;\;1 + -0.1111111111111111 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\\ \end{array} \]

Alternative 9
Error	8.45%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+46} \lor \neg \left(y \leq 3.7 \cdot 10^{+76}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.1111111111111111 \cdot \frac{1}{x}\\ \end{array} \]

Alternative 10
Error	8.43%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+46}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+76}:\\ \;\;\;\;1 + -0.1111111111111111 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \end{array} \]

Alternative 11
Error	8.41%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+46}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+76}:\\ \;\;\;\;1 + -0.1111111111111111 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \end{array} \]

Alternative 12
Error	8.45%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+46}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+76}:\\ \;\;\;\;1 + -0.1111111111111111 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \end{array} \]

Alternative 13
Error	34.59%
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-6}:\\ \;\;\;\;-0.1111111111111111 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14
Error	33.55%
Cost	448

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

Alternative 15
Error	34.57%
Cost	324

\[\begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 16
Error	33.53%
Cost	320

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

Alternative 17
Error	65.93%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023121 
(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)))))