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

Average Error: 0.2 → 0.2

Time: 7.8s

Precision: binary64

Cost: 7232

Math

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

↓

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

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 (+ (/ 1.0 (* x 9.0)) (/ (/ 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 - ((1.0 / (x * 9.0)) + ((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 - ((1.0d0 / (x * 9.0d0)) + ((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 - ((1.0 / (x * 9.0)) + ((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 - ((1.0 / (x * 9.0)) + ((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(1.0 - Float64(Float64(1.0 / Float64(x * 9.0)) + Float64(Float64(y / 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 - ((1.0 / (x * 9.0)) + ((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[(1.0 - N[(N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision] + N[(N[(y / 3.0), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $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

   \[\leadsto \color{blue}{1 - \left(\frac{1}{x \cdot 9} + \frac{\frac{y}{3}}{\sqrt{x}}\right)} \]
   Proof
   (-.f64 1 (+.f64 (/.f64 1 (*.f64 x 9)) (/.f64 (/.f64 y 3) (sqrt.f64 x)))): 0 points increase in error, 0 points decrease in error
   (-.f64 1 (+.f64 (/.f64 1 (*.f64 x 9)) (Rewrite<= associate-/r*_binary64 (/.f64 y (*.f64 3 (sqrt.f64 x)))))): 12 points increase in error, 7 points decrease in error
   (-.f64 1 (+.f64 (/.f64 1 (*.f64 x 9)) (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (/.f64 y (*.f64 3 (sqrt.f64 x)))))))): 0 points increase in error, 0 points decrease in error
   (-.f64 1 (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 1 (*.f64 x 9)) (neg.f64 (/.f64 y (*.f64 3 (sqrt.f64 x))))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 1 (/.f64 1 (*.f64 x 9))) (neg.f64 (/.f64 y (*.f64 3 (sqrt.f64 x)))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= sub-neg_binary64 (-.f64 (-.f64 1 (/.f64 1 (*.f64 x 9))) (/.f64 y (*.f64 3 (sqrt.f64 x))))): 0 points increase in error, 0 points decrease in error

3. Final simplification 0.2

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

Alternatives

Alternative 1
Error	3.5
Cost	7240

\[\begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+41}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+47}:\\ \;\;\;\;1 - 0.1111111111111111 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - y \cdot \left(0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)\\ \end{array} \]

Alternative 2
Error	3.4
Cost	7176

\[\begin{array}{l} t_0 := 1 + -0.3333333333333333 \cdot \left(y \cdot {x}^{-0.5}\right)\\ \mathbf{if}\;y \leq -3.9 \cdot 10^{+45}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.26 \cdot 10^{+47}:\\ \;\;\;\;1 - 0.1111111111111111 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.3
Cost	7104

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

Alternative 4
Error	0.2
Cost	7104

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

Alternative 5
Error	5.4
Cost	6984

\[\begin{array}{l} t_0 := \frac{y}{\sqrt{x}} \cdot -0.3333333333333333\\ \mathbf{if}\;y \leq -2.65 \cdot 10^{+48}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+64}:\\ \;\;\;\;1 - 0.1111111111111111 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	5.4
Cost	6984

\[\begin{array}{l} t_0 := y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{if}\;y \leq -2.55 \cdot 10^{+48}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+64}:\\ \;\;\;\;1 - 0.1111111111111111 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	5.4
Cost	6984

\[\begin{array}{l} \mathbf{if}\;y \leq -2.65 \cdot 10^{+48}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(y \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+62}:\\ \;\;\;\;1 - 0.1111111111111111 \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \]

Alternative 8
Error	20.9
Cost	448

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

Alternative 9
Error	21.5
Cost	324

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

Alternative 10
Error	20.9
Cost	320

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

Alternative 11
Error	41.9
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022329 
(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)))))