Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B

Average Error: 0.4 → 0.4

Time: 7.3s

Precision: binary64

Cost: 7232

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Fortran

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

↓

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

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Target

Original	0.4
Target	0.4
Herbie	0.4

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

Derivation

1. Initial program 0.4

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

2. Applied egg-rr 0.5

   \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \color{blue}{\left(\frac{1}{x \cdot 9} \cdot 4 - \left(\frac{1}{x \cdot 9} + \frac{1}{x \cdot 9} \cdot 2\right)\right)}\right) - 1\right) \]

3. Applied egg-rr 0.4

   \[\leadsto \color{blue}{3 \cdot \left(\sqrt{x} \cdot \left(\frac{1}{x \cdot 9} + \left(y + -1\right)\right)\right) + 0} \]

4. Simplified 0.4

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

   [Start] 0.4	\[ 3 \cdot \left(\sqrt{x} \cdot \left(\frac{1}{x \cdot 9} + \left(y + -1\right)\right)\right) + 0 \]
   rational_best-simplify-3 [=>] 0.4	\[ \color{blue}{3 \cdot \left(\sqrt{x} \cdot \left(\frac{1}{x \cdot 9} + \left(y + -1\right)\right)\right)} \]

5. Final simplification 0.4

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

Alternatives

Alternative 1
Error	23.8
Cost	7776

\[\begin{array}{l} t_0 := -3 \cdot \sqrt{x}\\ t_1 := \sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ t_2 := 3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{if}\;x \leq 2.75 \cdot 10^{-136}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-112}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-52}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{+107}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+232}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{+245}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	10.7
Cost	7636

\[\begin{array}{l} t_0 := 3 \cdot \left(\left(\frac{0.1111111111111111}{x} - 1\right) \cdot \sqrt{x}\right)\\ \mathbf{if}\;x \leq 2.75 \cdot 10^{-136}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-112}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-64}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-52}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;x \leq 180:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + 3 \cdot y\right)\\ \end{array} \]

Alternative 3
Error	10.8
Cost	7636

\[\begin{array}{l} t_0 := \sqrt{x} \cdot \left(3 \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right)\\ \mathbf{if}\;x \leq 2.75 \cdot 10^{-136}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-112}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-64}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-50}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;x \leq 155:\\ \;\;\;\;3 \cdot \left(\left(\frac{0.1111111111111111}{x} - 1\right) \cdot \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + 3 \cdot y\right)\\ \end{array} \]

Alternative 4
Error	10.8
Cost	7508

\[\begin{array}{l} t_0 := \sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{if}\;x \leq 2.75 \cdot 10^{-136}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-112}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-64}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-50}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + 3 \cdot y\right)\\ \end{array} \]

Alternative 5
Error	0.4
Cost	7104

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

Alternative 6
Error	26.8
Cost	6984

\[\begin{array}{l} t_0 := 3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{if}\;y \leq -0.0027:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-9}:\\ \;\;\;\;-3 \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	26.8
Cost	6984

\[\begin{array}{l} \mathbf{if}\;y \leq -0.0027:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-9}:\\ \;\;\;\;-3 \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \end{array} \]

Alternative 8
Error	46.8
Cost	6592

\[-3 \cdot \sqrt{x} \]

Reproduce

herbie shell --seed 2023096 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

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

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