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

Average Error: 0.4 → 0.4

Time: 13.7s

Precision: binary64

Cost: 7232

Math

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

↓

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

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) (+ y (+ (/ (/ 1.0 x) 9.0) -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) * (y + (((1.0 / x) / 9.0) + -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) * (y + (((1.0d0 / x) / 9.0d0) + (-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) * (y + (((1.0 / x) / 9.0) + -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) * (y + (((1.0 / x) / 9.0) + -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(y + Float64(Float64(Float64(1.0 / x) / 9.0) + -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) * (y + (((1.0 / x) / 9.0) + -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[(y + N[(N[(N[(1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision] + -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. Simplified 0.4

   \[\leadsto \color{blue}{3 \cdot \left(\sqrt{x} \cdot \left(y + \left(\frac{\frac{1}{x}}{9} - 1\right)\right)\right)} \]
   Proof
   (*.f64 3 (*.f64 (sqrt.f64 x) (+.f64 y (-.f64 (/.f64 (/.f64 1 x) 9) 1)))): 0 points increase in error, 0 points decrease in error
   (*.f64 3 (*.f64 (sqrt.f64 x) (+.f64 y (-.f64 (Rewrite<= associate-/r*_binary64 (/.f64 1 (*.f64 x 9))) 1)))): 0 points increase in error, 0 points decrease in error
   (*.f64 3 (*.f64 (sqrt.f64 x) (+.f64 y (Rewrite=> sub-neg_binary64 (+.f64 (/.f64 1 (*.f64 x 9)) (neg.f64 1)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 3 (*.f64 (sqrt.f64 x) (+.f64 y (Rewrite=> +-commutative_binary64 (+.f64 (neg.f64 1) (/.f64 1 (*.f64 x 9))))))): 23 points increase in error, 0 points decrease in error
   (*.f64 3 (*.f64 (sqrt.f64 x) (+.f64 y (+.f64 (Rewrite=> metadata-eval -1) (/.f64 1 (*.f64 x 9)))))): 0 points increase in error, 23 points decrease in error
   (*.f64 3 (*.f64 (sqrt.f64 x) (+.f64 y (+.f64 (Rewrite<= metadata-eval (-.f64 0 1)) (/.f64 1 (*.f64 x 9)))))): 24 points increase in error, 0 points decrease in error
   (*.f64 3 (*.f64 (sqrt.f64 x) (+.f64 y (Rewrite<= associate--r-_binary64 (-.f64 0 (-.f64 1 (/.f64 1 (*.f64 x 9)))))))): 0 points increase in error, 23 points decrease in error
   (*.f64 3 (*.f64 (sqrt.f64 x) (+.f64 y (Rewrite=> associate--r-_binary64 (+.f64 (-.f64 0 1) (/.f64 1 (*.f64 x 9))))))): 0 points increase in error, 1 points decrease in error
   (*.f64 3 (*.f64 (sqrt.f64 x) (+.f64 y (+.f64 (Rewrite=> metadata-eval -1) (/.f64 1 (*.f64 x 9)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 3 (*.f64 (sqrt.f64 x) (+.f64 y (+.f64 (Rewrite<= metadata-eval (neg.f64 1)) (/.f64 1 (*.f64 x 9)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 3 (*.f64 (sqrt.f64 x) (+.f64 y (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 1 (*.f64 x 9)) (neg.f64 1)))))): 24 points increase in error, 0 points decrease in error
   (*.f64 3 (*.f64 (sqrt.f64 x) (+.f64 y (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 1 (*.f64 x 9)) 1))))): 0 points increase in error, 23 points decrease in error
   (*.f64 3 (*.f64 (sqrt.f64 x) (Rewrite=> +-commutative_binary64 (+.f64 (-.f64 (/.f64 1 (*.f64 x 9)) 1) y)))): 0 points increase in error, 1 points decrease in error
   (*.f64 3 (*.f64 (sqrt.f64 x) (+.f64 (-.f64 (/.f64 1 (*.f64 x 9)) 1) (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 y)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 3 (*.f64 (sqrt.f64 x) (Rewrite<= sub-neg_binary64 (-.f64 (-.f64 (/.f64 1 (*.f64 x 9)) 1) (neg.f64 y))))): 0 points increase in error, 0 points decrease in error
   (*.f64 3 (*.f64 (sqrt.f64 x) (Rewrite=> sub-neg_binary64 (+.f64 (-.f64 (/.f64 1 (*.f64 x 9)) 1) (neg.f64 (neg.f64 y)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 3 (*.f64 (sqrt.f64 x) (+.f64 (-.f64 (/.f64 1 (*.f64 x 9)) 1) (Rewrite=> remove-double-neg_binary64 y)))): 1 points increase in error, 0 points decrease in error
   (*.f64 3 (*.f64 (sqrt.f64 x) (Rewrite<= +-commutative_binary64 (+.f64 y (-.f64 (/.f64 1 (*.f64 x 9)) 1))))): 0 points increase in error, 1 points decrease in error
   (*.f64 3 (*.f64 (sqrt.f64 x) (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 y (/.f64 1 (*.f64 x 9))) 1)))): 0 points increase in error, 0 points decrease in error
   (*.f64 3 (Rewrite<= *-rgt-identity_binary64 (*.f64 (*.f64 (sqrt.f64 x) (-.f64 (+.f64 y (/.f64 1 (*.f64 x 9))) 1)) 1))): 24 points increase in error, 0 points decrease in error
   (*.f64 3 (Rewrite=> associate-*l*_binary64 (*.f64 (sqrt.f64 x) (*.f64 (-.f64 (+.f64 y (/.f64 1 (*.f64 x 9))) 1) 1)))): 0 points increase in error, 23 points decrease in error
   (*.f64 3 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (sqrt.f64 x) (-.f64 (+.f64 y (/.f64 1 (*.f64 x 9))) 1)) 1))): 23 points increase in error, 0 points decrease in error
   (*.f64 3 (Rewrite=> *-rgt-identity_binary64 (*.f64 (sqrt.f64 x) (-.f64 (+.f64 y (/.f64 1 (*.f64 x 9))) 1)))): 0 points increase in error, 24 points decrease in error
   (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 3 (sqrt.f64 x)) (-.f64 (+.f64 y (/.f64 1 (*.f64 x 9))) 1))): 1 points increase in error, 0 points decrease in error

3. Final simplification 0.4

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

Alternatives

Alternative 1
Error	27.3
Cost	8040

\[\begin{array}{l} t_0 := \frac{{x}^{-0.5}}{3}\\ t_1 := 3 \cdot \left(\sqrt{x} \cdot y\right)\\ t_2 := \sqrt{x} \cdot -3\\ t_3 := \frac{0.3333333333333333}{\sqrt{x}}\\ \mathbf{if}\;y \leq -7.8 \cdot 10^{+120}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{+61}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq -70:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{-82}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq -2.4 \cdot 10^{-167}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-257}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{-179}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-74}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{-42}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+29}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	27.4
Cost	8040

\[\begin{array}{l} t_0 := 3 \cdot \left(\sqrt{x} \cdot y\right)\\ t_1 := \frac{{x}^{-0.5}}{3}\\ t_2 := \sqrt{x} \cdot -3\\ t_3 := \frac{0.3333333333333333}{\sqrt{x}}\\ \mathbf{if}\;y \leq -3.4 \cdot 10^{+120}:\\ \;\;\;\;y \cdot \left(3 \cdot \sqrt{x}\right)\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{+61}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq -70:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.18 \cdot 10^{-82}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq -4.5 \cdot 10^{-168}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-257}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-183}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-72}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-45}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+22}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	27.4
Cost	8040

\[\begin{array}{l} t_0 := \frac{\sqrt{\frac{9}{x}}}{9}\\ t_1 := 3 \cdot \left(\sqrt{x} \cdot y\right)\\ t_2 := \sqrt{x} \cdot -3\\ t_3 := \frac{0.3333333333333333}{\sqrt{x}}\\ \mathbf{if}\;y \leq -3.4 \cdot 10^{+120}:\\ \;\;\;\;y \cdot \left(3 \cdot \sqrt{x}\right)\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{+59}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq -70:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{-82}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq -1.15 \cdot 10^{-166}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-252}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-187}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-73}:\\ \;\;\;\;\frac{{x}^{-0.5}}{3}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-41}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+23}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	27.5
Cost	8040

\[\begin{array}{l} t_0 := \frac{\sqrt{\frac{9}{x}}}{9}\\ t_1 := 3 \cdot \left(\sqrt{x} \cdot y\right)\\ t_2 := \sqrt{x} \cdot -3\\ t_3 := \frac{0.3333333333333333}{\sqrt{x}}\\ \mathbf{if}\;y \leq -3.4 \cdot 10^{+120}:\\ \;\;\;\;y \cdot \left(3 \cdot \sqrt{x}\right)\\ \mathbf{elif}\;y \leq -1.85 \cdot 10^{+61}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq -70:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{-82}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{-171}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-247}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-181}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-72}:\\ \;\;\;\;\frac{\frac{3}{\sqrt{x}}}{9}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-41}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+31}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	10.8
Cost	7377

\[\begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+120}:\\ \;\;\;\;y \cdot \left(3 \cdot \sqrt{x}\right)\\ \mathbf{elif}\;y \leq -1.25 \cdot 10^{+61}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq -0.16 \lor \neg \left(y \leq 7.2 \cdot 10^{+24}\right):\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \end{array} \]

Alternative 6
Error	11.0
Cost	7376

\[\begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+120}:\\ \;\;\;\;y \cdot \left(3 \cdot \sqrt{x}\right)\\ \mathbf{elif}\;y \leq -9.5 \cdot 10^{+60}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq -2.25:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right)\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+33}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{3 \cdot y}{\frac{1}{\sqrt{x}}}\\ \end{array} \]

Alternative 7
Error	10.7
Cost	7376

\[\begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+120}:\\ \;\;\;\;y \cdot \left(3 \cdot \sqrt{x}\right)\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{+61}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right)\\ \mathbf{elif}\;y \leq -2.25:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right)\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+26}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{3 \cdot y}{\frac{1}{\sqrt{x}}}\\ \end{array} \]

Alternative 8
Error	0.4
Cost	7104

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

Alternative 9
Error	9.8
Cost	6980

\[\begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{-60}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right)\\ \end{array} \]

Alternative 10
Error	22.0
Cost	6724

\[\begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 11
Error	61.9
Cost	6592

\[\sqrt{x \cdot 9} \]

Alternative 12
Error	46.0
Cost	6592

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

Reproduce

herbie shell --seed 2022343 
(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)))