System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2

Average Error: 0.1 → 0.1

Time: 16.1s

Precision: binary64

Cost: 7296

Math

\[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]

↓

\[x \cdot 0.5 + \left(\left(1 + \log z\right) \cdot y + y \cdot \left(-z\right)\right) \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x 0.5) (* y (+ (- 1.0 z) (log z)))))

↓

(FPCore (x y z)
 :precision binary64
 (+ (* x 0.5) (+ (* (+ 1.0 (log z)) y) (* y (- z)))))

C

double code(double x, double y, double z) {
	return (x * 0.5) + (y * ((1.0 - z) + log(z)));
}

↓

double code(double x, double y, double z) {
	return (x * 0.5) + (((1.0 + log(z)) * y) + (y * -z));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * 0.5d0) + (y * ((1.0d0 - z) + log(z)))
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * 0.5d0) + (((1.0d0 + log(z)) * y) + (y * -z))
end function

Java

public static double code(double x, double y, double z) {
	return (x * 0.5) + (y * ((1.0 - z) + Math.log(z)));
}

↓

public static double code(double x, double y, double z) {
	return (x * 0.5) + (((1.0 + Math.log(z)) * y) + (y * -z));
}

Python

def code(x, y, z):
	return (x * 0.5) + (y * ((1.0 - z) + math.log(z)))

↓

def code(x, y, z):
	return (x * 0.5) + (((1.0 + math.log(z)) * y) + (y * -z))

Julia

function code(x, y, z)
	return Float64(Float64(x * 0.5) + Float64(y * Float64(Float64(1.0 - z) + log(z))))
end

↓

function code(x, y, z)
	return Float64(Float64(x * 0.5) + Float64(Float64(Float64(1.0 + log(z)) * y) + Float64(y * Float64(-z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * 0.5) + (y * ((1.0 - z) + log(z)));
end

↓

function tmp = code(x, y, z)
	tmp = (x * 0.5) + (((1.0 + log(z)) * y) + (y * -z));
end

Wolfram

code[x_, y_, z_] := N[(N[(x * 0.5), $MachinePrecision] + N[(y * N[(N[(1.0 - z), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := N[(N[(x * 0.5), $MachinePrecision] + N[(N[(N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] + N[(y * (-z)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	0.1
Target	0.1
Herbie	0.1

\[\left(y + 0.5 \cdot x\right) - y \cdot \left(z - \log z\right) \]

Derivation

1. Initial program 0.1

   \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]

2. Taylor expanded in y around 0 0.1

   \[\leadsto x \cdot 0.5 + \color{blue}{y \cdot \left(\left(1 + \log z\right) - z\right)} \]

3. Taylor expanded in z around 0 0.1

   \[\leadsto x \cdot 0.5 + \color{blue}{\left(-1 \cdot \left(y \cdot z\right) + \left(1 + \log z\right) \cdot y\right)} \]

4. Simplified 0.1

   \[\leadsto x \cdot 0.5 + \color{blue}{\left(\left(1 + \log z\right) \cdot y + y \cdot \left(-z\right)\right)} \]
   Proof

   [Start] 0.1	\[ x \cdot 0.5 + \left(-1 \cdot \left(y \cdot z\right) + \left(1 + \log z\right) \cdot y\right) \]
   rational.json-simplify-1 [=>] 0.1	\[ x \cdot 0.5 + \color{blue}{\left(\left(1 + \log z\right) \cdot y + -1 \cdot \left(y \cdot z\right)\right)} \]
   rational.json-simplify-43 [=>] 0.1	\[ x \cdot 0.5 + \left(\left(1 + \log z\right) \cdot y + \color{blue}{y \cdot \left(z \cdot -1\right)}\right) \]
   rational.json-simplify-8 [<=] 0.1	\[ x \cdot 0.5 + \left(\left(1 + \log z\right) \cdot y + y \cdot \color{blue}{\left(-z\right)}\right) \]

5. Final simplification 0.1

   \[\leadsto x \cdot 0.5 + \left(\left(1 + \log z\right) \cdot y + y \cdot \left(-z\right)\right) \]

Alternatives

Alternative 1
Error	18.4
Cost	7116

\[\begin{array}{l} t_0 := \left(1 + \log z\right) \cdot y\\ t_1 := x \cdot 0.5 + y \cdot \left(-z\right)\\ \mathbf{if}\;z \leq 1.26 \cdot 10^{-225}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-146}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-76}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	0.8
Cost	7108

\[\begin{array}{l} \mathbf{if}\;z \leq 0.115:\\ \;\;\;\;x \cdot 0.5 + \left(1 + \log z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(-z\right)\\ \end{array} \]

Alternative 3
Error	0.1
Cost	7104

\[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]

Alternative 4
Error	28.1
Cost	784

\[\begin{array}{l} t_0 := y \cdot \left(-z\right)\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-69}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{-38}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 9:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 5
Error	18.0
Cost	512

\[x \cdot 0.5 + y \cdot \left(-z\right) \]

Alternative 6
Error	35.1
Cost	192

\[0.5 \cdot x \]

Reproduce

herbie shell --seed 2023075 
(FPCore (x y z)
  :name "System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2"
  :precision binary64

  :herbie-target
  (- (+ y (* 0.5 x)) (* y (- z (log z))))

  (+ (* x 0.5) (* y (+ (- 1.0 z) (log z)))))