Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, B

Average Error: 0.2 → 0.1

Time: 3.5s

Precision: binary64

Cost: 448

Math

\[\left(x \cdot 3\right) \cdot y - z \]

↓

\[3 \cdot \left(y \cdot x\right) - z \]

FPCore

(FPCore (x y z) :precision binary64 (- (* (* x 3.0) y) z))

↓

(FPCore (x y z) :precision binary64 (- (* 3.0 (* y x)) z))

C

double code(double x, double y, double z) {
	return ((x * 3.0) * y) - z;
}

↓

double code(double x, double y, double z) {
	return (3.0 * (y * x)) - 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 * 3.0d0) * y) - 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 = (3.0d0 * (y * x)) - z
end function

Java

public static double code(double x, double y, double z) {
	return ((x * 3.0) * y) - z;
}

↓

public static double code(double x, double y, double z) {
	return (3.0 * (y * x)) - z;
}

Python

def code(x, y, z):
	return ((x * 3.0) * y) - z

↓

def code(x, y, z):
	return (3.0 * (y * x)) - z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x * 3.0) * y) - z)
end

↓

function code(x, y, z)
	return Float64(Float64(3.0 * Float64(y * x)) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x * 3.0) * y) - z;
end

↓

function tmp = code(x, y, z)
	tmp = (3.0 * (y * x)) - z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x * 3.0), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision]

↓

code[x_, y_, z_] := N[(N[(3.0 * N[(y * x), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]

Target

Original	0.2
Target	0.2
Herbie	0.1

\[x \cdot \left(3 \cdot y\right) - z \]

Derivation

1. Initial program 0.2

   \[\left(x \cdot 3\right) \cdot y - z \]

2. Simplified 0.2

   \[\leadsto \color{blue}{x \cdot \left(3 \cdot y\right) - z} \]
   Proof
   (-.f64 (*.f64 x (*.f64 3 y)) z): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x 3) y)) z): 18 points increase in error, 29 points decrease in error

3. Taylor expanded in x around 0 0.1

   \[\leadsto \color{blue}{3 \cdot \left(y \cdot x\right)} - z \]

4. Final simplification 0.1

   \[\leadsto 3 \cdot \left(y \cdot x\right) - z \]

Alternatives

Alternative 1
Error	16.8
Cost	848

\[\begin{array}{l} t_0 := 3 \cdot \left(y \cdot x\right)\\ \mathbf{if}\;z \leq -1.75 \cdot 10^{+14}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -5.2 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1.85 \cdot 10^{-37}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 5.3 \cdot 10^{-124}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 2
Error	16.8
Cost	848

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+14}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -2.15 \cdot 10^{-6}:\\ \;\;\;\;3 \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;z \leq -2.35 \cdot 10^{-39}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-122}:\\ \;\;\;\;y \cdot \left(3 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 3
Error	27.0
Cost	128

\[-z \]

Reproduce

herbie shell --seed 2022331 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (- (* x (* 3.0 y)) z)

  (- (* (* x 3.0) y) z))