Linear.Quaternion:$c/ from linear-1.19.1.3, B

Average Error: 16.9 → 0.0

Time: 4.4s

Precision: binary64

Cost: 448

Math

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

↓

\[y \cdot x - y \cdot z \]

FPCore

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

↓

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

C

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

↓

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

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Target

Original	16.9
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 16.9

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

2. Taylor expanded in x around 0 0.0

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

3. Simplified 0.0

   \[\leadsto \color{blue}{y \cdot x - y \cdot z} \]
   Proof

   [Start] 0.0	\[ y \cdot x + -1 \cdot \left(y \cdot z\right) \]
   *-commutative [=>] 0.0	\[ \color{blue}{x \cdot y} + -1 \cdot \left(y \cdot z\right) \]
   mul-1-neg [=>] 0.0	\[ x \cdot y + \color{blue}{\left(-y \cdot z\right)} \]
   sub-neg [<=] 0.0	\[ \color{blue}{x \cdot y - y \cdot z} \]
   *-commutative [<=] 0.0	\[ \color{blue}{y \cdot x} - y \cdot z \]

4. Final simplification 0.0

   \[\leadsto y \cdot x - y \cdot z \]

Alternatives

Alternative 1
Error	16.2
Cost	521

\[\begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+14} \lor \neg \left(z \leq 1.9 \cdot 10^{-110}\right):\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 2
Error	0.0
Cost	320

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

Alternative 3
Error	30.1
Cost	192

\[y \cdot x \]

Reproduce

herbie shell --seed 2023066 
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, B"
  :precision binary64

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

  (+ (- (- (* x y) (* y z)) (* y y)) (* y y)))