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

Average Error: 17.6 → 0.0

Time: 7.0s

Precision: binary64

Cost: 320

Math

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

↓

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

FPCore

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

↓

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

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 (x - z) * y;
}

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 = (x - z) * y
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 (x - z) * y;
}

Python

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

↓

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

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(x - z) * y)
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 = (x - z) * y;
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[(x - z), $MachinePrecision] * y), $MachinePrecision]

Target

Original	17.6
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 17.6

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

2. Simplified 17.4

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

   [Start] 17.6	\[ \left(\left(x \cdot y - y \cdot z\right) - y \cdot y\right) + y \cdot y \]
   rational.json-simplify-42 [=>] 17.6	\[ \color{blue}{\left(\left(x \cdot y - y \cdot y\right) - y \cdot z\right)} + y \cdot y \]
   rational.json-simplify-2 [=>] 17.6	\[ \left(\left(\color{blue}{y \cdot x} - y \cdot y\right) - y \cdot z\right) + y \cdot y \]
   rational.json-simplify-52 [=>] 17.6	\[ \left(\color{blue}{y \cdot \left(x - y\right)} - y \cdot z\right) + y \cdot y \]
   rational.json-simplify-2 [=>] 17.6	\[ \left(y \cdot \left(x - y\right) - \color{blue}{z \cdot y}\right) + y \cdot y \]
   rational.json-simplify-52 [=>] 17.6	\[ \color{blue}{y \cdot \left(\left(x - y\right) - z\right)} + y \cdot y \]
   rational.json-simplify-51 [=>] 17.4	\[ \color{blue}{y \cdot \left(y + \left(\left(x - y\right) - z\right)\right)} \]
   rational.json-simplify-42 [=>] 17.4	\[ y \cdot \left(y + \color{blue}{\left(\left(x - z\right) - y\right)}\right) \]

3. Taylor expanded in y around 0 0.0

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

4. Simplified 0.0

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

   [Start] 0.0	\[ y \cdot \left(x - z\right) \]
   rational.json-simplify-2 [=>] 0.0	\[ \color{blue}{\left(x - z\right) \cdot y} \]

5. Final simplification 0.0

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

Alternatives

Alternative 1
Error	15.4
Cost	520

\[\begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-54}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-81}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 2
Error	30.3
Cost	192

\[y \cdot x \]

Reproduce

herbie shell --seed 2023075 
(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)))