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

Average Error: 13.2 → 0.0

Time: 8.9s

Precision: binary64

Cost: 320

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Target

Original	13.2
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 13.2

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

2. Simplified 9.1

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

   [Start] 13.2	\[ \left(\left(x \cdot y - y \cdot y\right) + y \cdot y\right) - y \cdot z \]
   rational_best.json-simplify-2 [=>] 13.2	\[ \left(\left(\color{blue}{y \cdot x} - y \cdot y\right) + y \cdot y\right) - y \cdot z \]
   rational_best.json-simplify-48 [=>] 13.2	\[ \left(\color{blue}{y \cdot \left(x - y\right)} + y \cdot y\right) - y \cdot z \]
   rational_best.json-simplify-47 [=>] 9.1	\[ \color{blue}{y \cdot \left(y + \left(x - y\right)\right)} - y \cdot z \]
   rational_best.json-simplify-2 [=>] 9.1	\[ y \cdot \left(y + \left(x - y\right)\right) - \color{blue}{z \cdot y} \]
   rational_best.json-simplify-48 [=>] 9.1	\[ \color{blue}{y \cdot \left(\left(y + \left(x - y\right)\right) - z\right)} \]

3. Taylor expanded in y around 0 0.0

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

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	14.5
Cost	520

\[\begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-23}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-19}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 2
Error	29.9
Cost	192

\[y \cdot x \]

Reproduce

herbie shell --seed 2023088 
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, D"
  :precision binary64

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

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