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

Average Error: 0.1 → 0.1

Time: 12.2s

Precision: binary64

Cost: 576

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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) + (z * z)) + (z * z)) + (z * 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 = (z * (3.0d0 * z)) + (y * x)
end function

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Target

Original	0.1
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 0.1

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

2. Simplified 0.1

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

   [Start] 0.1	\[ \left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   rational_best-simplify-3 [=>] 0.1	\[ \color{blue}{z \cdot z + \left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right)} \]
   rational_best-simplify-3 [=>] 0.1	\[ z \cdot z + \color{blue}{\left(z \cdot z + \left(x \cdot y + z \cdot z\right)\right)} \]
   rational_best-simplify-3 [=>] 0.1	\[ z \cdot z + \left(z \cdot z + \color{blue}{\left(z \cdot z + x \cdot y\right)}\right) \]
   rational_best-simplify-47 [=>] 0.1	\[ z \cdot z + \color{blue}{\left(x \cdot y + \left(z \cdot z + z \cdot z\right)\right)} \]
   rational_best-simplify-74 [=>] 0.1	\[ z \cdot z + \left(x \cdot y + \color{blue}{z \cdot \left(z + z\right)}\right) \]

3. Applied egg-rr 9.6

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

4. Simplified 0.1

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

   [Start] 9.6	\[ z - \left(z - \left(x \cdot y + z \cdot \left(z + \left(z + z\right)\right)\right)\right) \]
   rational_best-simplify-66 [=>] 9.6	\[ z - \color{blue}{\left(\left(z - x \cdot y\right) + \left(-z \cdot \left(z + \left(z + z\right)\right)\right)\right)} \]
   rational_best-simplify-58 [=>] 9.6	\[ \color{blue}{\left(z - \left(-z \cdot \left(z + \left(z + z\right)\right)\right)\right) - \left(z - x \cdot y\right)} \]
   rational_best-simplify-69 [=>] 0.1	\[ \color{blue}{x \cdot y - \left(-z \cdot \left(z + \left(z + z\right)\right)\right)} \]
   rational_best-simplify-8 [<=] 0.1	\[ \color{blue}{\frac{x \cdot y}{1}} - \left(-z \cdot \left(z + \left(z + z\right)\right)\right) \]
   rational_best-simplify-44 [=>] 0.1	\[ \color{blue}{\frac{x \cdot y + x \cdot y}{1 + 1}} - \left(-z \cdot \left(z + \left(z + z\right)\right)\right) \]
   metadata-eval [=>] 0.1	\[ \frac{x \cdot y + x \cdot y}{\color{blue}{2}} - \left(-z \cdot \left(z + \left(z + z\right)\right)\right) \]
   rational_best-simplify-77 [=>] 0.1	\[ \color{blue}{\left(\frac{x \cdot y}{2} + \frac{x \cdot y}{2}\right)} - \left(-z \cdot \left(z + \left(z + z\right)\right)\right) \]
   rational_best-simplify-14 [=>] 0.1	\[ \left(\frac{x \cdot y}{2} + \frac{x \cdot y}{2}\right) - \color{blue}{\left(0 - z \cdot \left(z + \left(z + z\right)\right)\right)} \]
   rational_best-simplify-75 [=>] 0.1	\[ \left(\frac{x \cdot y}{2} + \frac{x \cdot y}{2}\right) - \left(0 - \color{blue}{\left(z \cdot z + \left(z + z\right) \cdot z\right)}\right) \]
   rational_best-simplify-1 [<=] 0.1	\[ \left(\frac{x \cdot y}{2} + \frac{x \cdot y}{2}\right) - \left(0 - \left(z \cdot z + \color{blue}{z \cdot \left(z + z\right)}\right)\right) \]
   rational_best-simplify-58 [=>] 0.1	\[ \left(\frac{x \cdot y}{2} + \frac{x \cdot y}{2}\right) - \color{blue}{\left(\left(0 - z \cdot \left(z + z\right)\right) - z \cdot z\right)} \]
   rational_best-simplify-75 [=>] 0.1	\[ \left(\frac{x \cdot y}{2} + \frac{x \cdot y}{2}\right) - \left(\left(0 - \color{blue}{\left(z \cdot z + z \cdot z\right)}\right) - z \cdot z\right) \]
   rational_best-simplify-57 [<=] 0.1	\[ \left(\frac{x \cdot y}{2} + \frac{x \cdot y}{2}\right) - \left(\color{blue}{\left(\left(0 - z \cdot z\right) - z \cdot z\right)} - z \cdot z\right) \]
   rational_best-simplify-14 [<=] 0.1	\[ \left(\frac{x \cdot y}{2} + \frac{x \cdot y}{2}\right) - \left(\left(\color{blue}{\left(-z \cdot z\right)} - z \cdot z\right) - z \cdot z\right) \]
   rational_best-simplify-84 [=>] 0.1	\[ \color{blue}{\left(\frac{x \cdot y}{2} + z \cdot z\right) + \left(\frac{x \cdot y}{2} - \left(\left(-z \cdot z\right) - z \cdot z\right)\right)} \]

5. Applied egg-rr 0.1

   \[\leadsto \color{blue}{\left(z \cdot \left(z \cdot 3\right) + 0\right)} + y \cdot x \]

6. Simplified 0.1

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

   [Start] 0.1	\[ \left(z \cdot \left(z \cdot 3\right) + 0\right) + y \cdot x \]
   rational_best-simplify-3 [=>] 0.1	\[ \color{blue}{\left(0 + z \cdot \left(z \cdot 3\right)\right)} + y \cdot x \]
   rational_best-simplify-6 [=>] 0.1	\[ \color{blue}{z \cdot \left(z \cdot 3\right)} + y \cdot x \]
   rational_best-simplify-1 [=>] 0.1	\[ z \cdot \color{blue}{\left(3 \cdot z\right)} + y \cdot x \]

7. Final simplification 0.1

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

Alternatives

Alternative 1
Error	11.2
Cost	584

\[\begin{array}{l} t_0 := 3 \cdot \left(z \cdot z\right)\\ \mathbf{if}\;z \leq -0.098:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-57}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	11.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -0.125:\\ \;\;\;\;3 \cdot \left(z \cdot z\right)\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-57}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(3 \cdot z\right)\\ \end{array} \]

Alternative 3
Error	23.8
Cost	192

\[y \cdot x \]

Reproduce

herbie shell --seed 2023101 
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, A"
  :precision binary64

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

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