Main:bigenough2 from A

Average Error: 0.0 → 0.0

Time: 3.4s

Precision: binary64

Cost: 576

Math

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

↓

\[y \cdot z + x \cdot \left(y + 1\right) \]

FPCore

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

↓

(FPCore (x y z) :precision binary64 (+ (* y z) (* x (+ y 1.0))))

C

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

↓

double code(double x, double y, double z) {
	return (y * z) + (x * (y + 1.0));
}

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 + x))
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 * z) + (x * (y + 1.0d0))
end function

Java

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

↓

public static double code(double x, double y, double z) {
	return (y * z) + (x * (y + 1.0));
}

Python

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

↓

def code(x, y, z):
	return (y * z) + (x * (y + 1.0))

Julia

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

↓

function code(x, y, z)
	return Float64(Float64(y * z) + Float64(x * Float64(y + 1.0)))
end

MATLAB

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

↓

function tmp = code(x, y, z)
	tmp = (y * z) + (x * (y + 1.0));
end

Wolfram

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

↓

code[x_, y_, z_] := N[(N[(y * z), $MachinePrecision] + N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.0

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

2. Taylor expanded in x around -inf 0.0

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

3. Simplified 0.0

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

   [Start] 0.0	\[ y \cdot z + -1 \cdot \left(\left(-1 \cdot y - 1\right) \cdot x\right) \]
   mul-1-neg [=>] 0.0	\[ y \cdot z + \color{blue}{\left(-\left(-1 \cdot y - 1\right) \cdot x\right)} \]
   unsub-neg [=>] 0.0	\[ \color{blue}{y \cdot z - \left(-1 \cdot y - 1\right) \cdot x} \]
   *-commutative [=>] 0.0	\[ y \cdot z - \color{blue}{x \cdot \left(-1 \cdot y - 1\right)} \]
   mul-1-neg [=>] 0.0	\[ y \cdot z - x \cdot \left(\color{blue}{\left(-y\right)} - 1\right) \]
   sub-neg [=>] 0.0	\[ y \cdot z - x \cdot \color{blue}{\left(\left(-y\right) + \left(-1\right)\right)} \]
   metadata-eval [=>] 0.0	\[ y \cdot z - x \cdot \left(\left(-y\right) + \color{blue}{-1}\right) \]
   +-commutative [=>] 0.0	\[ y \cdot z - x \cdot \color{blue}{\left(-1 + \left(-y\right)\right)} \]
   neg-sub0 [=>] 0.0	\[ y \cdot z - x \cdot \left(-1 + \color{blue}{\left(0 - y\right)}\right) \]
   associate-+r- [=>] 0.0	\[ y \cdot z - x \cdot \color{blue}{\left(\left(-1 + 0\right) - y\right)} \]
   metadata-eval [=>] 0.0	\[ y \cdot z - x \cdot \left(\color{blue}{-1} - y\right) \]

4. Final simplification 0.0

   \[\leadsto y \cdot z + x \cdot \left(y + 1\right) \]

Alternatives

Alternative 1
Error	23.8
Cost	720

\[\begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+75}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{+41}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -3.9 \cdot 10^{-84}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 2
Error	1.4
Cost	716

\[\begin{array}{l} t_0 := y \cdot \left(z + x\right)\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-20}:\\ \;\;\;\;y \cdot z + x\\ \mathbf{elif}\;y \leq 860000000:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	12.9
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -3.9 \cdot 10^{-84} \lor \neg \left(y \leq 0.0064\right):\\ \;\;\;\;y \cdot \left(z + x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	12.8
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -3.35 \cdot 10^{-84} \lor \neg \left(y \leq 860000000\right):\\ \;\;\;\;y \cdot \left(z + x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \end{array} \]

Alternative 5
Error	24.6
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 6
Error	0.0
Cost	448

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

Alternative 7
Error	34.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023031 
(FPCore (x y z)
  :name "Main:bigenough2 from A"
  :precision binary64
  (+ x (* y (+ z x))))