Main:bigenough2 from A

Average Error: 0.0 → 0.0

Time: 4.5s

Precision: binary64

Cost: 576

Math

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

↓

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

FPCore

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

↓

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

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 * (-1.0 - 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 * (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 * ((-1.0d0) - y))
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 * (-1.0 - y));
}

Python

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

↓

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

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(-1.0 - y)))
end

MATLAB

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

↓

function tmp = code(x, y, z)
	tmp = (y * z) - (x * (-1.0 - y));
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[(-1.0 - y), $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

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

Alternatives

Alternative 1
Error	26.4
Cost	1248

\[\begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+128}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{+42}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -0.0029:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-16}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-35}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-45}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-80}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-41}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	13.5
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-97} \lor \neg \left(y \leq 2.4 \cdot 10^{-59}\right):\\ \;\;\;\;y \cdot \left(z + x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	0.9
Cost	585

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

Alternative 4
Error	24.7
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1550:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 5
Error	0.0
Cost	448

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

Alternative 6
Error	35.2
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023060 
(FPCore (x y z)
  :name "Main:bigenough2 from A"
  :precision binary64
  (+ x (* y (+ z x))))