Expression 3, p15

Average Error: 0.0 → 0.0

Time: 9.6s

Precision: binary64

Cost: 448

\[0 \leq x \land x \leq 2\]

Math

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

↓

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

FPCore

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

↓

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

C

double code(double x) {
	return (x * (x * x)) + (x * x);
}

↓

double code(double x) {
	return ((x * x) + x) * x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * (x * x)) + (x * x)
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x * x) + x) * x
end function

Java

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

↓

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

Python

def code(x):
	return (x * (x * x)) + (x * x)

↓

def code(x):
	return ((x * x) + x) * x

Julia

function code(x)
	return Float64(Float64(x * Float64(x * x)) + Float64(x * x))
end

↓

function code(x)
	return Float64(Float64(Float64(x * x) + x) * x)
end

MATLAB

function tmp = code(x)
	tmp = (x * (x * x)) + (x * x);
end

↓

function tmp = code(x)
	tmp = ((x * x) + x) * x;
end

Wolfram

code[x_] := N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(N[(N[(x * x), $MachinePrecision] + x), $MachinePrecision] * x), $MachinePrecision]

Target

Original	0.0
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 0.0

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

2. Simplified 0.0

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, x, x\right)} \]
   Proof

3. Applied egg-rr 0.0

   \[\leadsto \color{blue}{\frac{\left({x}^{3} + 1\right) \cdot \left(x \cdot x\right)}{x \cdot x + \left(1 - x\right)}} \]

4. Taylor expanded in x around 0 0.0

   \[\leadsto \color{blue}{{x}^{2} + {x}^{3}} \]

5. Simplified 0.0

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

6. Applied egg-rr 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	448

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

Alternative 2
Error	1.5
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023010 
(FPCore (x)
  :name "Expression 3, p15"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 2.0))

  :herbie-target
  (* (* (+ 1.0 x) x) x)

  (+ (* x (* x x)) (* x x)))