Expression 3, p15

Percentage Accurate: 100.0% → 100.0%

Time: 6.5s

Alternatives: 4

Speedup: N/A×

Specification

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

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot x\right) + x \cdot x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot x\right) + x \cdot x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x + x \cdot x\right) \end{array} \]

FPCore

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

C

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))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[x \cdot \left(x \cdot x\right) + x \cdot x \]
2. Step-by-step derivation

   1. distribute-lft-out N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot x + x\right)}\right) \]

   3. distribute-lft1-in N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\left(x + 1\right) \cdot \color{blue}{x}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(x + 1\right)}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(x + 1\right)}\right)\right) \]

   6. +-lowering-+.f64 99.9%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(x + 1\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\left(x + 1\right) \cdot \color{blue}{x}\right)\right) \]

   2. distribute-lft1-in N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot x + \color{blue}{x}\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(x \cdot x\right), \color{blue}{x}\right)\right) \]

   4. *-lowering-*.f64 99.9%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), x\right)\right) \]

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

   \[\leadsto x \cdot \left(x + x \cdot x\right) \]
8. Add Preprocessing

Alternative 2: 100.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot \left(x + 1\right)\right) \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x * (x + 1.0d0))
end function

Java

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

Python

def code(x):
	return x * (x * (x + 1.0))

Julia

function code(x)
	return Float64(x * Float64(x * Float64(x + 1.0)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[x \cdot \left(x \cdot x\right) + x \cdot x \]
2. Step-by-step derivation

   1. distribute-lft-out N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot x + x\right)}\right) \]

   3. distribute-lft1-in N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\left(x + 1\right) \cdot \color{blue}{x}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(x + 1\right)}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(x + 1\right)}\right)\right) \]

   6. +-lowering-+.f64 99.9%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(x + 1\right)\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 3: 97.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ x \cdot x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return x * x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[x \cdot \left(x \cdot x\right) + x \cdot x \]
2. Step-by-step derivation

   1. distribute-lft-out N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot x + x\right)}\right) \]

   3. distribute-lft1-in N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\left(x + 1\right) \cdot \color{blue}{x}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(x + 1\right)}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(x + 1\right)}\right)\right) \]

   6. +-lowering-+.f64 99.9%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(x + 1\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2}} \]
6. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto x \cdot \color{blue}{x} \]

   2. *-lowering-*.f64 96.7%

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

7. Simplified 96.7%

   \[\leadsto \color{blue}{x \cdot x} \]
8. Add Preprocessing

Alternative 4: 4.2% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 99.9%

   \[x \cdot \left(x \cdot x\right) + x \cdot x \]
2. Step-by-step derivation

   1. distribute-lft-out N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot x + x\right)}\right) \]

   3. distribute-lft1-in N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\left(x + 1\right) \cdot \color{blue}{x}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(x + 1\right)}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(x + 1\right)}\right)\right) \]

   6. +-lowering-+.f64 99.9%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(x + 1\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\left(x + 1\right) \cdot \color{blue}{x}\right)\right) \]

   2. distribute-lft1-in N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot x + \color{blue}{x}\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(x \cdot x\right), \color{blue}{x}\right)\right) \]

   4. *-lowering-*.f64 99.9%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), x\right)\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto x \cdot \color{blue}{\left(x \cdot x + x\right)} \]
7. Step-by-step derivation

   1. distribute-lft1-in N/A

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

   2. associate-*r* N/A

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

   3. +-commutative N/A

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

   4. flip-+ N/A

      \[\leadsto \left(x \cdot \frac{1 \cdot 1 - x \cdot x}{1 - x}\right) \cdot x \]

   5. metadata-eval N/A

      \[\leadsto \left(x \cdot \frac{1 - x \cdot x}{1 - x}\right) \cdot x \]

   6. associate-/l* N/A

      \[\leadsto \frac{x \cdot \left(1 - x \cdot x\right)}{1 - x} \cdot x \]

   7. *-commutative N/A

      \[\leadsto \frac{\left(1 - x \cdot x\right) \cdot x}{1 - x} \cdot x \]

   8. *-commutative N/A

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

   9. clear-num N/A

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

   10. un-div-inv N/A

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

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{1 - x}{\left(1 - x \cdot x\right) \cdot x}\right)}\right) \]

   12. clear-num N/A

       \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{1}{\color{blue}{\frac{\left(1 - x \cdot x\right) \cdot x}{1 - x}}}\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{1}{\frac{x \cdot \left(1 - x \cdot x\right)}{\color{blue}{1} - x}}\right)\right) \]

   14. associate-/l* N/A

       \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{1}{x \cdot \color{blue}{\frac{1 - x \cdot x}{1 - x}}}\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{1}{x \cdot \frac{1 \cdot 1 - x \cdot x}{1 - x}}\right)\right) \]

   16. flip-+ N/A

       \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{1}{x \cdot \left(1 + \color{blue}{x}\right)}\right)\right) \]

   17. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{1}{x \cdot \left(x + \color{blue}{1}\right)}\right)\right) \]

   18. distribute-rgt-out N/A

       \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{1}{x \cdot x + \color{blue}{1 \cdot x}}\right)\right) \]

   19. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot x + 1 \cdot x\right)}\right)\right) \]

8. Applied egg-rr 99.7%

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

9. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{1 + x \cdot \left(x - 1\right)}{x}\right)}\right) \]
10. Step-by-step derivation

    1. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\left(1 + x \cdot \left(x - 1\right)\right), \color{blue}{x}\right)\right) \]

    2. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x - 1\right)\right)\right), x\right)\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x - 1\right)\right)\right), x\right)\right) \]

    4. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), x\right)\right) \]

    5. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x + -1\right)\right)\right), x\right)\right) \]

    6. +-lowering-+.f64 98.9%

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, -1\right)\right)\right), x\right)\right) \]

11. Simplified 98.9%

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

12. Taylor expanded in x around inf

    \[\leadsto \color{blue}{1} \]
13. Step-by-step derivation

14. Simplified 4.4%

    \[\leadsto \color{blue}{1} \]
15. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \left(\left(1 + x\right) \cdot x\right) \cdot x \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((1.0d0 + x) * x) * x
end function

Java

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

Python

def code(x):
	return ((1.0 + x) * x) * x

Julia

function code(x)
	return Float64(Float64(Float64(1.0 + x) * x) * x)
end

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024160 
(FPCore (x)
  :name "Expression 3, p15"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 2.0))

  :alt
  (! :herbie-platform default (* (* (+ 1 x) x) x))

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