Expanding a square

Percentage Accurate: 54.6% → 100.0%

Time: 2.1s

Alternatives: 3

Speedup: 1.8×

• 25.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

double code(double x) {
	return fma((x + 1.0), x, x);
}

Julia

function code(x)
	return fma(Float64(x + 1.0), x, x)
end

Wolfram

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

Derivation

1. Initial program 55.9%

   \[\left(x + 1\right) \cdot \left(x + 1\right) - 1 \]
2. Step-by-step derivation

   1. difference-of-sqr-1 55.9%

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

   2. *-commutative 55.9%

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

   3. associate--l+ 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \left(x + \color{blue}{0}\right) \cdot \left(\left(x + 1\right) + 1\right) \]

   5. +-rgt-identity 100.0%

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

   6. associate-+l+ 100.0%

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

   7. distribute-rgt-in 100.0%

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

   8. sqr-neg 100.0%

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

   9. distribute-rgt-neg-out 100.0%

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

   10. distribute-lft-neg-in 100.0%

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

   11. mul-1-neg 100.0%

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

   12. metadata-eval 100.0%

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

   13. distribute-rgt-out 100.0%

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

   14. metadata-eval 100.0%

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

   15. mul-1-neg 100.0%

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

   16. metadata-eval 100.0%

       \[\leadsto x \cdot \left(\left(-\left(-x\right)\right) + \color{blue}{2}\right) \]

   17. metadata-eval 100.0%

       \[\leadsto x \cdot \left(\left(-\left(-x\right)\right) + \color{blue}{\left(--2\right)}\right) \]

   18. metadata-eval 100.0%

       \[\leadsto x \cdot \left(\left(-\left(-x\right)\right) + \left(-\color{blue}{\left(-2\right)}\right)\right) \]

   19. metadata-eval 100.0%

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

   20. distribute-neg-in 100.0%

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

   21. distribute-neg-in 100.0%

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

   22. associate-+l+ 100.0%

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

   23. mul-1-neg 100.0%

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

   24. metadata-eval 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{x \cdot \left(x + 2\right)} \]
4. Step-by-step derivation

   1. *-commutative 100.0%

      \[\leadsto \color{blue}{\left(x + 2\right) \cdot x} \]

   2. metadata-eval 100.0%

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

   3. associate-+l+ 100.0%

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

   4. expm1-log1p-u 71.7%

      \[\leadsto \left(\left(x + 1\right) + 1\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)} \]

   5. expm1-udef 27.6%

      \[\leadsto \left(\left(x + 1\right) + 1\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(x\right)} - 1\right)} \]

   6. log1p-udef 27.6%

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

   7. add-exp-log 55.9%

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

   8. +-commutative 55.9%

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

   9. difference-of-sqr-1 55.9%

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

   10. distribute-lft1-in 55.9%

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

   11. associate--l+ 62.1%

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

   12. add-exp-log 34.7%

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

   13. +-commutative 34.7%

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

   14. log1p-udef 34.7%

       \[\leadsto x \cdot \left(x + 1\right) + \left(e^{\color{blue}{\mathsf{log1p}\left(x\right)}} - 1\right) \]

   15. expm1-udef 72.7%

       \[\leadsto x \cdot \left(x + 1\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)} \]

   16. expm1-log1p-u 100.0%

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

   17. *-commutative 100.0%

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

   18. fma-def 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(x + 1, x, x\right) \]

Alternative 2: 100.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x + 2\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (+ x 2.0)))

C

double code(double x) {
	return x * (x + 2.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x + 2.0d0)
end function

Java

public static double code(double x) {
	return x * (x + 2.0);
}

Python

def code(x):
	return x * (x + 2.0)

Julia

function code(x)
	return Float64(x * Float64(x + 2.0))
end

MATLAB

function tmp = code(x)
	tmp = x * (x + 2.0);
end

Wolfram

code[x_] := N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.9%

   \[\left(x + 1\right) \cdot \left(x + 1\right) - 1 \]
2. Step-by-step derivation

   1. difference-of-sqr-1 55.9%

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

   2. *-commutative 55.9%

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

   3. associate--l+ 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \left(x + \color{blue}{0}\right) \cdot \left(\left(x + 1\right) + 1\right) \]

   5. +-rgt-identity 100.0%

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

   6. associate-+l+ 100.0%

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

   7. distribute-rgt-in 100.0%

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

   8. sqr-neg 100.0%

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

   9. distribute-rgt-neg-out 100.0%

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

   10. distribute-lft-neg-in 100.0%

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

   11. mul-1-neg 100.0%

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

   12. metadata-eval 100.0%

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

   13. distribute-rgt-out 100.0%

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

   14. metadata-eval 100.0%

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

   15. mul-1-neg 100.0%

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

   16. metadata-eval 100.0%

       \[\leadsto x \cdot \left(\left(-\left(-x\right)\right) + \color{blue}{2}\right) \]

   17. metadata-eval 100.0%

       \[\leadsto x \cdot \left(\left(-\left(-x\right)\right) + \color{blue}{\left(--2\right)}\right) \]

   18. metadata-eval 100.0%

       \[\leadsto x \cdot \left(\left(-\left(-x\right)\right) + \left(-\color{blue}{\left(-2\right)}\right)\right) \]

   19. metadata-eval 100.0%

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

   20. distribute-neg-in 100.0%

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

   21. distribute-neg-in 100.0%

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

   22. associate-+l+ 100.0%

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

   23. mul-1-neg 100.0%

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

   24. metadata-eval 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{x \cdot \left(x + 2\right)} \]

4. Final simplification 100.0%

   \[\leadsto x \cdot \left(x + 2\right) \]

Alternative 3: 50.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ x \cdot 2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x 2.0))

C

double code(double x) {
	return x * 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * 2.0d0
end function

Java

public static double code(double x) {
	return x * 2.0;
}

Python

def code(x):
	return x * 2.0

Julia

function code(x)
	return Float64(x * 2.0)
end

MATLAB

function tmp = code(x)
	tmp = x * 2.0;
end

Wolfram

code[x_] := N[(x * 2.0), $MachinePrecision]

Derivation

1. Initial program 55.9%

   \[\left(x + 1\right) \cdot \left(x + 1\right) - 1 \]
2. Step-by-step derivation

   1. difference-of-sqr-1 55.9%

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

   2. *-commutative 55.9%

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

   3. associate--l+ 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \left(x + \color{blue}{0}\right) \cdot \left(\left(x + 1\right) + 1\right) \]

   5. +-rgt-identity 100.0%

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

   6. associate-+l+ 100.0%

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

   7. distribute-rgt-in 100.0%

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

   8. sqr-neg 100.0%

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

   9. distribute-rgt-neg-out 100.0%

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

   10. distribute-lft-neg-in 100.0%

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

   11. mul-1-neg 100.0%

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

   12. metadata-eval 100.0%

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

   13. distribute-rgt-out 100.0%

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

   14. metadata-eval 100.0%

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

   15. mul-1-neg 100.0%

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

   16. metadata-eval 100.0%

       \[\leadsto x \cdot \left(\left(-\left(-x\right)\right) + \color{blue}{2}\right) \]

   17. metadata-eval 100.0%

       \[\leadsto x \cdot \left(\left(-\left(-x\right)\right) + \color{blue}{\left(--2\right)}\right) \]

   18. metadata-eval 100.0%

       \[\leadsto x \cdot \left(\left(-\left(-x\right)\right) + \left(-\color{blue}{\left(-2\right)}\right)\right) \]

   19. metadata-eval 100.0%

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

   20. distribute-neg-in 100.0%

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

   21. distribute-neg-in 100.0%

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

   22. associate-+l+ 100.0%

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

   23. mul-1-neg 100.0%

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

   24. metadata-eval 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{x \cdot \left(x + 2\right)} \]

4. Taylor expanded in x around 0 49.1%

   \[\leadsto \color{blue}{2 \cdot x} \]

5. Final simplification 49.1%

   \[\leadsto x \cdot 2 \]

Reproduce

herbie shell --seed 2023333 
(FPCore (x)
  :name "Expanding a square"
  :precision binary64
  (- (* (+ x 1.0) (+ x 1.0)) 1.0))