Expression 4, p15

Percentage Accurate: 100.0% → 100.0%

Time: 5.2s

Alternatives: 5

Speedup: 1.0×

Specification

\[\left(5 \leq a \land a \leq 10\right) \land \left(0 \leq b \land b \leq 0.001\right)\]

Math

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

FPCore

(FPCore (a b) :precision binary64 (* (+ a b) (+ a b)))

C

double code(double a, double b) {
	return (a + b) * (a + b);
}

Fortran

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

Java

public static double code(double a, double b) {
	return (a + b) * (a + b);
}

Python

def code(a, b):
	return (a + b) * (a + b)

Julia

function code(a, b)
	return Float64(Float64(a + b) * Float64(a + b))
end

MATLAB

function tmp = code(a, b)
	tmp = (a + b) * (a + b);
end

Wolfram

code[a_, b_] := N[(N[(a + b), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 (* (+ a b) (+ a b)))

C

double code(double a, double b) {
	return (a + b) * (a + b);
}

Fortran

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

Java

public static double code(double a, double b) {
	return (a + b) * (a + b);
}

Python

def code(a, b):
	return (a + b) * (a + b)

Julia

function code(a, b)
	return Float64(Float64(a + b) * Float64(a + b))
end

MATLAB

function tmp = code(a, b)
	tmp = (a + b) * (a + b);
end

Wolfram

code[a_, b_] := N[(N[(a + b), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ a \cdot \left(b + \left(b + a\right)\right) + b \cdot b \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (+ (* a (+ b (+ b a))) (* b b)))

C

assert(a < b);
double code(double a, double b) {
	return (a * (b + (b + a))) + (b * b);
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (a * (b + (b + a))) + (b * b)
end function

Java

assert a < b;
public static double code(double a, double b) {
	return (a * (b + (b + a))) + (b * b);
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return (a * (b + (b + a))) + (b * b)

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(Float64(a * Float64(b + Float64(b + a))) + Float64(b * b))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = (a * (b + (b + a))) + (b * b);
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(N[(a * N[(b + N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(a + b\right) \cdot \left(a + b\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. distribute-rgt-in N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

   9. *-lowering-*.f64 100.0%

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

4. Applied egg-rr 100.0%

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

   1. distribute-lft-out N/A

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

   2. *-commutative N/A

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

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

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

   4. +-commutative N/A

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

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

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

   6. +-lowering-+.f64 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \left(b + a\right) \cdot \left(b + a\right) \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (* (+ b a) (+ b a)))

C

assert(a < b);
double code(double a, double b) {
	return (b + a) * (b + a);
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (b + a) * (b + a)
end function

Java

assert a < b;
public static double code(double a, double b) {
	return (b + a) * (b + a);
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return (b + a) * (b + a)

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(Float64(b + a) * Float64(b + a))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = (b + a) * (b + a);
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(N[(b + a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(a + b\right) \cdot \left(a + b\right) \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto \left(b + a\right) \cdot \left(b + a\right) \]
4. Add Preprocessing

Alternative 3: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ b \cdot \left(b + a \cdot 2\right) \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (* b (+ b (* a 2.0))))

C

assert(a < b);
double code(double a, double b) {
	return b * (b + (a * 2.0));
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = b * (b + (a * 2.0d0))
end function

Java

assert a < b;
public static double code(double a, double b) {
	return b * (b + (a * 2.0));
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return b * (b + (a * 2.0))

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(b * Float64(b + Float64(a * 2.0)))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = b * (b + (a * 2.0));
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(b * N[(b + N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(a + b\right) \cdot \left(a + b\right) \]
2. Add Preprocessing

3. Taylor expanded in a around 0

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

   1. fma-define N/A

      \[\leadsto \mathsf{fma}\left(2, \color{blue}{a \cdot b}, {b}^{2}\right) \]

   2. *-rgt-identity N/A

      \[\leadsto \mathsf{fma}\left(2, a \cdot \left(b \cdot \color{blue}{1}\right), {b}^{2}\right) \]

   3. *-inverses N/A

      \[\leadsto \mathsf{fma}\left(2, a \cdot \left(b \cdot \frac{b}{\color{blue}{b}}\right), {b}^{2}\right) \]

   4. associate-/l* N/A

      \[\leadsto \mathsf{fma}\left(2, a \cdot \frac{b \cdot b}{\color{blue}{b}}, {b}^{2}\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(2, a \cdot \frac{{b}^{2}}{b}, {b}^{2}\right) \]

   6. associate-/l* N/A

      \[\leadsto \mathsf{fma}\left(2, \frac{a \cdot {b}^{2}}{\color{blue}{b}}, {b}^{2}\right) \]

   7. associate-*l/ N/A

      \[\leadsto \mathsf{fma}\left(2, \frac{a}{b} \cdot \color{blue}{{b}^{2}}, {b}^{2}\right) \]

   8. fma-define N/A

      \[\leadsto 2 \cdot \left(\frac{a}{b} \cdot {b}^{2}\right) + \color{blue}{{b}^{2}} \]

   9. associate-*l* N/A

      \[\leadsto \left(2 \cdot \frac{a}{b}\right) \cdot {b}^{2} + {\color{blue}{b}}^{2} \]

   10. *-lft-identity N/A

       \[\leadsto \left(2 \cdot \frac{a}{b}\right) \cdot {b}^{2} + 1 \cdot \color{blue}{{b}^{2}} \]

   11. distribute-rgt-in N/A

       \[\leadsto {b}^{2} \cdot \color{blue}{\left(2 \cdot \frac{a}{b} + 1\right)} \]

   12. +-commutative N/A

       \[\leadsto {b}^{2} \cdot \left(1 + \color{blue}{2 \cdot \frac{a}{b}}\right) \]

   13. unpow2 N/A

       \[\leadsto \left(b \cdot b\right) \cdot \left(\color{blue}{1} + 2 \cdot \frac{a}{b}\right) \]

   14. associate-*l* N/A

       \[\leadsto b \cdot \color{blue}{\left(b \cdot \left(1 + 2 \cdot \frac{a}{b}\right)\right)} \]

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

       \[\leadsto \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot \left(1 + 2 \cdot \frac{a}{b}\right)\right)}\right) \]

   16. distribute-lft-in N/A

       \[\leadsto \mathsf{*.f64}\left(b, \left(b \cdot 1 + \color{blue}{b \cdot \left(2 \cdot \frac{a}{b}\right)}\right)\right) \]

   17. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f64}\left(b, \left(b + \color{blue}{b} \cdot \left(2 \cdot \frac{a}{b}\right)\right)\right) \]

   18. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(b, \left(b + \left(2 \cdot \frac{a}{b}\right) \cdot \color{blue}{b}\right)\right) \]

   19. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(b, \left(b + \frac{2 \cdot a}{b} \cdot b\right)\right) \]

   20. associate-*l/ N/A

       \[\leadsto \mathsf{*.f64}\left(b, \left(b + \frac{\left(2 \cdot a\right) \cdot b}{\color{blue}{b}}\right)\right) \]

   21. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(b, \left(b + \left(2 \cdot a\right) \cdot \color{blue}{\frac{b}{b}}\right)\right) \]

   22. *-inverses N/A

       \[\leadsto \mathsf{*.f64}\left(b, \left(b + \left(2 \cdot a\right) \cdot 1\right)\right) \]

   23. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f64}\left(b, \left(b + 2 \cdot \color{blue}{a}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(b, \color{blue}{\left(2 \cdot a\right)}\right)\right) \]

   25. *-lowering-*.f64 5.1%

       \[\leadsto \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(b, \mathsf{*.f64}\left(2, \color{blue}{a}\right)\right)\right) \]

5. Simplified 5.1%

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

6. Final simplification 5.1%

   \[\leadsto b \cdot \left(b + a \cdot 2\right) \]
7. Add Preprocessing

Alternative 4: 98.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ b \cdot \left(b + a\right) \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (* b (+ b a)))

C

assert(a < b);
double code(double a, double b) {
	return b * (b + a);
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = b * (b + a)
end function

Java

assert a < b;
public static double code(double a, double b) {
	return b * (b + a);
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return b * (b + a)

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(b * Float64(b + a))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = b * (b + a);
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(b * N[(b + a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(a + b\right) \cdot \left(a + b\right) \]
2. Add Preprocessing

3. Taylor expanded in a around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, b\right), \color{blue}{b}\right) \]
4. Step-by-step derivation

5. Simplified 5.1%

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

6. Final simplification 5.1%

   \[\leadsto b \cdot \left(b + a\right) \]
7. Add Preprocessing

Alternative 5: 98.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ b \cdot b \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (* b b))

C

assert(a < b);
double code(double a, double b) {
	return b * b;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = b * b
end function

Java

assert a < b;
public static double code(double a, double b) {
	return b * b;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return b * b

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(b * b)
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = b * b;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(b * b), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(a + b\right) \cdot \left(a + b\right) \]
2. Add Preprocessing

3. Taylor expanded in a around 0

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 4.0%

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

5. Simplified 4.0%

   \[\leadsto \color{blue}{b \cdot b} \]
6. Add Preprocessing

Developer Target 1: 100.0% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 (+ (+ (+ (* b a) (* b b)) (* b a)) (* a a)))

C

double code(double a, double b) {
	return (((b * a) + (b * b)) + (b * a)) + (a * a);
}

Fortran

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

Java

public static double code(double a, double b) {
	return (((b * a) + (b * b)) + (b * a)) + (a * a);
}

Python

def code(a, b):
	return (((b * a) + (b * b)) + (b * a)) + (a * a)

Julia

function code(a, b)
	return Float64(Float64(Float64(Float64(b * a) + Float64(b * b)) + Float64(b * a)) + Float64(a * a))
end

MATLAB

function tmp = code(a, b)
	tmp = (((b * a) + (b * b)) + (b * a)) + (a * a);
end

Wolfram

code[a_, b_] := N[(N[(N[(N[(b * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(b * a), $MachinePrecision]), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024149 
(FPCore (a b)
  :name "Expression 4, p15"
  :precision binary64
  :pre (and (and (<= 5.0 a) (<= a 10.0)) (and (<= 0.0 b) (<= b 0.001)))

  :alt
  (! :herbie-platform default (+ (+ (+ (* b a) (* b b)) (* b a)) (* a a)))

  (* (+ a b) (+ a b)))