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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(N[(a + b), $MachinePrecision] * b + N[(N[(a + b), $MachinePrecision] * a), $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. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 100.0

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(a + b, b, \left(a + b\right) \cdot a\right) \]
6. Add Preprocessing

Alternative 2: 100.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(N[(2.0 * a + b), $MachinePrecision] * b + N[(a * 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 b around 0

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

5. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(2, a, b\right), b, a \cdot a\right)} \]
6. Add Preprocessing

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 4: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{{b}^{2} \cdot \left(1 + 2 \cdot \frac{a}{b}\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. distribute-lft-in N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. associate-*r/ N/A

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

   10. associate-*l/ N/A

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

   11. *-inverses N/A

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

   12. *-lft-identity N/A

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

   13. *-commutative N/A

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

   14. *-rgt-identity N/A

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

   15. lower-fma.f64 5.2

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

5. Applied rewrites 5.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, a, b\right) \cdot b} \]
6. Add Preprocessing

Alternative 5: 98.7% accurate, 2.0× 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 b around inf

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

   1. unpow2 N/A

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

   2. lower-*.f64 4.1

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

5. Applied rewrites 4.1%

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

Developer Target 1: 100.0% accurate, 0.4× 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 2024242 
(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)))