Expression 4, p15

Percentage Accurate: 100.0% → 100.0%

Time: 4.1s

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, 0.1× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 (+ (* b (+ b (* 2.0 a))) (pow a 2.0)))

C

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

Fortran

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

Java

public static double code(double a, double b) {
	return (b * (b + (2.0 * a))) + Math.pow(a, 2.0);
}

Python

def code(a, b):
	return (b * (b + (2.0 * a))) + math.pow(a, 2.0)

Julia

function code(a, b)
	return Float64(Float64(b * Float64(b + Float64(2.0 * a))) + (a ^ 2.0))
end

MATLAB

function tmp = code(a, b)
	tmp = (b * (b + (2.0 * a))) + (a ^ 2.0);
end

Wolfram

code[a_, b_] := N[(N[(b * N[(b + N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in a around 0 100.0%

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

   1. +-commutative 100.0%

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

   2. associate-+r+ 100.0%

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

   3. +-commutative 100.0%

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

   4. unpow2 100.0%

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

   5. associate-*r* 100.0%

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

   6. distribute-rgt-out 100.0%

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

5. Simplified 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. +-commutative 99.9%

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

   2. distribute-rgt-in 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 3: 5.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 (* b (+ b (* 2.0 a))))

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return b * (b + (2.0 * a))

Julia

function code(a, b)
	return Float64(b * Float64(b + Float64(2.0 * a)))
end

MATLAB

function tmp = code(a, b)
	tmp = b * (b + (2.0 * a));
end

Wolfram

code[a_, b_] := N[(b * N[(b + N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in a around 0 5.4%

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

   1. +-commutative 5.4%

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

   2. unpow2 5.4%

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

   3. associate-*r* 5.4%

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

   4. distribute-rgt-out 5.4%

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

5. Simplified 5.4%

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

6. Final simplification 5.4%

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

Alternative 4: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 5: 5.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ b \cdot \left(2 \cdot a\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (* b (* 2.0 a)))

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return b * (2.0 * a)

Julia

function code(a, b)
	return Float64(b * Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b)
	tmp = b * (2.0 * a);
end

Wolfram

code[a_, b_] := N[(b * N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in a around 0 5.4%

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

   1. +-commutative 5.4%

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

   2. unpow2 5.4%

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

   3. associate-*r* 5.4%

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

   4. distribute-rgt-out 5.4%

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

5. Simplified 5.4%

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

6. Taylor expanded in b around 0 5.4%

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

   1. associate-*r* 5.4%

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

   2. *-commutative 5.4%

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

8. Simplified 5.4%

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

9. Final simplification 5.4%

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

Developer target: 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 2024040 
(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)))

  :herbie-target
  (+ (+ (+ (* b a) (* b b)) (* b a)) (* a a))

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