Expression 4, p15

Percentage Accurate: 100.0% → 100.0%

Time: 7.2s

Alternatives: 4

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

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

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 (+ a b))))

C

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

Julia

a, b = sort([a, b])
function code(a, b)
	return fma(Float64(a + b), b, Float64(a * Float64(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] * b + N[(a * N[(a + b), $MachinePrecision]), $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. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

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

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

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

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

   7. +-lowering-+.f64 100.0%

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

4. Applied egg-rr 100.0%

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

Alternative 2: 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 3: 98.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(b * N[(a + b), $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 5.2%

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

6. Final simplification 5.2%

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

Alternative 4: 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 a around 0

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

   1. +-rgt-identity N/A

      \[\leadsto {b}^{2} + \color{blue}{0} \]

   2. unpow2 N/A

      \[\leadsto b \cdot b + 0 \]

   3. accelerator-lowering-fma.f64 4.1%

      \[\leadsto \mathsf{fma.f64}\left(b, \color{blue}{b}, 0\right) \]

5. Simplified 4.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(b, b, 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. *-lowering-*.f64 4.1%

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

7. Applied egg-rr 4.1%

   \[\leadsto \color{blue}{b \cdot b} \]
8. 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 2024193 
(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)))