Expression 4, p15

Percentage Accurate: 100.0% → 100.0%

Time: 2.0s

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b)
use fmin_fmax_functions
    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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b)
use fmin_fmax_functions
    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

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

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. 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. distribute-rgt-in N/A

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

   4. add-flip N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. distribute-lft-in N/A

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

   8. associate--l+ N/A

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

   9. *-commutative N/A

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

   10. add-flip N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. distribute-rgt-in N/A

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

   14. lower-fma.f64 N/A

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

   15. distribute-rgt-in N/A

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

   16. +-commutative N/A

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. *-commutative N/A

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

   20. lower-fma.f64 N/A

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

   21. lift-+.f64 N/A

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

   22. +-commutative N/A

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

   23. lower-+.f64 N/A

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

   24. lower-*.f64 100.0

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

3. Applied rewrites 100.0%

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

   1. lift-fma.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-+r+ N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. distribute-rgt-out N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-+.f64 100.0

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 100.0

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

5. Applied rewrites 100.0%

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 100.0

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

   4. lift-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. associate-+r+ N/A

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

   7. count-2 N/A

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

   8. lower-fma.f64 100.0

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

7. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(2, a, b\right), b, a \cdot a\right)} \]
8. 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.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b)
use fmin_fmax_functions
    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

Alternative 3: 98.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 100.0%

   \[\left(a + b\right) \cdot \left(a + b\right) \]
2. 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. distribute-rgt-in N/A

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

   4. add-flip N/A

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

   5. sub-flip N/A

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

   6. *-commutative N/A

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

   7. distribute-rgt-neg-in N/A

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

   8. distribute-rgt-neg-out N/A

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

   9. remove-double-neg N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 100.0

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 100.0

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in a around 0

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in a around 0

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

9. Applied rewrites 98.7%

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

    1. lift-fma.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. distribute-lft-out N/A

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

    4. lift-+.f64 N/A

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

    5. *-commutative N/A

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

    6. lower-*.f64 98.7

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

11. Applied rewrites 98.7%

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

Alternative 4: 98.7% accurate, 2.4× 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.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b)
use fmin_fmax_functions
    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. Taylor expanded in a around 0

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in a around 0

   \[\leadsto b \cdot \color{blue}{b} \]
6. Step-by-step derivation

7. Applied rewrites 98.7%

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b)
use fmin_fmax_functions
    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 2025150 
(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 c (+ (+ (+ (* b a) (* b b)) (* b a)) (* a a)))

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