Linear.V4:$cdot from linear-1.19.1.3, C

Percentage Accurate: 96.0% → 97.9%

Time: 5.8s

Alternatives: 10

Speedup: 1.3×

• 42.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (+ (+ (+ (* x y) (* z t)) (* a b)) (* c i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((x * y) + (z * t)) + (a * b)) + (c * i);
}

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(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = (((x * y) + (z * t)) + (a * b)) + (c * i)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((x * y) + (z * t)) + (a * b)) + (c * i);
}

Python

def code(x, y, z, t, a, b, c, i):
	return (((x * y) + (z * t)) + (a * b)) + (c * i)

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(Float64(Float64(Float64(x * y) + Float64(z * t)) + Float64(a * b)) + Float64(c * i))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = (((x * y) + (z * t)) + (a * b)) + (c * i);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision] + N[(c * i), $MachinePrecision]), $MachinePrecision]

Initial Program: 96.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (+ (+ (+ (* x y) (* z t)) (* a b)) (* c i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((x * y) + (z * t)) + (a * b)) + (c * i);
}

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(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = (((x * y) + (z * t)) + (a * b)) + (c * i)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((x * y) + (z * t)) + (a * b)) + (c * i);
}

Python

def code(x, y, z, t, a, b, c, i):
	return (((x * y) + (z * t)) + (a * b)) + (c * i)

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(Float64(Float64(Float64(x * y) + Float64(z * t)) + Float64(a * b)) + Float64(c * i))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = (((x * y) + (z * t)) + (a * b)) + (c * i);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision] + N[(c * i), $MachinePrecision]), $MachinePrecision]

Alternative 1: 97.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(z, t, \mathsf{fma}\left(y, x, \mathsf{fma}\left(i, c, b \cdot a\right)\right)\right) \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (fma z t (fma y x (fma i c (* b a)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return fma(z, t, fma(y, x, fma(i, c, (b * a))));
}

Julia

function code(x, y, z, t, a, b, c, i)
	return fma(z, t, fma(y, x, fma(i, c, Float64(b * a))))
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(z * t + N[(y * x + N[(i * c + N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.5%

   \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + \color{blue}{c \cdot i} \]

   3. fp-cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) - \left(\mathsf{neg}\left(c\right)\right) \cdot i} \]

   4. fp-cancel-sub-sign-inv N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(c\right)\right)\right)\right) \cdot i} \]

   5. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(c\right)\right)\right)\right) \cdot i \]

   6. lift-+.f64 N/A

      \[\leadsto \left(\color{blue}{\left(x \cdot y + z \cdot t\right)} + a \cdot b\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(c\right)\right)\right)\right) \cdot i \]

   7. +-commutative N/A

      \[\leadsto \left(\color{blue}{\left(z \cdot t + x \cdot y\right)} + a \cdot b\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(c\right)\right)\right)\right) \cdot i \]

   8. associate-+l+ N/A

      \[\leadsto \color{blue}{\left(z \cdot t + \left(x \cdot y + a \cdot b\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(c\right)\right)\right)\right) \cdot i \]

   9. associate-+l+ N/A

      \[\leadsto \color{blue}{z \cdot t + \left(\left(x \cdot y + a \cdot b\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(c\right)\right)\right)\right) \cdot i\right)} \]

   10. lift-*.f64 N/A

       \[\leadsto \color{blue}{z \cdot t} + \left(\left(x \cdot y + a \cdot b\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(c\right)\right)\right)\right) \cdot i\right) \]

   11. remove-double-neg N/A

       \[\leadsto z \cdot t + \left(\left(x \cdot y + a \cdot b\right) + \color{blue}{c} \cdot i\right) \]

   12. lift-*.f64 N/A

       \[\leadsto z \cdot t + \left(\left(x \cdot y + a \cdot b\right) + \color{blue}{c \cdot i}\right) \]

   13. associate-+r+ N/A

       \[\leadsto z \cdot t + \color{blue}{\left(x \cdot y + \left(a \cdot b + c \cdot i\right)\right)} \]

   14. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(z, t, x \cdot y + \left(a \cdot b + c \cdot i\right)\right)} \]

   15. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(z, t, \color{blue}{x \cdot y} + \left(a \cdot b + c \cdot i\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(z, t, \color{blue}{y \cdot x} + \left(a \cdot b + c \cdot i\right)\right) \]

   17. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(z, t, \color{blue}{\mathsf{fma}\left(y, x, a \cdot b + c \cdot i\right)}\right) \]

   18. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(z, t, \mathsf{fma}\left(y, x, \color{blue}{c \cdot i + a \cdot b}\right)\right) \]

   19. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(z, t, \mathsf{fma}\left(y, x, \color{blue}{c \cdot i} + a \cdot b\right)\right) \]

   20. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(z, t, \mathsf{fma}\left(y, x, \color{blue}{i \cdot c} + a \cdot b\right)\right) \]

4. Applied rewrites 98.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, t, \mathsf{fma}\left(y, x, \mathsf{fma}\left(i, c, b \cdot a\right)\right)\right)} \]
5. Add Preprocessing

Alternative 2: 43.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -2 \cdot 10^{+172}:\\ \;\;\;\;i \cdot c\\ \mathbf{elif}\;c \cdot i \leq 2 \cdot 10^{-199}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;c \cdot i \leq 100000000000:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;c \cdot i \leq 2 \cdot 10^{+49}:\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;i \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -2e+172)
   (* i c)
   (if (<= (* c i) 2e-199)
     (* t z)
     (if (<= (* c i) 100000000000.0)
       (* b a)
       (if (<= (* c i) 2e+49) (* t z) (* i c))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((c * i) <= -2e+172) {
		tmp = i * c;
	} else if ((c * i) <= 2e-199) {
		tmp = t * z;
	} else if ((c * i) <= 100000000000.0) {
		tmp = b * a;
	} else if ((c * i) <= 2e+49) {
		tmp = t * z;
	} else {
		tmp = i * c;
	}
	return tmp;
}

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(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if ((c * i) <= (-2d+172)) then
        tmp = i * c
    else if ((c * i) <= 2d-199) then
        tmp = t * z
    else if ((c * i) <= 100000000000.0d0) then
        tmp = b * a
    else if ((c * i) <= 2d+49) then
        tmp = t * z
    else
        tmp = i * c
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((c * i) <= -2e+172) {
		tmp = i * c;
	} else if ((c * i) <= 2e-199) {
		tmp = t * z;
	} else if ((c * i) <= 100000000000.0) {
		tmp = b * a;
	} else if ((c * i) <= 2e+49) {
		tmp = t * z;
	} else {
		tmp = i * c;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (c * i) <= -2e+172:
		tmp = i * c
	elif (c * i) <= 2e-199:
		tmp = t * z
	elif (c * i) <= 100000000000.0:
		tmp = b * a
	elif (c * i) <= 2e+49:
		tmp = t * z
	else:
		tmp = i * c
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(c * i) <= -2e+172)
		tmp = Float64(i * c);
	elseif (Float64(c * i) <= 2e-199)
		tmp = Float64(t * z);
	elseif (Float64(c * i) <= 100000000000.0)
		tmp = Float64(b * a);
	elseif (Float64(c * i) <= 2e+49)
		tmp = Float64(t * z);
	else
		tmp = Float64(i * c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((c * i) <= -2e+172)
		tmp = i * c;
	elseif ((c * i) <= 2e-199)
		tmp = t * z;
	elseif ((c * i) <= 100000000000.0)
		tmp = b * a;
	elseif ((c * i) <= 2e+49)
		tmp = t * z;
	else
		tmp = i * c;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(c * i), $MachinePrecision], -2e+172], N[(i * c), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 2e-199], N[(t * z), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 100000000000.0], N[(b * a), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 2e+49], N[(t * z), $MachinePrecision], N[(i * c), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c i) < -2.0000000000000002e172 or 1.99999999999999989e49 < (*.f64 c i)

   1. Initial program 93.7%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot i} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{i \cdot c} \]

      2. lower-*.f64 67.3

         \[\leadsto \color{blue}{i \cdot c} \]

   5. Applied rewrites 67.3%

      \[\leadsto \color{blue}{i \cdot c} \]

   if -2.0000000000000002e172 < (*.f64 c i) < 1.99999999999999996e-199 or 1e11 < (*.f64 c i) < 1.99999999999999989e49

   1. Initial program 98.4%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{a \cdot b + \left(c \cdot i + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{c \cdot i - \left(\mathsf{neg}\left(y\right)\right) \cdot x}\right) \]

      5. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \color{blue}{\left(\mathsf{neg}\left(y \cdot x\right)\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \left(\mathsf{neg}\left(\color{blue}{x \cdot y}\right)\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \color{blue}{-1 \cdot \left(x \cdot y\right)}\right) \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{c \cdot i + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{i \cdot c} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(b, a, i \cdot c + \color{blue}{1} \cdot \left(x \cdot y\right)\right) \]

      11. *-lft-identity N/A

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

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{\mathsf{fma}\left(i, c, x \cdot y\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, \color{blue}{y \cdot x}\right)\right) \]

      14. lower-*.f64 66.6

          \[\leadsto \mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, \color{blue}{y \cdot x}\right)\right) \]

   5. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, y \cdot x\right)\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto a \cdot b + \color{blue}{x \cdot y} \]
   7. Step-by-step derivation

   8. Applied rewrites 61.2%

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

   9. Taylor expanded in x around 0

      \[\leadsto a \cdot b \]
   10. Step-by-step derivation

   11. Applied rewrites 24.9%

       \[\leadsto b \cdot a \]

   12. Taylor expanded in z around inf

       \[\leadsto \color{blue}{t \cdot z} \]
   13. Step-by-step derivation

       1. lower-*.f64 36.3

          \[\leadsto \color{blue}{t \cdot z} \]

   14. Applied rewrites 36.3%

       \[\leadsto \color{blue}{t \cdot z} \]

   if 1.99999999999999996e-199 < (*.f64 c i) < 1e11

   1. Initial program 97.2%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{a \cdot b + \left(c \cdot i + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{c \cdot i - \left(\mathsf{neg}\left(y\right)\right) \cdot x}\right) \]

      5. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \color{blue}{\left(\mathsf{neg}\left(y \cdot x\right)\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \left(\mathsf{neg}\left(\color{blue}{x \cdot y}\right)\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \color{blue}{-1 \cdot \left(x \cdot y\right)}\right) \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{c \cdot i + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{i \cdot c} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(b, a, i \cdot c + \color{blue}{1} \cdot \left(x \cdot y\right)\right) \]

      11. *-lft-identity N/A

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

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{\mathsf{fma}\left(i, c, x \cdot y\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, \color{blue}{y \cdot x}\right)\right) \]

      14. lower-*.f64 84.0

          \[\leadsto \mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, \color{blue}{y \cdot x}\right)\right) \]

   5. Applied rewrites 84.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, y \cdot x\right)\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto a \cdot b + \color{blue}{x \cdot y} \]
   7. Step-by-step derivation

   8. Applied rewrites 82.6%

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

   9. Taylor expanded in x around 0

      \[\leadsto a \cdot b \]
   10. Step-by-step derivation

   11. Applied rewrites 44.4%

       \[\leadsto b \cdot a \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 75.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot y + z \cdot t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+147} \lor \neg \left(t\_1 \leq 10^{+98}\right):\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, b \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* x y) (* z t))))
   (if (or (<= t_1 -1e+147) (not (<= t_1 1e+98)))
     (fma t z (* y x))
     (fma i c (* b a)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (x * y) + (z * t);
	double tmp;
	if ((t_1 <= -1e+147) || !(t_1 <= 1e+98)) {
		tmp = fma(t, z, (y * x));
	} else {
		tmp = fma(i, c, (b * a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(x * y) + Float64(z * t))
	tmp = 0.0
	if ((t_1 <= -1e+147) || !(t_1 <= 1e+98))
		tmp = fma(t, z, Float64(y * x));
	else
		tmp = fma(i, c, Float64(b * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e+147], N[Not[LessEqual[t$95$1, 1e+98]], $MachinePrecision]], N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(i * c + N[(b * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x y) (*.f64 z t)) < -9.9999999999999998e146 or 9.99999999999999998e97 < (+.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 93.5%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c \cdot i + \left(t \cdot z + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto c \cdot i + \color{blue}{1 \cdot \left(t \cdot z + x \cdot y\right)} \]

      2. metadata-eval N/A

         \[\leadsto c \cdot i + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \left(t \cdot z + x \cdot y\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{i \cdot c} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(t \cdot z + x \cdot y\right) \]

      4. metadata-eval N/A

         \[\leadsto i \cdot c + \color{blue}{1} \cdot \left(t \cdot z + x \cdot y\right) \]

      5. *-lft-identity N/A

         \[\leadsto i \cdot c + \color{blue}{\left(t \cdot z + x \cdot y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(i, c, t \cdot z + x \cdot y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(i, c, \color{blue}{\mathsf{fma}\left(t, z, x \cdot y\right)}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(i, c, \mathsf{fma}\left(t, z, \color{blue}{y \cdot x}\right)\right) \]

      9. lower-*.f64 90.1

         \[\leadsto \mathsf{fma}\left(i, c, \mathsf{fma}\left(t, z, \color{blue}{y \cdot x}\right)\right) \]

   5. Applied rewrites 90.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, c, \mathsf{fma}\left(t, z, y \cdot x\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(i, c, t \cdot z\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 50.0%

      \[\leadsto \mathsf{fma}\left(i, c, t \cdot z\right) \]

   9. Taylor expanded in z around 0

      \[\leadsto c \cdot i + \color{blue}{x \cdot y} \]
   10. Step-by-step derivation

   11. Applied rewrites 56.8%

       \[\leadsto \mathsf{fma}\left(i, \color{blue}{c}, y \cdot x\right) \]

   12. Taylor expanded in c around 0

       \[\leadsto t \cdot z + \color{blue}{x \cdot y} \]
   13. Step-by-step derivation

   14. Applied rewrites 81.0%

       \[\leadsto \mathsf{fma}\left(t, \color{blue}{z}, y \cdot x\right) \]

   if -9.9999999999999998e146 < (+.f64 (*.f64 x y) (*.f64 z t)) < 9.99999999999999998e97

   1. Initial program 100.0%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{a \cdot b + \left(c \cdot i + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{c \cdot i - \left(\mathsf{neg}\left(y\right)\right) \cdot x}\right) \]

      5. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \color{blue}{\left(\mathsf{neg}\left(y \cdot x\right)\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \left(\mathsf{neg}\left(\color{blue}{x \cdot y}\right)\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \color{blue}{-1 \cdot \left(x \cdot y\right)}\right) \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{c \cdot i + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{i \cdot c} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(b, a, i \cdot c + \color{blue}{1} \cdot \left(x \cdot y\right)\right) \]

      11. *-lft-identity N/A

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

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{\mathsf{fma}\left(i, c, x \cdot y\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, \color{blue}{y \cdot x}\right)\right) \]

      14. lower-*.f64 91.4

          \[\leadsto \mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, \color{blue}{y \cdot x}\right)\right) \]

   5. Applied rewrites 91.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, y \cdot x\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto a \cdot b + \color{blue}{c \cdot i} \]
   7. Step-by-step derivation

   8. Applied rewrites 78.9%

      \[\leadsto \mathsf{fma}\left(i, \color{blue}{c}, b \cdot a\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y + z \cdot t \leq -1 \cdot 10^{+147} \lor \neg \left(x \cdot y + z \cdot t \leq 10^{+98}\right):\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, b \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+126} \lor \neg \left(a \cdot b \leq 10^{+98}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, y \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, \mathsf{fma}\left(t, z, y \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* a b) -5e+126) (not (<= (* a b) 1e+98)))
   (fma b a (fma i c (* y x)))
   (fma i c (fma t z (* y x)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((a * b) <= -5e+126) || !((a * b) <= 1e+98)) {
		tmp = fma(b, a, fma(i, c, (y * x)));
	} else {
		tmp = fma(i, c, fma(t, z, (y * x)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(a * b) <= -5e+126) || !(Float64(a * b) <= 1e+98))
		tmp = fma(b, a, fma(i, c, Float64(y * x)));
	else
		tmp = fma(i, c, fma(t, z, Float64(y * x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(a * b), $MachinePrecision], -5e+126], N[Not[LessEqual[N[(a * b), $MachinePrecision], 1e+98]], $MachinePrecision]], N[(b * a + N[(i * c + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(i * c + N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -4.99999999999999977e126 or 9.99999999999999998e97 < (*.f64 a b)

   1. Initial program 96.1%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{a \cdot b + \left(c \cdot i + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{c \cdot i - \left(\mathsf{neg}\left(y\right)\right) \cdot x}\right) \]

      5. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \color{blue}{\left(\mathsf{neg}\left(y \cdot x\right)\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \left(\mathsf{neg}\left(\color{blue}{x \cdot y}\right)\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \color{blue}{-1 \cdot \left(x \cdot y\right)}\right) \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{c \cdot i + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{i \cdot c} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(b, a, i \cdot c + \color{blue}{1} \cdot \left(x \cdot y\right)\right) \]

      11. *-lft-identity N/A

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

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{\mathsf{fma}\left(i, c, x \cdot y\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, \color{blue}{y \cdot x}\right)\right) \]

      14. lower-*.f64 91.3

          \[\leadsto \mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, \color{blue}{y \cdot x}\right)\right) \]

   5. Applied rewrites 91.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, y \cdot x\right)\right)} \]

   if -4.99999999999999977e126 < (*.f64 a b) < 9.99999999999999998e97

   1. Initial program 96.6%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c \cdot i + \left(t \cdot z + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto c \cdot i + \color{blue}{1 \cdot \left(t \cdot z + x \cdot y\right)} \]

      2. metadata-eval N/A

         \[\leadsto c \cdot i + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \left(t \cdot z + x \cdot y\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{i \cdot c} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(t \cdot z + x \cdot y\right) \]

      4. metadata-eval N/A

         \[\leadsto i \cdot c + \color{blue}{1} \cdot \left(t \cdot z + x \cdot y\right) \]

      5. *-lft-identity N/A

         \[\leadsto i \cdot c + \color{blue}{\left(t \cdot z + x \cdot y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(i, c, t \cdot z + x \cdot y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(i, c, \color{blue}{\mathsf{fma}\left(t, z, x \cdot y\right)}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(i, c, \mathsf{fma}\left(t, z, \color{blue}{y \cdot x}\right)\right) \]

      9. lower-*.f64 95.1

         \[\leadsto \mathsf{fma}\left(i, c, \mathsf{fma}\left(t, z, \color{blue}{y \cdot x}\right)\right) \]

   5. Applied rewrites 95.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, c, \mathsf{fma}\left(t, z, y \cdot x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+126} \lor \neg \left(a \cdot b \leq 10^{+98}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, y \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, \mathsf{fma}\left(t, z, y \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -4 \cdot 10^{+155} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+181}\right):\\ \;\;\;\;\mathsf{fma}\left(i, c, t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, y \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* z t) -4e+155) (not (<= (* z t) 2e+181)))
   (fma i c (* t z))
   (fma b a (fma i c (* y x)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((z * t) <= -4e+155) || !((z * t) <= 2e+181)) {
		tmp = fma(i, c, (t * z));
	} else {
		tmp = fma(b, a, fma(i, c, (y * x)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(z * t) <= -4e+155) || !(Float64(z * t) <= 2e+181))
		tmp = fma(i, c, Float64(t * z));
	else
		tmp = fma(b, a, fma(i, c, Float64(y * x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -4e+155], N[Not[LessEqual[N[(z * t), $MachinePrecision], 2e+181]], $MachinePrecision]], N[(i * c + N[(t * z), $MachinePrecision]), $MachinePrecision], N[(b * a + N[(i * c + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -4.00000000000000003e155 or 1.9999999999999998e181 < (*.f64 z t)

   1. Initial program 89.2%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c \cdot i + \left(t \cdot z + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto c \cdot i + \color{blue}{1 \cdot \left(t \cdot z + x \cdot y\right)} \]

      2. metadata-eval N/A

         \[\leadsto c \cdot i + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \left(t \cdot z + x \cdot y\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{i \cdot c} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(t \cdot z + x \cdot y\right) \]

      4. metadata-eval N/A

         \[\leadsto i \cdot c + \color{blue}{1} \cdot \left(t \cdot z + x \cdot y\right) \]

      5. *-lft-identity N/A

         \[\leadsto i \cdot c + \color{blue}{\left(t \cdot z + x \cdot y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(i, c, t \cdot z + x \cdot y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(i, c, \color{blue}{\mathsf{fma}\left(t, z, x \cdot y\right)}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(i, c, \mathsf{fma}\left(t, z, \color{blue}{y \cdot x}\right)\right) \]

      9. lower-*.f64 94.0

         \[\leadsto \mathsf{fma}\left(i, c, \mathsf{fma}\left(t, z, \color{blue}{y \cdot x}\right)\right) \]

   5. Applied rewrites 94.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, c, \mathsf{fma}\left(t, z, y \cdot x\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(i, c, t \cdot z\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 89.0%

      \[\leadsto \mathsf{fma}\left(i, c, t \cdot z\right) \]

   if -4.00000000000000003e155 < (*.f64 z t) < 1.9999999999999998e181

   1. Initial program 98.9%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{a \cdot b + \left(c \cdot i + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{c \cdot i - \left(\mathsf{neg}\left(y\right)\right) \cdot x}\right) \]

      5. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \color{blue}{\left(\mathsf{neg}\left(y \cdot x\right)\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \left(\mathsf{neg}\left(\color{blue}{x \cdot y}\right)\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \color{blue}{-1 \cdot \left(x \cdot y\right)}\right) \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{c \cdot i + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{i \cdot c} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(b, a, i \cdot c + \color{blue}{1} \cdot \left(x \cdot y\right)\right) \]

      11. *-lft-identity N/A

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

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{\mathsf{fma}\left(i, c, x \cdot y\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, \color{blue}{y \cdot x}\right)\right) \]

      14. lower-*.f64 92.7

          \[\leadsto \mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, \color{blue}{y \cdot x}\right)\right) \]

   5. Applied rewrites 92.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, y \cdot x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -4 \cdot 10^{+155} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+181}\right):\\ \;\;\;\;\mathsf{fma}\left(i, c, t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, y \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+75} \lor \neg \left(x \cdot y \leq 10^{+90}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, b \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* x y) -1e+75) (not (<= (* x y) 1e+90)))
   (fma b a (* y x))
   (fma i c (* b a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((x * y) <= -1e+75) || !((x * y) <= 1e+90)) {
		tmp = fma(b, a, (y * x));
	} else {
		tmp = fma(i, c, (b * a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(x * y) <= -1e+75) || !(Float64(x * y) <= 1e+90))
		tmp = fma(b, a, Float64(y * x));
	else
		tmp = fma(i, c, Float64(b * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -1e+75], N[Not[LessEqual[N[(x * y), $MachinePrecision], 1e+90]], $MachinePrecision]], N[(b * a + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(i * c + N[(b * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -9.99999999999999927e74 or 9.99999999999999966e89 < (*.f64 x y)

   1. Initial program 94.8%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{a \cdot b + \left(c \cdot i + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{c \cdot i - \left(\mathsf{neg}\left(y\right)\right) \cdot x}\right) \]

      5. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \color{blue}{\left(\mathsf{neg}\left(y \cdot x\right)\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \left(\mathsf{neg}\left(\color{blue}{x \cdot y}\right)\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \color{blue}{-1 \cdot \left(x \cdot y\right)}\right) \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{c \cdot i + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{i \cdot c} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(b, a, i \cdot c + \color{blue}{1} \cdot \left(x \cdot y\right)\right) \]

      11. *-lft-identity N/A

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

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{\mathsf{fma}\left(i, c, x \cdot y\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, \color{blue}{y \cdot x}\right)\right) \]

      14. lower-*.f64 84.7

          \[\leadsto \mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, \color{blue}{y \cdot x}\right)\right) \]

   5. Applied rewrites 84.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, y \cdot x\right)\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto a \cdot b + \color{blue}{x \cdot y} \]
   7. Step-by-step derivation

   8. Applied rewrites 79.5%

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

   if -9.99999999999999927e74 < (*.f64 x y) < 9.99999999999999966e89

   1. Initial program 97.5%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{a \cdot b + \left(c \cdot i + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{c \cdot i - \left(\mathsf{neg}\left(y\right)\right) \cdot x}\right) \]

      5. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \color{blue}{\left(\mathsf{neg}\left(y \cdot x\right)\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \left(\mathsf{neg}\left(\color{blue}{x \cdot y}\right)\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \color{blue}{-1 \cdot \left(x \cdot y\right)}\right) \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{c \cdot i + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{i \cdot c} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(b, a, i \cdot c + \color{blue}{1} \cdot \left(x \cdot y\right)\right) \]

      11. *-lft-identity N/A

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

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{\mathsf{fma}\left(i, c, x \cdot y\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, \color{blue}{y \cdot x}\right)\right) \]

      14. lower-*.f64 72.7

          \[\leadsto \mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, \color{blue}{y \cdot x}\right)\right) \]

   5. Applied rewrites 72.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, y \cdot x\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto a \cdot b + \color{blue}{c \cdot i} \]
   7. Step-by-step derivation

   8. Applied rewrites 66.5%

      \[\leadsto \mathsf{fma}\left(i, \color{blue}{c}, b \cdot a\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+75} \lor \neg \left(x \cdot y \leq 10^{+90}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, b \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -1 \cdot 10^{+167} \lor \neg \left(c \cdot i \leq 5 \cdot 10^{+145}\right):\\ \;\;\;\;i \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* c i) -1e+167) (not (<= (* c i) 5e+145)))
   (* i c)
   (fma b a (* y x))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((c * i) <= -1e+167) || !((c * i) <= 5e+145)) {
		tmp = i * c;
	} else {
		tmp = fma(b, a, (y * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(c * i) <= -1e+167) || !(Float64(c * i) <= 5e+145))
		tmp = Float64(i * c);
	else
		tmp = fma(b, a, Float64(y * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(c * i), $MachinePrecision], -1e+167], N[Not[LessEqual[N[(c * i), $MachinePrecision], 5e+145]], $MachinePrecision]], N[(i * c), $MachinePrecision], N[(b * a + N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c i) < -1e167 or 4.99999999999999967e145 < (*.f64 c i)

   1. Initial program 93.7%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot i} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{i \cdot c} \]

      2. lower-*.f64 74.2

         \[\leadsto \color{blue}{i \cdot c} \]

   5. Applied rewrites 74.2%

      \[\leadsto \color{blue}{i \cdot c} \]

   if -1e167 < (*.f64 c i) < 4.99999999999999967e145

   1. Initial program 97.7%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{a \cdot b + \left(c \cdot i + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{c \cdot i - \left(\mathsf{neg}\left(y\right)\right) \cdot x}\right) \]

      5. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \color{blue}{\left(\mathsf{neg}\left(y \cdot x\right)\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \left(\mathsf{neg}\left(\color{blue}{x \cdot y}\right)\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \color{blue}{-1 \cdot \left(x \cdot y\right)}\right) \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{c \cdot i + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{i \cdot c} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(b, a, i \cdot c + \color{blue}{1} \cdot \left(x \cdot y\right)\right) \]

      11. *-lft-identity N/A

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

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{\mathsf{fma}\left(i, c, x \cdot y\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, \color{blue}{y \cdot x}\right)\right) \]

      14. lower-*.f64 73.2

          \[\leadsto \mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, \color{blue}{y \cdot x}\right)\right) \]

   5. Applied rewrites 73.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, y \cdot x\right)\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto a \cdot b + \color{blue}{x \cdot y} \]
   7. Step-by-step derivation

   8. Applied rewrites 66.4%

      \[\leadsto \mathsf{fma}\left(b, \color{blue}{a}, y \cdot x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -1 \cdot 10^{+167} \lor \neg \left(c \cdot i \leq 5 \cdot 10^{+145}\right):\\ \;\;\;\;i \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(b, a, y \cdot x\right)\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+103}:\\ \;\;\;\;\mathsf{fma}\left(i, c, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, b \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* a b) -1e+80)
   (fma b a (* y x))
   (if (<= (* a b) 2e+103) (fma i c (* y x)) (fma i c (* b a)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((a * b) <= -1e+80) {
		tmp = fma(b, a, (y * x));
	} else if ((a * b) <= 2e+103) {
		tmp = fma(i, c, (y * x));
	} else {
		tmp = fma(i, c, (b * a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(a * b) <= -1e+80)
		tmp = fma(b, a, Float64(y * x));
	elseif (Float64(a * b) <= 2e+103)
		tmp = fma(i, c, Float64(y * x));
	else
		tmp = fma(i, c, Float64(b * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(a * b), $MachinePrecision], -1e+80], N[(b * a + N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 2e+103], N[(i * c + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(i * c + N[(b * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -1e80

   1. Initial program 97.7%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{a \cdot b + \left(c \cdot i + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{c \cdot i - \left(\mathsf{neg}\left(y\right)\right) \cdot x}\right) \]

      5. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \color{blue}{\left(\mathsf{neg}\left(y \cdot x\right)\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \left(\mathsf{neg}\left(\color{blue}{x \cdot y}\right)\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \color{blue}{-1 \cdot \left(x \cdot y\right)}\right) \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{c \cdot i + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{i \cdot c} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(b, a, i \cdot c + \color{blue}{1} \cdot \left(x \cdot y\right)\right) \]

      11. *-lft-identity N/A

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

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{\mathsf{fma}\left(i, c, x \cdot y\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, \color{blue}{y \cdot x}\right)\right) \]

      14. lower-*.f64 87.0

          \[\leadsto \mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, \color{blue}{y \cdot x}\right)\right) \]

   5. Applied rewrites 87.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, y \cdot x\right)\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto a \cdot b + \color{blue}{x \cdot y} \]
   7. Step-by-step derivation

   8. Applied rewrites 76.3%

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

   if -1e80 < (*.f64 a b) < 2e103

   1. Initial program 96.5%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c \cdot i + \left(t \cdot z + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto c \cdot i + \color{blue}{1 \cdot \left(t \cdot z + x \cdot y\right)} \]

      2. metadata-eval N/A

         \[\leadsto c \cdot i + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \left(t \cdot z + x \cdot y\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{i \cdot c} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(t \cdot z + x \cdot y\right) \]

      4. metadata-eval N/A

         \[\leadsto i \cdot c + \color{blue}{1} \cdot \left(t \cdot z + x \cdot y\right) \]

      5. *-lft-identity N/A

         \[\leadsto i \cdot c + \color{blue}{\left(t \cdot z + x \cdot y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(i, c, t \cdot z + x \cdot y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(i, c, \color{blue}{\mathsf{fma}\left(t, z, x \cdot y\right)}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(i, c, \mathsf{fma}\left(t, z, \color{blue}{y \cdot x}\right)\right) \]

      9. lower-*.f64 96.1

         \[\leadsto \mathsf{fma}\left(i, c, \mathsf{fma}\left(t, z, \color{blue}{y \cdot x}\right)\right) \]

   5. Applied rewrites 96.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, c, \mathsf{fma}\left(t, z, y \cdot x\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(i, c, t \cdot z\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 62.4%

      \[\leadsto \mathsf{fma}\left(i, c, t \cdot z\right) \]

   9. Taylor expanded in z around 0

      \[\leadsto c \cdot i + \color{blue}{x \cdot y} \]
   10. Step-by-step derivation

   11. Applied rewrites 69.4%

       \[\leadsto \mathsf{fma}\left(i, \color{blue}{c}, y \cdot x\right) \]

   if 2e103 < (*.f64 a b)

   1. Initial program 94.6%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{a \cdot b + \left(c \cdot i + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{c \cdot i - \left(\mathsf{neg}\left(y\right)\right) \cdot x}\right) \]

      5. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \color{blue}{\left(\mathsf{neg}\left(y \cdot x\right)\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \left(\mathsf{neg}\left(\color{blue}{x \cdot y}\right)\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \color{blue}{-1 \cdot \left(x \cdot y\right)}\right) \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{c \cdot i + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{i \cdot c} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(b, a, i \cdot c + \color{blue}{1} \cdot \left(x \cdot y\right)\right) \]

      11. *-lft-identity N/A

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

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{\mathsf{fma}\left(i, c, x \cdot y\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, \color{blue}{y \cdot x}\right)\right) \]

      14. lower-*.f64 89.4

          \[\leadsto \mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, \color{blue}{y \cdot x}\right)\right) \]

   5. Applied rewrites 89.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, y \cdot x\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto a \cdot b + \color{blue}{c \cdot i} \]
   7. Step-by-step derivation

   8. Applied rewrites 84.1%

      \[\leadsto \mathsf{fma}\left(i, \color{blue}{c}, b \cdot a\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 43.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+77} \lor \neg \left(a \cdot b \leq 10^{+108}\right):\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;i \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* a b) -5e+77) (not (<= (* a b) 1e+108))) (* b a) (* i c)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((a * b) <= -5e+77) || !((a * b) <= 1e+108)) {
		tmp = b * a;
	} else {
		tmp = i * c;
	}
	return tmp;
}

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(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if (((a * b) <= (-5d+77)) .or. (.not. ((a * b) <= 1d+108))) then
        tmp = b * a
    else
        tmp = i * c
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((a * b) <= -5e+77) || !((a * b) <= 1e+108)) {
		tmp = b * a;
	} else {
		tmp = i * c;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if ((a * b) <= -5e+77) or not ((a * b) <= 1e+108):
		tmp = b * a
	else:
		tmp = i * c
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(a * b) <= -5e+77) || !(Float64(a * b) <= 1e+108))
		tmp = Float64(b * a);
	else
		tmp = Float64(i * c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (((a * b) <= -5e+77) || ~(((a * b) <= 1e+108)))
		tmp = b * a;
	else
		tmp = i * c;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(a * b), $MachinePrecision], -5e+77], N[Not[LessEqual[N[(a * b), $MachinePrecision], 1e+108]], $MachinePrecision]], N[(b * a), $MachinePrecision], N[(i * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -5.00000000000000004e77 or 1e108 < (*.f64 a b)

   1. Initial program 96.3%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{a \cdot b + \left(c \cdot i + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{c \cdot i - \left(\mathsf{neg}\left(y\right)\right) \cdot x}\right) \]

      5. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \color{blue}{\left(\mathsf{neg}\left(y \cdot x\right)\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \left(\mathsf{neg}\left(\color{blue}{x \cdot y}\right)\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \color{blue}{-1 \cdot \left(x \cdot y\right)}\right) \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{c \cdot i + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{i \cdot c} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(b, a, i \cdot c + \color{blue}{1} \cdot \left(x \cdot y\right)\right) \]

      11. *-lft-identity N/A

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

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{\mathsf{fma}\left(i, c, x \cdot y\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, \color{blue}{y \cdot x}\right)\right) \]

      14. lower-*.f64 88.1

          \[\leadsto \mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, \color{blue}{y \cdot x}\right)\right) \]

   5. Applied rewrites 88.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, y \cdot x\right)\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto a \cdot b + \color{blue}{x \cdot y} \]
   7. Step-by-step derivation

   8. Applied rewrites 77.5%

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

   9. Taylor expanded in x around 0

      \[\leadsto a \cdot b \]
   10. Step-by-step derivation

   11. Applied rewrites 60.4%

       \[\leadsto b \cdot a \]

   if -5.00000000000000004e77 < (*.f64 a b) < 1e108

   1. Initial program 96.5%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot i} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{i \cdot c} \]

      2. lower-*.f64 36.1

         \[\leadsto \color{blue}{i \cdot c} \]

   5. Applied rewrites 36.1%

      \[\leadsto \color{blue}{i \cdot c} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+77} \lor \neg \left(a \cdot b \leq 10^{+108}\right):\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;i \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 27.5% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ b \cdot a \end{array} \]

FPCore

(FPCore (x y z t a b c i) :precision binary64 (* b a))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return b * 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(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = b * a
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return b * a;
}

Python

def code(x, y, z, t, a, b, c, i):
	return b * a

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(b * a)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = b * a;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(b * a), $MachinePrecision]

Derivation

1. Initial program 96.5%

   \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{a \cdot b + \left(c \cdot i + x \cdot y\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. *-commutative N/A

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

   4. fp-cancel-sign-sub-inv N/A

      \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{c \cdot i - \left(\mathsf{neg}\left(y\right)\right) \cdot x}\right) \]

   5. distribute-lft-neg-out N/A

      \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \color{blue}{\left(\mathsf{neg}\left(y \cdot x\right)\right)}\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \left(\mathsf{neg}\left(\color{blue}{x \cdot y}\right)\right)\right) \]

   7. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \color{blue}{-1 \cdot \left(x \cdot y\right)}\right) \]

   8. fp-cancel-sub-sign-inv N/A

      \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{c \cdot i + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)}\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{i \cdot c} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(b, a, i \cdot c + \color{blue}{1} \cdot \left(x \cdot y\right)\right) \]

   11. *-lft-identity N/A

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

   12. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{\mathsf{fma}\left(i, c, x \cdot y\right)}\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, \color{blue}{y \cdot x}\right)\right) \]

   14. lower-*.f64 77.2

       \[\leadsto \mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, \color{blue}{y \cdot x}\right)\right) \]

5. Applied rewrites 77.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, y \cdot x\right)\right)} \]

6. Taylor expanded in c around 0

   \[\leadsto a \cdot b + \color{blue}{x \cdot y} \]
7. Step-by-step derivation

8. Applied rewrites 50.6%

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

9. Taylor expanded in x around 0

   \[\leadsto a \cdot b \]
10. Step-by-step derivation

11. Applied rewrites 22.6%

    \[\leadsto b \cdot a \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024364 
(FPCore (x y z t a b c i)
  :name "Linear.V4:$cdot from linear-1.19.1.3, C"
  :precision binary64
  (+ (+ (+ (* x y) (* z t)) (* a b)) (* c i)))