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

Percentage Accurate: 95.9% → 97.8%

Time: 5.6s

Alternatives: 12

Speedup: 1.3×

• 41.1% 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: 95.9% 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: 97.8% 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 94.9%

   \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
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 96.9%

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

Alternative 2: 42.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{+156}:\\ \;\;\;\;i \cdot c\\ \mathbf{elif}\;c \cdot i \leq -5 \cdot 10^{+28}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;c \cdot i \leq -5 \cdot 10^{-55}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;c \cdot i \leq -2 \cdot 10^{-313}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;c \cdot i \leq 5 \cdot 10^{-262}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;c \cdot i \leq 5 \cdot 10^{+92}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;i \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -5e+156)
   (* i c)
   (if (<= (* c i) -5e+28)
     (* b a)
     (if (<= (* c i) -5e-55)
       (* y x)
       (if (<= (* c i) -2e-313)
         (* t z)
         (if (<= (* c i) 5e-262)
           (* b a)
           (if (<= (* c i) 5e+92) (* y x) (* 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) <= -5e+156) {
		tmp = i * c;
	} else if ((c * i) <= -5e+28) {
		tmp = b * a;
	} else if ((c * i) <= -5e-55) {
		tmp = y * x;
	} else if ((c * i) <= -2e-313) {
		tmp = t * z;
	} else if ((c * i) <= 5e-262) {
		tmp = b * a;
	} else if ((c * i) <= 5e+92) {
		tmp = y * x;
	} 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) <= (-5d+156)) then
        tmp = i * c
    else if ((c * i) <= (-5d+28)) then
        tmp = b * a
    else if ((c * i) <= (-5d-55)) then
        tmp = y * x
    else if ((c * i) <= (-2d-313)) then
        tmp = t * z
    else if ((c * i) <= 5d-262) then
        tmp = b * a
    else if ((c * i) <= 5d+92) then
        tmp = y * x
    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) <= -5e+156) {
		tmp = i * c;
	} else if ((c * i) <= -5e+28) {
		tmp = b * a;
	} else if ((c * i) <= -5e-55) {
		tmp = y * x;
	} else if ((c * i) <= -2e-313) {
		tmp = t * z;
	} else if ((c * i) <= 5e-262) {
		tmp = b * a;
	} else if ((c * i) <= 5e+92) {
		tmp = y * x;
	} else {
		tmp = i * c;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (c * i) <= -5e+156:
		tmp = i * c
	elif (c * i) <= -5e+28:
		tmp = b * a
	elif (c * i) <= -5e-55:
		tmp = y * x
	elif (c * i) <= -2e-313:
		tmp = t * z
	elif (c * i) <= 5e-262:
		tmp = b * a
	elif (c * i) <= 5e+92:
		tmp = y * x
	else:
		tmp = i * c
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(c * i) <= -5e+156)
		tmp = Float64(i * c);
	elseif (Float64(c * i) <= -5e+28)
		tmp = Float64(b * a);
	elseif (Float64(c * i) <= -5e-55)
		tmp = Float64(y * x);
	elseif (Float64(c * i) <= -2e-313)
		tmp = Float64(t * z);
	elseif (Float64(c * i) <= 5e-262)
		tmp = Float64(b * a);
	elseif (Float64(c * i) <= 5e+92)
		tmp = Float64(y * x);
	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) <= -5e+156)
		tmp = i * c;
	elseif ((c * i) <= -5e+28)
		tmp = b * a;
	elseif ((c * i) <= -5e-55)
		tmp = y * x;
	elseif ((c * i) <= -2e-313)
		tmp = t * z;
	elseif ((c * i) <= 5e-262)
		tmp = b * a;
	elseif ((c * i) <= 5e+92)
		tmp = y * x;
	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], -5e+156], N[(i * c), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], -5e+28], N[(b * a), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], -5e-55], N[(y * x), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], -2e-313], N[(t * z), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 5e-262], N[(b * a), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 5e+92], N[(y * x), $MachinePrecision], N[(i * c), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 c i) < -4.99999999999999992e156 or 5.00000000000000022e92 < (*.f64 c i)

   1. Initial program 90.8%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 90.7

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

   5. Applied rewrites 90.7%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 24.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 15.0%

       \[\leadsto b \cdot a \]

   12. Taylor expanded in c around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 73.8

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

   14. Applied rewrites 73.8%

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

   if -4.99999999999999992e156 < (*.f64 c i) < -4.99999999999999957e28 or -1.99999999998e-313 < (*.f64 c i) < 4.99999999999999992e-262

   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 79.7

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

   5. Applied rewrites 79.7%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 71.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 50.0%

       \[\leadsto b \cdot a \]

   if -4.99999999999999957e28 < (*.f64 c i) < -5.0000000000000002e-55 or 4.99999999999999992e-262 < (*.f64 c i) < 5.00000000000000022e92

   1. Initial program 98.7%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 75.7

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 69.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 25.8%

       \[\leadsto b \cdot a \]

   12. Taylor expanded in x around inf

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

       1. *-commutative N/A

          \[\leadsto \color{blue}{y \cdot x} \]

       2. lower-*.f64 47.7

          \[\leadsto \color{blue}{y \cdot x} \]

   14. Applied rewrites 47.7%

       \[\leadsto \color{blue}{y \cdot x} \]

   if -5.0000000000000002e-55 < (*.f64 c i) < -1.99999999998e-313

   1. Initial program 95.7%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 47.5

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

   5. Applied rewrites 47.5%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 47.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 24.3%

       \[\leadsto b \cdot a \]

   12. Taylor expanded in z around inf

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

       1. lower-*.f64 54.9

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

   14. Applied rewrites 54.9%

       \[\leadsto \color{blue}{t \cdot z} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 66.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, a, y \cdot x\right)\\ \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(b, a, i \cdot c\right)\\ \mathbf{elif}\;c \cdot i \leq -5 \cdot 10^{-55}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \cdot i \leq -2 \cdot 10^{-313}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \mathbf{elif}\;c \cdot i \leq 5 \cdot 10^{+92}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma b a (* y x))))
   (if (<= (* c i) -5e+74)
     (fma b a (* i c))
     (if (<= (* c i) -5e-55)
       t_1
       (if (<= (* c i) -2e-313)
         (fma b a (* t z))
         (if (<= (* c i) 5e+92) t_1 (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 t_1 = fma(b, a, (y * x));
	double tmp;
	if ((c * i) <= -5e+74) {
		tmp = fma(b, a, (i * c));
	} else if ((c * i) <= -5e-55) {
		tmp = t_1;
	} else if ((c * i) <= -2e-313) {
		tmp = fma(b, a, (t * z));
	} else if ((c * i) <= 5e+92) {
		tmp = t_1;
	} else {
		tmp = fma(i, c, (y * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(b, a, Float64(y * x))
	tmp = 0.0
	if (Float64(c * i) <= -5e+74)
		tmp = fma(b, a, Float64(i * c));
	elseif (Float64(c * i) <= -5e-55)
		tmp = t_1;
	elseif (Float64(c * i) <= -2e-313)
		tmp = fma(b, a, Float64(t * z));
	elseif (Float64(c * i) <= 5e+92)
		tmp = t_1;
	else
		tmp = fma(i, c, Float64(y * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(b * a + N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(c * i), $MachinePrecision], -5e+74], N[(b * a + N[(i * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], -5e-55], t$95$1, If[LessEqual[N[(c * i), $MachinePrecision], -2e-313], N[(b * a + N[(t * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 5e+92], t$95$1, N[(i * c + N[(y * x), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 c i) < -4.99999999999999963e74

   1. Initial program 92.5%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 90.2

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 86.9%

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

   if -4.99999999999999963e74 < (*.f64 c i) < -5.0000000000000002e-55 or -1.99999999998e-313 < (*.f64 c i) < 5.00000000000000022e92

   1. Initial program 96.7%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 76.8

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

   5. Applied rewrites 76.8%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 71.3%

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

   if -5.0000000000000002e-55 < (*.f64 c i) < -1.99999999998e-313

   1. Initial program 95.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 77.4

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 77.4%

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

   if 5.00000000000000022e92 < (*.f64 c i)

   1. Initial program 91.6%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 89.5

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

   5. Applied rewrites 89.5%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 29.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 16.6%

       \[\leadsto b \cdot a \]

   12. Taylor expanded in a around 0

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

   14. Applied rewrites 79.9%

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

Alternative 4: 62.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, a, y \cdot x\right)\\ \mathbf{if}\;c \cdot i \leq -1 \cdot 10^{+157}:\\ \;\;\;\;i \cdot c\\ \mathbf{elif}\;c \cdot i \leq -5 \cdot 10^{-55}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \cdot i \leq -2 \cdot 10^{-313}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \mathbf{elif}\;c \cdot i \leq 5 \cdot 10^{+92}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;i \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma b a (* y x))))
   (if (<= (* c i) -1e+157)
     (* i c)
     (if (<= (* c i) -5e-55)
       t_1
       (if (<= (* c i) -2e-313)
         (fma b a (* t z))
         (if (<= (* c i) 5e+92) t_1 (* i c)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(b, a, (y * x));
	double tmp;
	if ((c * i) <= -1e+157) {
		tmp = i * c;
	} else if ((c * i) <= -5e-55) {
		tmp = t_1;
	} else if ((c * i) <= -2e-313) {
		tmp = fma(b, a, (t * z));
	} else if ((c * i) <= 5e+92) {
		tmp = t_1;
	} else {
		tmp = i * c;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(b, a, Float64(y * x))
	tmp = 0.0
	if (Float64(c * i) <= -1e+157)
		tmp = Float64(i * c);
	elseif (Float64(c * i) <= -5e-55)
		tmp = t_1;
	elseif (Float64(c * i) <= -2e-313)
		tmp = fma(b, a, Float64(t * z));
	elseif (Float64(c * i) <= 5e+92)
		tmp = t_1;
	else
		tmp = Float64(i * c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(b * a + N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(c * i), $MachinePrecision], -1e+157], N[(i * c), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], -5e-55], t$95$1, If[LessEqual[N[(c * i), $MachinePrecision], -2e-313], N[(b * a + N[(t * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 5e+92], t$95$1, N[(i * c), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c i) < -9.99999999999999983e156 or 5.00000000000000022e92 < (*.f64 c i)

   1. Initial program 92.0%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 90.6

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 23.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 15.2%

       \[\leadsto b \cdot a \]

   12. Taylor expanded in c around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 74.7

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

   14. Applied rewrites 74.7%

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

   if -9.99999999999999983e156 < (*.f64 c i) < -5.0000000000000002e-55 or -1.99999999998e-313 < (*.f64 c i) < 5.00000000000000022e92

   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 77.6

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 70.5%

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

   if -5.0000000000000002e-55 < (*.f64 c i) < -1.99999999998e-313

   1. Initial program 95.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 77.4

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 77.4%

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

Alternative 5: 42.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{+156}:\\ \;\;\;\;i \cdot c\\ \mathbf{elif}\;c \cdot i \leq -5 \cdot 10^{+38}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;c \cdot i \leq -2 \cdot 10^{-313}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;c \cdot i \leq 5 \cdot 10^{+92}:\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;i \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -5e+156)
   (* i c)
   (if (<= (* c i) -5e+38)
     (* b a)
     (if (<= (* c i) -2e-313)
       (* t z)
       (if (<= (* c i) 5e+92) (* 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 ((c * i) <= -5e+156) {
		tmp = i * c;
	} else if ((c * i) <= -5e+38) {
		tmp = b * a;
	} else if ((c * i) <= -2e-313) {
		tmp = t * z;
	} else if ((c * i) <= 5e+92) {
		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 ((c * i) <= (-5d+156)) then
        tmp = i * c
    else if ((c * i) <= (-5d+38)) then
        tmp = b * a
    else if ((c * i) <= (-2d-313)) then
        tmp = t * z
    else if ((c * i) <= 5d+92) 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 ((c * i) <= -5e+156) {
		tmp = i * c;
	} else if ((c * i) <= -5e+38) {
		tmp = b * a;
	} else if ((c * i) <= -2e-313) {
		tmp = t * z;
	} else if ((c * i) <= 5e+92) {
		tmp = b * a;
	} else {
		tmp = i * c;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (c * i) <= -5e+156:
		tmp = i * c
	elif (c * i) <= -5e+38:
		tmp = b * a
	elif (c * i) <= -2e-313:
		tmp = t * z
	elif (c * i) <= 5e+92:
		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(c * i) <= -5e+156)
		tmp = Float64(i * c);
	elseif (Float64(c * i) <= -5e+38)
		tmp = Float64(b * a);
	elseif (Float64(c * i) <= -2e-313)
		tmp = Float64(t * z);
	elseif (Float64(c * i) <= 5e+92)
		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 ((c * i) <= -5e+156)
		tmp = i * c;
	elseif ((c * i) <= -5e+38)
		tmp = b * a;
	elseif ((c * i) <= -2e-313)
		tmp = t * z;
	elseif ((c * i) <= 5e+92)
		tmp = b * a;
	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], -5e+156], N[(i * c), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], -5e+38], N[(b * a), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], -2e-313], N[(t * z), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 5e+92], N[(b * a), $MachinePrecision], N[(i * c), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c i) < -4.99999999999999992e156 or 5.00000000000000022e92 < (*.f64 c i)

   1. Initial program 90.8%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 90.7

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

   5. Applied rewrites 90.7%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 24.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 15.0%

       \[\leadsto b \cdot a \]

   12. Taylor expanded in c around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 73.8

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

   14. Applied rewrites 73.8%

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

   if -4.99999999999999992e156 < (*.f64 c i) < -4.9999999999999997e38 or -1.99999999998e-313 < (*.f64 c i) < 5.00000000000000022e92

   1. Initial program 97.4%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 77.5

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

   5. Applied rewrites 77.5%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 71.7%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 39.4%

       \[\leadsto b \cdot a \]

   if -4.9999999999999997e38 < (*.f64 c i) < -1.99999999998e-313

   1. Initial program 95.2%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 55.4

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

   5. Applied rewrites 55.4%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 50.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 21.6%

       \[\leadsto b \cdot a \]

   12. Taylor expanded in z around inf

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

       1. lower-*.f64 46.9

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

   14. Applied rewrites 46.9%

       \[\leadsto \color{blue}{t \cdot z} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 88.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -20000000 \lor \neg \left(x \cdot y \leq 5 \cdot 10^{+170}\right):\\ \;\;\;\;\mathsf{fma}\left(i, c, \mathsf{fma}\left(t, z, y \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, t \cdot z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* x y) -20000000.0) (not (<= (* x y) 5e+170)))
   (fma i c (fma t z (* y x)))
   (fma b a (fma i c (* t z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((x * y) <= -20000000.0) || !((x * y) <= 5e+170)) {
		tmp = fma(i, c, fma(t, z, (y * x)));
	} else {
		tmp = fma(b, a, fma(i, c, (t * z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(x * y) <= -20000000.0) || !(Float64(x * y) <= 5e+170))
		tmp = fma(i, c, fma(t, z, Float64(y * x)));
	else
		tmp = fma(b, a, fma(i, c, Float64(t * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -20000000.0], N[Not[LessEqual[N[(x * y), $MachinePrecision], 5e+170]], $MachinePrecision]], N[(i * c + N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * a + N[(i * c + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2e7 or 4.99999999999999977e170 < (*.f64 x y)

   1. Initial program 91.2%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 90.6

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

   5. Applied rewrites 90.6%

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

   if -2e7 < (*.f64 x y) < 4.99999999999999977e170

   1. Initial program 97.4%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 96.0

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

   5. Applied rewrites 96.0%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -20000000 \lor \neg \left(x \cdot y \leq 5 \cdot 10^{+170}\right):\\ \;\;\;\;\mathsf{fma}\left(i, c, \mathsf{fma}\left(t, z, y \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, t \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+34} \lor \neg \left(x \cdot y \leq 10^{+192}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, y \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, t \cdot z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* x y) -4e+34) (not (<= (* x y) 1e+192)))
   (fma b a (fma i c (* y x)))
   (fma b a (fma i c (* t z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((x * y) <= -4e+34) || !((x * y) <= 1e+192)) {
		tmp = fma(b, a, fma(i, c, (y * x)));
	} else {
		tmp = fma(b, a, fma(i, c, (t * z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(x * y) <= -4e+34) || !(Float64(x * y) <= 1e+192))
		tmp = fma(b, a, fma(i, c, Float64(y * x)));
	else
		tmp = fma(b, a, fma(i, c, Float64(t * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -4e+34], N[Not[LessEqual[N[(x * y), $MachinePrecision], 1e+192]], $MachinePrecision]], N[(b * a + N[(i * c + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * a + N[(i * c + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -3.99999999999999978e34 or 1.00000000000000004e192 < (*.f64 x y)

   1. Initial program 90.2%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 87.2

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

   5. Applied rewrites 87.2%

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

   if -3.99999999999999978e34 < (*.f64 x y) < 1.00000000000000004e192

   1. Initial program 97.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 94.3

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

   5. Applied rewrites 94.3%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+34} \lor \neg \left(x \cdot y \leq 10^{+192}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, y \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, t \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+34} \lor \neg \left(x \cdot y \leq 10^{+192}\right):\\ \;\;\;\;\mathsf{fma}\left(i, c, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, t \cdot z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* x y) -4e+34) (not (<= (* x y) 1e+192)))
   (fma i c (* y x))
   (fma b a (fma i c (* t z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((x * y) <= -4e+34) || !((x * y) <= 1e+192)) {
		tmp = fma(i, c, (y * x));
	} else {
		tmp = fma(b, a, fma(i, c, (t * z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(x * y) <= -4e+34) || !(Float64(x * y) <= 1e+192))
		tmp = fma(i, c, Float64(y * x));
	else
		tmp = fma(b, a, fma(i, c, Float64(t * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -4e+34], N[Not[LessEqual[N[(x * y), $MachinePrecision], 1e+192]], $MachinePrecision]], N[(i * c + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(b * a + N[(i * c + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -3.99999999999999978e34 or 1.00000000000000004e192 < (*.f64 x y)

   1. Initial program 90.2%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 87.2

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

   5. Applied rewrites 87.2%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 75.7%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 13.5%

       \[\leadsto b \cdot a \]

   12. Taylor expanded in a around 0

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

   14. Applied rewrites 78.9%

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

   if -3.99999999999999978e34 < (*.f64 x y) < 1.00000000000000004e192

   1. Initial program 97.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 94.3

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

   5. Applied rewrites 94.3%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+34} \lor \neg \left(x \cdot y \leq 10^{+192}\right):\\ \;\;\;\;\mathsf{fma}\left(i, c, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, t \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+119} \lor \neg \left(a \cdot b \leq 5 \cdot 10^{+67}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* a b) -2e+119) (not (<= (* a b) 5e+67)))
   (fma b a (* t z))
   (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 (((a * b) <= -2e+119) || !((a * b) <= 5e+67)) {
		tmp = fma(b, a, (t * z));
	} else {
		tmp = fma(i, c, (y * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(a * b) <= -2e+119) || !(Float64(a * b) <= 5e+67))
		tmp = fma(b, a, Float64(t * z));
	else
		tmp = fma(i, c, Float64(y * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(a * b), $MachinePrecision], -2e+119], N[Not[LessEqual[N[(a * b), $MachinePrecision], 5e+67]], $MachinePrecision]], N[(b * a + N[(t * z), $MachinePrecision]), $MachinePrecision], N[(i * c + N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -1.99999999999999989e119 or 4.99999999999999976e67 < (*.f64 a b)

   1. Initial program 86.2%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 85.5

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

   5. Applied rewrites 85.5%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 80.5%

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

   if -1.99999999999999989e119 < (*.f64 a b) < 4.99999999999999976e67

   1. Initial program 99.4%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 73.6

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

   5. Applied rewrites 73.6%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 40.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 7.1%

       \[\leadsto b \cdot a \]

   12. Taylor expanded in a around 0

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

   14. Applied rewrites 69.0%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+119} \lor \neg \left(a \cdot b \leq 5 \cdot 10^{+67}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+34} \lor \neg \left(x \cdot y \leq 10^{+192}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* x y) -4e+34) (not (<= (* x y) 1e+192)))
   (* y x)
   (fma b a (* t z))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((x * y) <= -4e+34) || !((x * y) <= 1e+192)) {
		tmp = y * x;
	} else {
		tmp = fma(b, a, (t * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(x * y) <= -4e+34) || !(Float64(x * y) <= 1e+192))
		tmp = Float64(y * x);
	else
		tmp = fma(b, a, Float64(t * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -4e+34], N[Not[LessEqual[N[(x * y), $MachinePrecision], 1e+192]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(b * a + N[(t * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -3.99999999999999978e34 or 1.00000000000000004e192 < (*.f64 x y)

   1. Initial program 90.2%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 87.2

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

   5. Applied rewrites 87.2%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 75.7%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 13.5%

       \[\leadsto b \cdot a \]

   12. Taylor expanded in x around inf

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

       1. *-commutative N/A

          \[\leadsto \color{blue}{y \cdot x} \]

       2. lower-*.f64 67.6

          \[\leadsto \color{blue}{y \cdot x} \]

   14. Applied rewrites 67.6%

       \[\leadsto \color{blue}{y \cdot x} \]

   if -3.99999999999999978e34 < (*.f64 x y) < 1.00000000000000004e192

   1. Initial program 97.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 94.3

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

   5. Applied rewrites 94.3%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 65.0%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+34} \lor \neg \left(x \cdot y \leq 10^{+192}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -0.0001 \lor \neg \left(a \cdot b \leq 5 \cdot 10^{+67}\right):\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;i \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* a b) -0.0001) (not (<= (* a b) 5e+67))) (* 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) <= -0.0001) || !((a * b) <= 5e+67)) {
		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) <= (-0.0001d0)) .or. (.not. ((a * b) <= 5d+67))) 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) <= -0.0001) || !((a * b) <= 5e+67)) {
		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) <= -0.0001) or not ((a * b) <= 5e+67):
		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) <= -0.0001) || !(Float64(a * b) <= 5e+67))
		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) <= -0.0001) || ~(((a * b) <= 5e+67)))
		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], -0.0001], N[Not[LessEqual[N[(a * b), $MachinePrecision], 5e+67]], $MachinePrecision]], N[(b * a), $MachinePrecision], N[(i * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -1.00000000000000005e-4 or 4.99999999999999976e67 < (*.f64 a b)

   1. Initial program 89.7%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 81.4

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

   5. Applied rewrites 81.4%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 74.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 57.2%

       \[\leadsto b \cdot a \]

   if -1.00000000000000005e-4 < (*.f64 a b) < 4.99999999999999976e67

   1. Initial program 99.3%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 71.4

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

   5. Applied rewrites 71.4%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 34.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 3.1%

       \[\leadsto b \cdot a \]

   12. Taylor expanded in c around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 39.1

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

   14. Applied rewrites 39.1%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -0.0001 \lor \neg \left(a \cdot b \leq 5 \cdot 10^{+67}\right):\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;i \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 27.1% 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 94.9%

   \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
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 76.0

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

5. Applied rewrites 76.0%

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

6. Taylor expanded in c around 0

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

8. Applied rewrites 52.5%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 27.8%

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

Reproduce

herbie shell --seed 2024354 
(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)))