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

Percentage Accurate: 96.1% → 98.1%

Time: 5.3s

Alternatives: 14

Speedup: 1.3×

• 41.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.1% 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: 98.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.9%

   \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
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 99.2%

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

Alternative 2: 66.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, y, a \cdot b\right)\\ t_2 := \mathsf{fma}\left(i, c, a \cdot b\right)\\ \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{+116}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;c \cdot i \leq -1 \cdot 10^{-96}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \cdot i \leq 0:\\ \;\;\;\;\mathsf{fma}\left(z, t, a \cdot b\right)\\ \mathbf{elif}\;c \cdot i \leq 2 \cdot 10^{+70}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma x y (* a b))) (t_2 (fma i c (* a b))))
   (if (<= (* c i) -5e+116)
     t_2
     (if (<= (* c i) -1e-96)
       t_1
       (if (<= (* c i) 0.0)
         (fma z t (* a b))
         (if (<= (* c i) 2e+70) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(x, y, (a * b));
	double t_2 = fma(i, c, (a * b));
	double tmp;
	if ((c * i) <= -5e+116) {
		tmp = t_2;
	} else if ((c * i) <= -1e-96) {
		tmp = t_1;
	} else if ((c * i) <= 0.0) {
		tmp = fma(z, t, (a * b));
	} else if ((c * i) <= 2e+70) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(x, y, Float64(a * b))
	t_2 = fma(i, c, Float64(a * b))
	tmp = 0.0
	if (Float64(c * i) <= -5e+116)
		tmp = t_2;
	elseif (Float64(c * i) <= -1e-96)
		tmp = t_1;
	elseif (Float64(c * i) <= 0.0)
		tmp = fma(z, t, Float64(a * b));
	elseif (Float64(c * i) <= 2e+70)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x * y + N[(a * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(i * c + N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(c * i), $MachinePrecision], -5e+116], t$95$2, If[LessEqual[N[(c * i), $MachinePrecision], -1e-96], t$95$1, If[LessEqual[N[(c * i), $MachinePrecision], 0.0], N[(z * t + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 2e+70], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c i) < -5.00000000000000025e116 or 2.00000000000000015e70 < (*.f64 c i)

   1. Initial program 91.1%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 79.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 82.7

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

   7. Applied rewrites 82.7%

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

   if -5.00000000000000025e116 < (*.f64 c i) < -9.9999999999999991e-97 or -0.0 < (*.f64 c i) < 2.00000000000000015e70

   1. Initial program 100.0%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing
   3. 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 100.0%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 76.3%

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

   8. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(x, y, a \cdot b\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 72.5%

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

   if -9.9999999999999991e-97 < (*.f64 c i) < -0.0

   1. Initial program 100.0%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing
   3. 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 100.0%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 77.1%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \mathbf{elif}\;c \cdot i \leq -1 \cdot 10^{-96}:\\ \;\;\;\;\mathsf{fma}\left(x, y, a \cdot b\right)\\ \mathbf{elif}\;c \cdot i \leq 0:\\ \;\;\;\;\mathsf{fma}\left(z, t, a \cdot b\right)\\ \mathbf{elif}\;c \cdot i \leq 2 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(x, y, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot y + z \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+179} \lor \neg \left(t\_1 \leq 10^{+135}\right):\\ \;\;\;\;\mathsf{fma}\left(z, t, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* x y) (* z t))))
   (if (or (<= t_1 -5e+179) (not (<= t_1 1e+135)))
     (fma z t (* x y))
     (fma i c (* a b)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (x * y) + (z * t);
	double tmp;
	if ((t_1 <= -5e+179) || !(t_1 <= 1e+135)) {
		tmp = fma(z, t, (x * y));
	} else {
		tmp = fma(i, c, (a * b));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(x * y) + Float64(z * t))
	tmp = 0.0
	if ((t_1 <= -5e+179) || !(t_1 <= 1e+135))
		tmp = fma(z, t, Float64(x * y));
	else
		tmp = fma(i, c, Float64(a * b));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e+179], N[Not[LessEqual[t$95$1, 1e+135]], $MachinePrecision]], N[(z * t + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(i * c + N[(a * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x y) (*.f64 z t)) < -5e179 or 9.99999999999999962e134 < (+.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 94.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

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

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. associate-+l+ N/A

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

      10. lift-*.f64 N/A

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

      11. remove-double-neg N/A

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

      12. lift-*.f64 N/A

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

      13. associate-+r+ N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(z, t, \color{blue}{x \cdot y}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 77.4%

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

   if -5e179 < (+.f64 (*.f64 x y) (*.f64 z t)) < 9.99999999999999962e134

   1. Initial program 99.2%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 76.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 77.5

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

   7. Applied rewrites 77.5%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y + z \cdot t \leq -5 \cdot 10^{+179} \lor \neg \left(x \cdot y + z \cdot t \leq 10^{+135}\right):\\ \;\;\;\;\mathsf{fma}\left(z, t, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{+116} \lor \neg \left(c \cdot i \leq 2 \cdot 10^{+70}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(t, z, i \cdot c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(t, z, y \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* c i) -5e+116) (not (<= (* c i) 2e+70)))
   (fma b a (fma t z (* i c)))
   (fma b a (fma t z (* y x)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((c * i) <= -5e+116) || !((c * i) <= 2e+70)) {
		tmp = fma(b, a, fma(t, z, (i * c)));
	} else {
		tmp = fma(b, a, fma(t, z, (y * x)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(c * i) <= -5e+116) || !(Float64(c * i) <= 2e+70))
		tmp = fma(b, a, fma(t, z, Float64(i * c)));
	else
		tmp = fma(b, a, fma(t, z, Float64(y * x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(c * i), $MachinePrecision], -5e+116], N[Not[LessEqual[N[(c * i), $MachinePrecision], 2e+70]], $MachinePrecision]], N[(b * a + N[(t * z + N[(i * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * a + N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c i) < -5.00000000000000025e116 or 2.00000000000000015e70 < (*.f64 c i)

   1. Initial program 91.1%

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

   5. Applied rewrites 91.3%

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

   if -5.00000000000000025e116 < (*.f64 c i) < 2.00000000000000015e70

   1. Initial program 100.0%

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

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 97.2%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{+116} \lor \neg \left(c \cdot i \leq 2 \cdot 10^{+70}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(t, z, i \cdot c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(t, z, y \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+179}:\\ \;\;\;\;\mathsf{fma}\left(z, t, a \cdot b\right)\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, c \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(t, z, i \cdot c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* z t) -5e+179)
   (fma z t (* a b))
   (if (<= (* z t) 5e+41)
     (fma x y (fma a b (* c i)))
     (fma b a (fma t z (* i c))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((z * t) <= -5e+179) {
		tmp = fma(z, t, (a * b));
	} else if ((z * t) <= 5e+41) {
		tmp = fma(x, y, fma(a, b, (c * i)));
	} else {
		tmp = fma(b, a, fma(t, z, (i * c)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(z * t) <= -5e+179)
		tmp = fma(z, t, Float64(a * b));
	elseif (Float64(z * t) <= 5e+41)
		tmp = fma(x, y, fma(a, b, Float64(c * i)));
	else
		tmp = fma(b, a, fma(t, z, Float64(i * c)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(z * t), $MachinePrecision], -5e+179], N[(z * t + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+41], N[(x * y + N[(a * b + N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * a + N[(t * z + N[(i * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -5e179

   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. 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 100.0%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 96.3%

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

   if -5e179 < (*.f64 z t) < 5.00000000000000022e41

   1. Initial program 98.3%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 99.4%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 92.6%

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

   if 5.00000000000000022e41 < (*.f64 z t)

   1. Initial program 92.7%

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

   5. Applied rewrites 91.2%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+179}:\\ \;\;\;\;\mathsf{fma}\left(z, t, a \cdot b\right)\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, c \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(t, z, i \cdot c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+179}:\\ \;\;\;\;\mathsf{fma}\left(z, t, a \cdot b\right)\\ \mathbf{elif}\;z \cdot t \leq 10^{+230}:\\ \;\;\;\;\mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, c \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, c \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* z t) -5e+179)
   (fma z t (* a b))
   (if (<= (* z t) 1e+230) (fma x y (fma a b (* c i))) (fma z t (* c i)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((z * t) <= -5e+179) {
		tmp = fma(z, t, (a * b));
	} else if ((z * t) <= 1e+230) {
		tmp = fma(x, y, fma(a, b, (c * i)));
	} else {
		tmp = fma(z, t, (c * i));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(z * t) <= -5e+179)
		tmp = fma(z, t, Float64(a * b));
	elseif (Float64(z * t) <= 1e+230)
		tmp = fma(x, y, fma(a, b, Float64(c * i)));
	else
		tmp = fma(z, t, Float64(c * i));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(z * t), $MachinePrecision], -5e+179], N[(z * t + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+230], N[(x * y + N[(a * b + N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * t + N[(c * i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -5e179

   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. 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 100.0%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 96.3%

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

   if -5e179 < (*.f64 z t) < 1.0000000000000001e230

   1. Initial program 98.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

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

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. associate-+l+ N/A

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

      10. lift-*.f64 N/A

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

      11. remove-double-neg N/A

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

      12. lift-*.f64 N/A

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

      13. associate-+r+ N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 89.3%

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

   if 1.0000000000000001e230 < (*.f64 z t)

   1. Initial program 86.7%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 96.7%

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

   5. Taylor expanded in c around inf

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

   7. Applied rewrites 88.7%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+179}:\\ \;\;\;\;\mathsf{fma}\left(z, t, a \cdot b\right)\\ \mathbf{elif}\;z \cdot t \leq 10^{+230}:\\ \;\;\;\;\mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, c \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, c \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+121}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;a \cdot b \leq 10^{-167}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+139}:\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* a b) -5e+121)
   (* b a)
   (if (<= (* a b) 1e-167) (* y x) (if (<= (* a b) 2e+139) (* t z) (* b a)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((a * b) <= -5e+121) {
		tmp = b * a;
	} else if ((a * b) <= 1e-167) {
		tmp = y * x;
	} else if ((a * b) <= 2e+139) {
		tmp = t * z;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if ((a * b) <= (-5d+121)) then
        tmp = b * a
    else if ((a * b) <= 1d-167) then
        tmp = y * x
    else if ((a * b) <= 2d+139) then
        tmp = t * z
    else
        tmp = b * a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((a * b) <= -5e+121) {
		tmp = b * a;
	} else if ((a * b) <= 1e-167) {
		tmp = y * x;
	} else if ((a * b) <= 2e+139) {
		tmp = t * z;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (a * b) <= -5e+121:
		tmp = b * a
	elif (a * b) <= 1e-167:
		tmp = y * x
	elif (a * b) <= 2e+139:
		tmp = t * z
	else:
		tmp = b * a
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(a * b) <= -5e+121)
		tmp = Float64(b * a);
	elseif (Float64(a * b) <= 1e-167)
		tmp = Float64(y * x);
	elseif (Float64(a * b) <= 2e+139)
		tmp = Float64(t * z);
	else
		tmp = Float64(b * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((a * b) <= -5e+121)
		tmp = b * a;
	elseif ((a * b) <= 1e-167)
		tmp = y * x;
	elseif ((a * b) <= 2e+139)
		tmp = t * z;
	else
		tmp = b * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(a * b), $MachinePrecision], -5e+121], N[(b * a), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 1e-167], N[(y * x), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 2e+139], N[(t * z), $MachinePrecision], N[(b * a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -5.00000000000000007e121 or 2.00000000000000007e139 < (*.f64 a b)

   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 a around inf

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

   5. Applied rewrites 70.6%

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

   if -5.00000000000000007e121 < (*.f64 a b) < 1e-167

   1. Initial program 98.4%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 44.2%

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

   if 1e-167 < (*.f64 a b) < 2.00000000000000007e139

   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 inf

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

   5. Applied rewrites 44.3%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+121}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;a \cdot b \leq 10^{-167}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+139}:\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+63}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{-146}:\\ \;\;\;\;i \cdot c\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+139}:\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* a b) -5e+63)
   (* b a)
   (if (<= (* a b) 2e-146) (* i c) (if (<= (* a b) 2e+139) (* t z) (* b a)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((a * b) <= -5e+63) {
		tmp = b * a;
	} else if ((a * b) <= 2e-146) {
		tmp = i * c;
	} else if ((a * b) <= 2e+139) {
		tmp = t * z;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if ((a * b) <= (-5d+63)) then
        tmp = b * a
    else if ((a * b) <= 2d-146) then
        tmp = i * c
    else if ((a * b) <= 2d+139) then
        tmp = t * z
    else
        tmp = b * a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((a * b) <= -5e+63) {
		tmp = b * a;
	} else if ((a * b) <= 2e-146) {
		tmp = i * c;
	} else if ((a * b) <= 2e+139) {
		tmp = t * z;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (a * b) <= -5e+63:
		tmp = b * a
	elif (a * b) <= 2e-146:
		tmp = i * c
	elif (a * b) <= 2e+139:
		tmp = t * z
	else:
		tmp = b * a
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(a * b) <= -5e+63)
		tmp = Float64(b * a);
	elseif (Float64(a * b) <= 2e-146)
		tmp = Float64(i * c);
	elseif (Float64(a * b) <= 2e+139)
		tmp = Float64(t * z);
	else
		tmp = Float64(b * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((a * b) <= -5e+63)
		tmp = b * a;
	elseif ((a * b) <= 2e-146)
		tmp = i * c;
	elseif ((a * b) <= 2e+139)
		tmp = t * z;
	else
		tmp = b * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(a * b), $MachinePrecision], -5e+63], N[(b * a), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 2e-146], N[(i * c), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 2e+139], N[(t * z), $MachinePrecision], N[(b * a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -5.00000000000000011e63 or 2.00000000000000007e139 < (*.f64 a b)

   1. Initial program 95.8%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 64.8%

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

   if -5.00000000000000011e63 < (*.f64 a b) < 2.00000000000000005e-146

   1. Initial program 98.2%

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

   3. Taylor expanded in c around inf

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

   5. Applied rewrites 36.9%

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

   if 2.00000000000000005e-146 < (*.f64 a b) < 2.00000000000000007e139

   1. Initial program 96.0%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 44.6%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+63}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{-146}:\\ \;\;\;\;i \cdot c\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+139}:\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{+116} \lor \neg \left(c \cdot i \leq 2 \cdot 10^{+70}\right):\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* c i) -5e+116) (not (<= (* c i) 2e+70)))
   (fma i c (* a b))
   (fma x y (* a b))))

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+116) || !((c * i) <= 2e+70)) {
		tmp = fma(i, c, (a * b));
	} else {
		tmp = fma(x, y, (a * b));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(c * i) <= -5e+116) || !(Float64(c * i) <= 2e+70))
		tmp = fma(i, c, Float64(a * b));
	else
		tmp = fma(x, y, Float64(a * b));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(c * i), $MachinePrecision], -5e+116], N[Not[LessEqual[N[(c * i), $MachinePrecision], 2e+70]], $MachinePrecision]], N[(i * c + N[(a * b), $MachinePrecision]), $MachinePrecision], N[(x * y + N[(a * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c i) < -5.00000000000000025e116 or 2.00000000000000015e70 < (*.f64 c i)

   1. Initial program 91.1%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 79.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 82.7

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

   7. Applied rewrites 82.7%

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

   if -5.00000000000000025e116 < (*.f64 c i) < 2.00000000000000015e70

   1. Initial program 100.0%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing
   3. 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 100.0%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 70.9%

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

   8. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(x, y, a \cdot b\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 68.4%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{+116} \lor \neg \left(c \cdot i \leq 2 \cdot 10^{+70}\right):\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{+116} \lor \neg \left(c \cdot i \leq 2 \cdot 10^{+70}\right):\\ \;\;\;\;\mathsf{fma}\left(a, b, c \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* c i) -5e+116) (not (<= (* c i) 2e+70)))
   (fma a b (* c i))
   (fma x y (* a b))))

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+116) || !((c * i) <= 2e+70)) {
		tmp = fma(a, b, (c * i));
	} else {
		tmp = fma(x, y, (a * b));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(c * i) <= -5e+116) || !(Float64(c * i) <= 2e+70))
		tmp = fma(a, b, Float64(c * i));
	else
		tmp = fma(x, y, Float64(a * b));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(c * i), $MachinePrecision], -5e+116], N[Not[LessEqual[N[(c * i), $MachinePrecision], 2e+70]], $MachinePrecision]], N[(a * b + N[(c * i), $MachinePrecision]), $MachinePrecision], N[(x * y + N[(a * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c i) < -5.00000000000000025e116 or 2.00000000000000015e70 < (*.f64 c i)

   1. Initial program 91.1%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 97.8%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 87.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 80.4%

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

   if -5.00000000000000025e116 < (*.f64 c i) < 2.00000000000000015e70

   1. Initial program 100.0%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing
   3. 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 100.0%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 70.9%

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

   8. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(x, y, a \cdot b\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 68.4%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{+116} \lor \neg \left(c \cdot i \leq 2 \cdot 10^{+70}\right):\\ \;\;\;\;\mathsf{fma}\left(a, b, c \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty \lor \neg \left(x \cdot y \leq 2 \cdot 10^{+131}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, c \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* x y) (- INFINITY)) (not (<= (* x y) 2e+131)))
   (* y x)
   (fma a b (* c i))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((x * y) <= -((double) INFINITY)) || !((x * y) <= 2e+131)) {
		tmp = y * x;
	} else {
		tmp = fma(a, b, (c * i));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(x * y) <= Float64(-Inf)) || !(Float64(x * y) <= 2e+131))
		tmp = Float64(y * x);
	else
		tmp = fma(a, b, Float64(c * i));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], (-Infinity)], N[Not[LessEqual[N[(x * y), $MachinePrecision], 2e+131]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(a * b + N[(c * i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -inf.0 or 1.9999999999999998e131 < (*.f64 x y)

   1. Initial program 93.4%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 84.8%

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

   if -inf.0 < (*.f64 x y) < 1.9999999999999998e131

   1. Initial program 97.9%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 100.0%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 72.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 62.4%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty \lor \neg \left(x \cdot y \leq 2 \cdot 10^{+131}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, c \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+108}:\\ \;\;\;\;\mathsf{fma}\left(z, t, a \cdot b\right)\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(x, y, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, c \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* z t) -1e+108)
   (fma z t (* a b))
   (if (<= (* z t) 5e+41) (fma x y (* a b)) (fma z t (* c i)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((z * t) <= -1e+108) {
		tmp = fma(z, t, (a * b));
	} else if ((z * t) <= 5e+41) {
		tmp = fma(x, y, (a * b));
	} else {
		tmp = fma(z, t, (c * i));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(z * t) <= -1e+108)
		tmp = fma(z, t, Float64(a * b));
	elseif (Float64(z * t) <= 5e+41)
		tmp = fma(x, y, Float64(a * b));
	else
		tmp = fma(z, t, Float64(c * i));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(z * t), $MachinePrecision], -1e+108], N[(z * t + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+41], N[(x * y + N[(a * b), $MachinePrecision]), $MachinePrecision], N[(z * t + N[(c * i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -1e108

   1. Initial program 97.5%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   2. Add Preprocessing
   3. 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 100.0%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 80.5%

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

   if -1e108 < (*.f64 z t) < 5.00000000000000022e41

   1. Initial program 98.1%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 99.4%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 94.4%

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

   8. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(x, y, a \cdot b\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 71.7%

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

   if 5.00000000000000022e41 < (*.f64 z t)

   1. Initial program 92.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

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

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. associate-+l+ N/A

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

      10. lift-*.f64 N/A

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

      11. remove-double-neg N/A

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

      12. lift-*.f64 N/A

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

      13. associate-+r+ N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in c around inf

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

   7. Applied rewrites 78.0%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+108}:\\ \;\;\;\;\mathsf{fma}\left(z, t, a \cdot b\right)\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(x, y, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, c \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 42.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -1 \cdot 10^{+109} \lor \neg \left(c \cdot i \leq 4 \cdot 10^{+75}\right):\\ \;\;\;\;i \cdot c\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* c i) -1e+109) (not (<= (* c i) 4e+75))) (* i c) (* b a)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((c * i) <= -1e+109) || !((c * i) <= 4e+75)) {
		tmp = i * c;
	} else {
		tmp = b * a;
	}
	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) <= (-1d+109)) .or. (.not. ((c * i) <= 4d+75))) then
        tmp = i * c
    else
        tmp = b * a
    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) <= -1e+109) || !((c * i) <= 4e+75)) {
		tmp = i * c;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if ((c * i) <= -1e+109) or not ((c * i) <= 4e+75):
		tmp = i * c
	else:
		tmp = b * a
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(c * i) <= -1e+109) || !(Float64(c * i) <= 4e+75))
		tmp = Float64(i * c);
	else
		tmp = Float64(b * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (((c * i) <= -1e+109) || ~(((c * i) <= 4e+75)))
		tmp = i * c;
	else
		tmp = b * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(c * i), $MachinePrecision], -1e+109], N[Not[LessEqual[N[(c * i), $MachinePrecision], 4e+75]], $MachinePrecision]], N[(i * c), $MachinePrecision], N[(b * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c i) < -9.99999999999999982e108 or 3.99999999999999971e75 < (*.f64 c i)

   1. Initial program 91.1%

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

   3. Taylor expanded in c around inf

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

   5. Applied rewrites 68.6%

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

   if -9.99999999999999982e108 < (*.f64 c i) < 3.99999999999999971e75

   1. Initial program 100.0%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 33.7%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -1 \cdot 10^{+109} \lor \neg \left(c \cdot i \leq 4 \cdot 10^{+75}\right):\\ \;\;\;\;i \cdot c\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 27.2% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.9%

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

3. Taylor expanded in a around inf

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

5. Applied rewrites 27.9%

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

6. Final simplification 27.9%

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

Reproduce

herbie shell --seed 2025018 
(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)))