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

Percentage Accurate: 95.2% → 97.6%

Time: 5.1s

Alternatives: 13

Speedup: 1.3×

• 43.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.6% accurate, 1.3× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.9%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. lift-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. associate-+l+ N/A

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

   7. associate--l+ N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

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

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

   11. lift-*.f64 N/A

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

   12. lower-fma.f64 N/A

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

   13. associate-+r+ N/A

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

   14. lift-*.f64 N/A

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

   15. remove-double-neg N/A

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

   16. remove-double-neg N/A

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

   17. *-commutative N/A

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

   18. lower-fma.f64 N/A

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

   19. +-commutative N/A

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

4. Applied rewrites 98.4%

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

Alternative 2: 73.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, x, z \cdot t\right)\\ t_2 := x \cdot y + z \cdot t\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+292}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+153}:\\ \;\;\;\;t \cdot z + c \cdot i\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(y, x, a \cdot b\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(b, a, c \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y x (* z t))) (t_2 (+ (* x y) (* z t))))
   (if (<= t_2 -2e+292)
     t_1
     (if (<= t_2 -2e+153)
       (+ (* t z) (* c i))
       (if (<= t_2 -2e+50)
         (fma y x (* a b))
         (if (<= t_2 1e+69) (fma b a (* c i)) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(y, x, (z * t));
	double t_2 = (x * y) + (z * t);
	double tmp;
	if (t_2 <= -2e+292) {
		tmp = t_1;
	} else if (t_2 <= -2e+153) {
		tmp = (t * z) + (c * i);
	} else if (t_2 <= -2e+50) {
		tmp = fma(y, x, (a * b));
	} else if (t_2 <= 1e+69) {
		tmp = fma(b, a, (c * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, x, Float64(z * t))
	t_2 = Float64(Float64(x * y) + Float64(z * t))
	tmp = 0.0
	if (t_2 <= -2e+292)
		tmp = t_1;
	elseif (t_2 <= -2e+153)
		tmp = Float64(Float64(t * z) + Float64(c * i));
	elseif (t_2 <= -2e+50)
		tmp = fma(y, x, Float64(a * b));
	elseif (t_2 <= 1e+69)
		tmp = fma(b, a, Float64(c * i));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(y * x + N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+292], t$95$1, If[LessEqual[t$95$2, -2e+153], N[(N[(t * z), $MachinePrecision] + N[(c * i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2e+50], N[(y * x + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+69], N[(b * a + N[(c * i), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (*.f64 x y) (*.f64 z t)) < -2e292 or 1.0000000000000001e69 < (+.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 89.9%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. associate--l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. associate-+r+ N/A

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

      14. lift-*.f64 N/A

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

      15. remove-double-neg N/A

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

      16. remove-double-neg N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. +-commutative N/A

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 77.7%

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

   if -2e292 < (+.f64 (*.f64 x y) (*.f64 z t)) < -2e153

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 74.5%

      \[\leadsto \color{blue}{t \cdot z} + c \cdot i \]

   if -2e153 < (+.f64 (*.f64 x y) (*.f64 z t)) < -2.0000000000000002e50

   1. Initial program 95.0%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. associate--l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. associate-+r+ N/A

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

      14. lift-*.f64 N/A

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

      15. remove-double-neg N/A

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

      16. remove-double-neg N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. +-commutative N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 80.9%

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

   if -2.0000000000000002e50 < (+.f64 (*.f64 x y) (*.f64 z t)) < 1.0000000000000001e69

   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 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 94.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 89.0%

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

Alternative 3: 73.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, x, z \cdot t\right)\\ t_2 := x \cdot y + z \cdot t\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+292}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(i, c, t \cdot z\right)\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(y, x, a \cdot b\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(b, a, c \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y x (* z t))) (t_2 (+ (* x y) (* z t))))
   (if (<= t_2 -2e+292)
     t_1
     (if (<= t_2 -2e+153)
       (fma i c (* t z))
       (if (<= t_2 -2e+50)
         (fma y x (* a b))
         (if (<= t_2 1e+69) (fma b a (* c i)) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(y, x, (z * t));
	double t_2 = (x * y) + (z * t);
	double tmp;
	if (t_2 <= -2e+292) {
		tmp = t_1;
	} else if (t_2 <= -2e+153) {
		tmp = fma(i, c, (t * z));
	} else if (t_2 <= -2e+50) {
		tmp = fma(y, x, (a * b));
	} else if (t_2 <= 1e+69) {
		tmp = fma(b, a, (c * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, x, Float64(z * t))
	t_2 = Float64(Float64(x * y) + Float64(z * t))
	tmp = 0.0
	if (t_2 <= -2e+292)
		tmp = t_1;
	elseif (t_2 <= -2e+153)
		tmp = fma(i, c, Float64(t * z));
	elseif (t_2 <= -2e+50)
		tmp = fma(y, x, Float64(a * b));
	elseif (t_2 <= 1e+69)
		tmp = fma(b, a, Float64(c * i));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(y * x + N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+292], t$95$1, If[LessEqual[t$95$2, -2e+153], N[(i * c + N[(t * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2e+50], N[(y * x + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+69], N[(b * a + N[(c * i), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (*.f64 x y) (*.f64 z t)) < -2e292 or 1.0000000000000001e69 < (+.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 89.9%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. associate--l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. associate-+r+ N/A

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

      14. lift-*.f64 N/A

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

      15. remove-double-neg N/A

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

      16. remove-double-neg N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. +-commutative N/A

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 77.7%

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

   if -2e292 < (+.f64 (*.f64 x y) (*.f64 z t)) < -2e153

   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

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

   5. Applied rewrites 46.3%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 74.5%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{z \cdot t + c \cdot i} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{c \cdot i + z \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot i} + z \cdot t \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{i \cdot c} + z \cdot t \]

      5. lower-fma.f64 74.5

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

   10. Applied rewrites 74.5%

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

   if -2e153 < (+.f64 (*.f64 x y) (*.f64 z t)) < -2.0000000000000002e50

   1. Initial program 95.0%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. associate--l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. associate-+r+ N/A

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

      14. lift-*.f64 N/A

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

      15. remove-double-neg N/A

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

      16. remove-double-neg N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. +-commutative N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 80.9%

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

   if -2.0000000000000002e50 < (+.f64 (*.f64 x y) (*.f64 z t)) < 1.0000000000000001e69

   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 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 94.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 89.0%

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

Alternative 4: 66.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(i, c, t \cdot z\right)\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-105}:\\ \;\;\;\;\mathsf{fma}\left(y, x, a \cdot b\right)\\ \mathbf{elif}\;z \cdot t \leq 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(b, a, c \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, z \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* z t) -1e+147)
   (fma i c (* t z))
   (if (<= (* z t) 2e-105)
     (fma y x (* a b))
     (if (<= (* z t) 1e+69) (fma b a (* c i)) (fma b a (* z t))))))

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+147) {
		tmp = fma(i, c, (t * z));
	} else if ((z * t) <= 2e-105) {
		tmp = fma(y, x, (a * b));
	} else if ((z * t) <= 1e+69) {
		tmp = fma(b, a, (c * i));
	} else {
		tmp = fma(b, a, (z * t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(z * t) <= -1e+147)
		tmp = fma(i, c, Float64(t * z));
	elseif (Float64(z * t) <= 2e-105)
		tmp = fma(y, x, Float64(a * b));
	elseif (Float64(z * t) <= 1e+69)
		tmp = fma(b, a, Float64(c * i));
	else
		tmp = fma(b, a, Float64(z * t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(z * t), $MachinePrecision], -1e+147], N[(i * c + N[(t * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e-105], N[(y * x + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+69], N[(b * a + N[(c * i), $MachinePrecision]), $MachinePrecision], N[(b * a + N[(z * t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 z t) < -9.9999999999999998e146

   1. Initial program 91.2%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 33.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 85.6%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{z \cdot t + c \cdot i} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{c \cdot i + z \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot i} + z \cdot t \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{i \cdot c} + z \cdot t \]

      5. lower-fma.f64 85.6

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

   10. Applied rewrites 85.6%

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

   if -9.9999999999999998e146 < (*.f64 z t) < 1.99999999999999993e-105

   1. Initial program 97.8%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. associate--l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. associate-+r+ N/A

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

      14. lift-*.f64 N/A

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

      15. remove-double-neg N/A

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

      16. remove-double-neg N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. +-commutative N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 72.2%

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

   if 1.99999999999999993e-105 < (*.f64 z t) < 1.0000000000000001e69

   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 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 86.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 79.3%

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

   if 1.0000000000000001e69 < (*.f64 z t)

   1. Initial program 87.9%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 82.4%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 74.2%

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

Alternative 5: 65.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -2 \cdot 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(i, c, t \cdot z\right)\\ \mathbf{elif}\;c \cdot i \leq 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(b, a, z \cdot t\right)\\ \mathbf{elif}\;c \cdot i \leq 5 \cdot 10^{+161}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, c \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -2e+120)
   (fma i c (* t z))
   (if (<= (* c i) 1e+81)
     (fma b a (* z t))
     (if (<= (* c i) 5e+161) (* y x) (fma b a (* c i))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((c * i) <= -2e+120) {
		tmp = fma(i, c, (t * z));
	} else if ((c * i) <= 1e+81) {
		tmp = fma(b, a, (z * t));
	} else if ((c * i) <= 5e+161) {
		tmp = y * x;
	} else {
		tmp = fma(b, a, (c * i));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(c * i) <= -2e+120)
		tmp = fma(i, c, Float64(t * z));
	elseif (Float64(c * i) <= 1e+81)
		tmp = fma(b, a, Float64(z * t));
	elseif (Float64(c * i) <= 5e+161)
		tmp = Float64(y * x);
	else
		tmp = fma(b, a, Float64(c * i));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(c * i), $MachinePrecision], -2e+120], N[(i * c + N[(t * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 1e+81], N[(b * a + N[(z * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 5e+161], N[(y * x), $MachinePrecision], N[(b * a + N[(c * i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 c i) < -2e120

   1. Initial program 92.3%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 75.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 81.5%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{z \cdot t + c \cdot i} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{c \cdot i + z \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot i} + z \cdot t \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{i \cdot c} + z \cdot t \]

      5. lower-fma.f64 81.5

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

   10. Applied rewrites 81.5%

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

   if -2e120 < (*.f64 c i) < 9.99999999999999921e80

   1. Initial program 97.0%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 70.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 66.2%

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

   if 9.99999999999999921e80 < (*.f64 c i) < 4.9999999999999997e161

   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 x around inf

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

   5. Applied rewrites 75.3%

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

   if 4.9999999999999997e161 < (*.f64 c i)

   1. Initial program 86.8%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 89.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 80.9%

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

Alternative 6: 65.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, a, c \cdot i\right)\\ \mathbf{if}\;c \cdot i \leq -2 \cdot 10^{+148}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \cdot i \leq 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(b, a, z \cdot t\right)\\ \mathbf{elif}\;c \cdot i \leq 5 \cdot 10^{+161}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma b a (* c i))))
   (if (<= (* c i) -2e+148)
     t_1
     (if (<= (* c i) 1e+81)
       (fma b a (* z t))
       (if (<= (* c i) 5e+161) (* y x) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(b, a, (c * i));
	double tmp;
	if ((c * i) <= -2e+148) {
		tmp = t_1;
	} else if ((c * i) <= 1e+81) {
		tmp = fma(b, a, (z * t));
	} else if ((c * i) <= 5e+161) {
		tmp = y * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(b, a, Float64(c * i))
	tmp = 0.0
	if (Float64(c * i) <= -2e+148)
		tmp = t_1;
	elseif (Float64(c * i) <= 1e+81)
		tmp = fma(b, a, Float64(z * t));
	elseif (Float64(c * i) <= 5e+161)
		tmp = Float64(y * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(b * a + N[(c * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(c * i), $MachinePrecision], -2e+148], t$95$1, If[LessEqual[N[(c * i), $MachinePrecision], 1e+81], N[(b * a + N[(z * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 5e+161], N[(y * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c i) < -2.0000000000000001e148 or 4.9999999999999997e161 < (*.f64 c i)

   1. Initial program 89.2%

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

   3. Taylor expanded in 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 88.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 80.2%

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

   if -2.0000000000000001e148 < (*.f64 c i) < 9.99999999999999921e80

   1. Initial program 97.0%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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 70.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 65.6%

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

   if 9.99999999999999921e80 < (*.f64 c i) < 4.9999999999999997e161

   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 x around inf

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

   5. Applied rewrites 75.3%

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

Alternative 7: 88.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -1 \cdot 10^{+104} \lor \neg \left(c \cdot i \leq 5 \cdot 10^{+161}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(t, z, i \cdot c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(t, z, y \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* c i) -1e+104) (not (<= (* c i) 5e+161)))
   (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) <= -1e+104) || !((c * i) <= 5e+161)) {
		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) <= -1e+104) || !(Float64(c * i) <= 5e+161))
		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], -1e+104], N[Not[LessEqual[N[(c * i), $MachinePrecision], 5e+161]], $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) < -1e104 or 4.9999999999999997e161 < (*.f64 c i)

   1. Initial program 88.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 87.7%

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

   if -1e104 < (*.f64 c i) < 4.9999999999999997e161

   1. Initial program 97.7%

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

   3. Taylor expanded in 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 95.6%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -1 \cdot 10^{+104} \lor \neg \left(c \cdot i \leq 5 \cdot 10^{+161}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(t, z, i \cdot c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(t, z, y \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(y, x, a \cdot b\right)\\ \mathbf{elif}\;x \cdot y \leq 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(t, z, i \cdot c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* x y) -2e+75)
   (fma y x (* a b))
   (if (<= (* x y) 1e+114) (fma b a (fma t z (* i c))) (fma y x (* z t)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((x * y) <= -2e+75) {
		tmp = fma(y, x, (a * b));
	} else if ((x * y) <= 1e+114) {
		tmp = fma(b, a, fma(t, z, (i * c)));
	} else {
		tmp = fma(y, x, (z * t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(x * y) <= -2e+75)
		tmp = fma(y, x, Float64(a * b));
	elseif (Float64(x * y) <= 1e+114)
		tmp = fma(b, a, fma(t, z, Float64(i * c)));
	else
		tmp = fma(y, x, Float64(z * t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e+75], N[(y * x + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e+114], N[(b * a + N[(t * z + N[(i * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x + N[(z * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1.99999999999999985e75

   1. Initial program 89.6%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. associate--l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. associate-+r+ N/A

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

      14. lift-*.f64 N/A

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

      15. remove-double-neg N/A

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

      16. remove-double-neg N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. +-commutative N/A

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

   4. Applied rewrites 95.8%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 73.9%

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

   if -1.99999999999999985e75 < (*.f64 x y) < 1e114

   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. 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 94.0%

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

   if 1e114 < (*.f64 x y)

   1. Initial program 89.8%

      \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
   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. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. associate--l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. associate-+r+ N/A

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

      14. lift-*.f64 N/A

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

      15. remove-double-neg N/A

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

      16. remove-double-neg N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. +-commutative N/A

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 84.3%

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

Alternative 9: 42.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+147}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;z \cdot t \leq 10^{-320}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+42}:\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* z t) -1e+147)
   (* t z)
   (if (<= (* z t) 1e-320) (* y x) (if (<= (* z t) 5e+42) (* b a) (* t z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((z * t) <= -1e+147) {
		tmp = t * z;
	} else if ((z * t) <= 1e-320) {
		tmp = y * x;
	} else if ((z * t) <= 5e+42) {
		tmp = b * a;
	} else {
		tmp = t * z;
	}
	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 ((z * t) <= (-1d+147)) then
        tmp = t * z
    else if ((z * t) <= 1d-320) then
        tmp = y * x
    else if ((z * t) <= 5d+42) then
        tmp = b * a
    else
        tmp = t * z
    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 ((z * t) <= -1e+147) {
		tmp = t * z;
	} else if ((z * t) <= 1e-320) {
		tmp = y * x;
	} else if ((z * t) <= 5e+42) {
		tmp = b * a;
	} else {
		tmp = t * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (z * t) <= -1e+147:
		tmp = t * z
	elif (z * t) <= 1e-320:
		tmp = y * x
	elif (z * t) <= 5e+42:
		tmp = b * a
	else:
		tmp = t * z
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(z * t) <= -1e+147)
		tmp = Float64(t * z);
	elseif (Float64(z * t) <= 1e-320)
		tmp = Float64(y * x);
	elseif (Float64(z * t) <= 5e+42)
		tmp = Float64(b * a);
	else
		tmp = Float64(t * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((z * t) <= -1e+147)
		tmp = t * z;
	elseif ((z * t) <= 1e-320)
		tmp = y * x;
	elseif ((z * t) <= 5e+42)
		tmp = b * a;
	else
		tmp = t * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(z * t), $MachinePrecision], -1e+147], N[(t * z), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e-320], N[(y * x), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+42], N[(b * a), $MachinePrecision], N[(t * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -9.9999999999999998e146 or 5.00000000000000007e42 < (*.f64 z t)

   1. Initial program 89.7%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 61.6%

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

   if -9.9999999999999998e146 < (*.f64 z t) < 9.99989e-321

   1. Initial program 97.1%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 44.7%

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

   if 9.99989e-321 < (*.f64 z t) < 5.00000000000000007e42

   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 46.9%

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

Alternative 10: 41.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -2 \cdot 10^{+120}:\\ \;\;\;\;i \cdot c\\ \mathbf{elif}\;c \cdot i \leq -5 \cdot 10^{-303}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;c \cdot i \leq 5 \cdot 10^{+231}:\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;i \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -2e+120)
   (* i c)
   (if (<= (* c i) -5e-303) (* b a) (if (<= (* c i) 5e+231) (* t z) (* i c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((c * i) <= -2e+120) {
		tmp = i * c;
	} else if ((c * i) <= -5e-303) {
		tmp = b * a;
	} else if ((c * i) <= 5e+231) {
		tmp = t * z;
	} else {
		tmp = i * c;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if ((c * i) <= (-2d+120)) then
        tmp = i * c
    else if ((c * i) <= (-5d-303)) then
        tmp = b * a
    else if ((c * i) <= 5d+231) then
        tmp = t * z
    else
        tmp = i * c
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((c * i) <= -2e+120) {
		tmp = i * c;
	} else if ((c * i) <= -5e-303) {
		tmp = b * a;
	} else if ((c * i) <= 5e+231) {
		tmp = t * z;
	} else {
		tmp = i * c;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (c * i) <= -2e+120:
		tmp = i * c
	elif (c * i) <= -5e-303:
		tmp = b * a
	elif (c * i) <= 5e+231:
		tmp = t * z
	else:
		tmp = i * c
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(c * i) <= -2e+120)
		tmp = Float64(i * c);
	elseif (Float64(c * i) <= -5e-303)
		tmp = Float64(b * a);
	elseif (Float64(c * i) <= 5e+231)
		tmp = Float64(t * z);
	else
		tmp = Float64(i * c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((c * i) <= -2e+120)
		tmp = i * c;
	elseif ((c * i) <= -5e-303)
		tmp = b * a;
	elseif ((c * i) <= 5e+231)
		tmp = t * z;
	else
		tmp = i * c;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(c * i), $MachinePrecision], -2e+120], N[(i * c), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], -5e-303], N[(b * a), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 5e+231], N[(t * z), $MachinePrecision], N[(i * c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c i) < -2e120 or 5.00000000000000028e231 < (*.f64 c i)

   1. Initial program 90.6%

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

   3. Taylor expanded in c around inf

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

   5. Applied rewrites 73.3%

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

   if -2e120 < (*.f64 c i) < -4.9999999999999998e-303

   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 a around inf

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

   5. Applied rewrites 43.4%

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

   if -4.9999999999999998e-303 < (*.f64 c i) < 5.00000000000000028e231

   1. Initial program 96.4%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 37.7%

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

Alternative 11: 62.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+75} \lor \neg \left(x \cdot y \leq 10^{+114}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, c \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* x y) -2e+75) (not (<= (* x y) 1e+114)))
   (* y x)
   (fma b a (* 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) <= -2e+75) || !((x * y) <= 1e+114)) {
		tmp = y * x;
	} else {
		tmp = fma(b, a, (c * i));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(x * y) <= -2e+75) || !(Float64(x * y) <= 1e+114))
		tmp = Float64(y * x);
	else
		tmp = fma(b, a, Float64(c * i));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -2e+75], N[Not[LessEqual[N[(x * y), $MachinePrecision], 1e+114]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(b * a + N[(c * i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.99999999999999985e75 or 1e114 < (*.f64 x y)

   1. Initial program 89.7%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 63.7%

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

   if -1.99999999999999985e75 < (*.f64 x y) < 1e114

   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. 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 94.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 63.9%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+75} \lor \neg \left(x \cdot y \leq 10^{+114}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, c \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 41.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -2 \cdot 10^{+120} \lor \neg \left(c \cdot i \leq 10^{+291}\right):\\ \;\;\;\;i \cdot c\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* c i) -2e+120) (not (<= (* c i) 1e+291))) (* 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) <= -2e+120) || !((c * i) <= 1e+291)) {
		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) <= (-2d+120)) .or. (.not. ((c * i) <= 1d+291))) 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) <= -2e+120) || !((c * i) <= 1e+291)) {
		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) <= -2e+120) or not ((c * i) <= 1e+291):
		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) <= -2e+120) || !(Float64(c * i) <= 1e+291))
		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) <= -2e+120) || ~(((c * i) <= 1e+291)))
		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], -2e+120], N[Not[LessEqual[N[(c * i), $MachinePrecision], 1e+291]], $MachinePrecision]], N[(i * c), $MachinePrecision], N[(b * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c i) < -2e120 or 9.9999999999999996e290 < (*.f64 c i)

   1. Initial program 88.9%

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

   3. Taylor expanded in c around inf

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

   5. Applied rewrites 80.0%

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

   if -2e120 < (*.f64 c i) < 9.9999999999999996e290

   1. Initial program 96.5%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 34.6%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -2 \cdot 10^{+120} \lor \neg \left(c \cdot i \leq 10^{+291}\right):\\ \;\;\;\;i \cdot c\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 27.9% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.9%

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

3. Taylor expanded in a around inf

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

5. Applied rewrites 29.4%

   \[\leadsto \color{blue}{b \cdot a} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025022 
(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)))