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

Percentage Accurate: 95.6% → 97.8%

Time: 7.8s

Alternatives: 14

Speedup: 0.5×

• 41.8% 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: 95.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (+ (+ (+ (* x y) (* z t)) (* a b)) (* c i)) INFINITY)
   (fma i c (fma b a (fma t z (* y x))))
   (fma i c (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 (((((x * y) + (z * t)) + (a * b)) + (c * i)) <= ((double) INFINITY)) {
		tmp = fma(i, c, fma(b, a, fma(t, z, (y * x))));
	} else {
		tmp = fma(i, c, fma(b, a, (z * t)));
	}
	return tmp;
}

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (*.f64 x y) (*.f64 z t)) (*.f64 a b)) (*.f64 c i)) < +inf.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

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   if +inf.0 < (+.f64 (+.f64 (+.f64 (*.f64 x y) (*.f64 z t)) (*.f64 a b)) (*.f64 c i))

   1. Initial program 0.0%

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

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 46.3

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

   4. Applied rewrites 46.3%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 49.4

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

   7. Applied rewrites 49.4%

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

Alternative 2: 42.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+188}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{+42}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-150}:\\ \;\;\;\;i \cdot c\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+96}:\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* x y) -5e+188)
   (* y x)
   (if (<= (* x y) -2e+42)
     (* b a)
     (if (<= (* x y) 5e-150)
       (* i c)
       (if (<= (* x y) 2e+96) (* 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 ((x * y) <= -5e+188) {
		tmp = y * x;
	} else if ((x * y) <= -2e+42) {
		tmp = b * a;
	} else if ((x * y) <= 5e-150) {
		tmp = i * c;
	} else if ((x * y) <= 2e+96) {
		tmp = t * z;
	} else {
		tmp = y * x;
	}
	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 ((x * y) <= (-5d+188)) then
        tmp = y * x
    else if ((x * y) <= (-2d+42)) then
        tmp = b * a
    else if ((x * y) <= 5d-150) then
        tmp = i * c
    else if ((x * y) <= 2d+96) then
        tmp = t * z
    else
        tmp = y * x
    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 ((x * y) <= -5e+188) {
		tmp = y * x;
	} else if ((x * y) <= -2e+42) {
		tmp = b * a;
	} else if ((x * y) <= 5e-150) {
		tmp = i * c;
	} else if ((x * y) <= 2e+96) {
		tmp = t * z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (x * y) <= -5e+188:
		tmp = y * x
	elif (x * y) <= -2e+42:
		tmp = b * a
	elif (x * y) <= 5e-150:
		tmp = i * c
	elif (x * y) <= 2e+96:
		tmp = t * z
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(x * y) <= -5e+188)
		tmp = Float64(y * x);
	elseif (Float64(x * y) <= -2e+42)
		tmp = Float64(b * a);
	elseif (Float64(x * y) <= 5e-150)
		tmp = Float64(i * c);
	elseif (Float64(x * y) <= 2e+96)
		tmp = Float64(t * z);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((x * y) <= -5e+188)
		tmp = y * x;
	elseif ((x * y) <= -2e+42)
		tmp = b * a;
	elseif ((x * y) <= 5e-150)
		tmp = i * c;
	elseif ((x * y) <= 2e+96)
		tmp = t * z;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(x * y), $MachinePrecision], -5e+188], N[(y * x), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -2e+42], N[(b * a), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e-150], N[(i * c), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e+96], N[(t * z), $MachinePrecision], N[(y * x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -5.0000000000000001e188 or 2.0000000000000001e96 < (*.f64 x y)

   1. Initial program 91.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 65.4

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

   5. Applied rewrites 65.4%

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

   if -5.0000000000000001e188 < (*.f64 x y) < -2.00000000000000009e42

   1. Initial program 98.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 24.9

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

   5. Applied rewrites 24.9%

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

   if -2.00000000000000009e42 < (*.f64 x y) < 4.9999999999999999e-150

   1. Initial program 97.2%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 35.3

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

   5. Applied rewrites 35.3%

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

   if 4.9999999999999999e-150 < (*.f64 x y) < 2.0000000000000001e96

   1. Initial program 97.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 31.1

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

   5. Applied rewrites 31.1%

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

Alternative 3: 77.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, y, t \cdot z\right)\\ t_2 := x \cdot y + z \cdot t\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+158}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+129}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma x y (* t z))) (t_2 (+ (* x y) (* z t))))
   (if (<= t_2 -2e+158) t_1 (if (<= t_2 2e+129) (fma i c (* a b)) 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(x, y, (t * z));
	double t_2 = (x * y) + (z * t);
	double tmp;
	if (t_2 <= -2e+158) {
		tmp = t_1;
	} else if (t_2 <= 2e+129) {
		tmp = fma(i, c, (a * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(x, y, Float64(t * z))
	t_2 = Float64(Float64(x * y) + Float64(z * t))
	tmp = 0.0
	if (t_2 <= -2e+158)
		tmp = t_1;
	elseif (t_2 <= 2e+129)
		tmp = fma(i, c, Float64(a * b));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x * y + N[(t * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+158], t$95$1, If[LessEqual[t$95$2, 2e+129], N[(i * c + N[(a * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x y) (*.f64 z t)) < -1.99999999999999991e158 or 2e129 < (+.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 92.2%

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

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 95.7

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

   4. Applied rewrites 95.7%

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

   5. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

      12. *-commutative N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-*.f64 87.3

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

   7. Applied rewrites 87.3%

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

   8. Taylor expanded in z around inf

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

      1. lower-*.f64 77.2

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

   10. Applied rewrites 77.2%

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

   if -1.99999999999999991e158 < (+.f64 (*.f64 x y) (*.f64 z t)) < 2e129

   1. Initial program 99.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 76.3

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

   5. Applied rewrites 76.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative 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 76.8

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 76.8

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

   7. Applied rewrites 76.8%

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

Alternative 4: 88.8% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma x y (fma z t (* a b)))))
   (if (<= (* a b) -2e+116)
     t_1
     (if (<= (* a b) 5e-10) (fma i c (fma x y (* z t))) 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(x, y, fma(z, t, (a * b)));
	double tmp;
	if ((a * b) <= -2e+116) {
		tmp = t_1;
	} else if ((a * b) <= 5e-10) {
		tmp = fma(i, c, fma(x, y, (z * t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(x, y, fma(z, t, Float64(a * b)))
	tmp = 0.0
	if (Float64(a * b) <= -2e+116)
		tmp = t_1;
	elseif (Float64(a * b) <= 5e-10)
		tmp = fma(i, c, fma(x, y, Float64(z * t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x * y + N[(z * t + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -2e+116], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], 5e-10], N[(i * c + N[(x * y + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -2.00000000000000003e116 or 5.00000000000000031e-10 < (*.f64 a b)

   1. Initial program 92.9%

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

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 96.3

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

   4. Applied rewrites 96.3%

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

   5. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

      12. *-commutative N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-*.f64 83.6

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

   7. Applied rewrites 83.6%

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

   if -2.00000000000000003e116 < (*.f64 a b) < 5.00000000000000031e-10

   1. Initial program 97.6%

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

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 98.7

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 92.7

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

   7. Applied rewrites 92.7%

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

Alternative 5: 88.8% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma x y (fma z t (* a b)))))
   (if (<= (* a b) -2e+116)
     t_1
     (if (<= (* a b) 5e-10) (fma i c (fma t z (* 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(x, y, fma(z, t, (a * b)));
	double tmp;
	if ((a * b) <= -2e+116) {
		tmp = t_1;
	} else if ((a * b) <= 5e-10) {
		tmp = fma(i, c, fma(t, z, (y * x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(x, y, fma(z, t, Float64(a * b)))
	tmp = 0.0
	if (Float64(a * b) <= -2e+116)
		tmp = t_1;
	elseif (Float64(a * b) <= 5e-10)
		tmp = fma(i, c, fma(t, z, 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[(x * y + N[(z * t + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -2e+116], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], 5e-10], N[(i * c + N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -2.00000000000000003e116 or 5.00000000000000031e-10 < (*.f64 a b)

   1. Initial program 92.9%

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

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 96.3

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

   4. Applied rewrites 96.3%

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

   5. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

      12. *-commutative N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-*.f64 83.6

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

   7. Applied rewrites 83.6%

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

   if -2.00000000000000003e116 < (*.f64 a b) < 5.00000000000000031e-10

   1. Initial program 97.6%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 92.7

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

   5. Applied rewrites 92.7%

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

Alternative 6: 86.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, z, y \cdot x\right)\\ \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{+238}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(i, c, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma t z (* y x))))
   (if (<= (* a b) -1e+238)
     (fma b a (* t z))
     (if (<= (* a b) 5e-10) (fma i c t_1) (fma b a 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(t, z, (y * x));
	double tmp;
	if ((a * b) <= -1e+238) {
		tmp = fma(b, a, (t * z));
	} else if ((a * b) <= 5e-10) {
		tmp = fma(i, c, t_1);
	} else {
		tmp = fma(b, a, t_1);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(t, z, Float64(y * x))
	tmp = 0.0
	if (Float64(a * b) <= -1e+238)
		tmp = fma(b, a, Float64(t * z));
	elseif (Float64(a * b) <= 5e-10)
		tmp = fma(i, c, t_1);
	else
		tmp = fma(b, a, t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -1e+238], N[(b * a + N[(t * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 5e-10], N[(i * c + t$95$1), $MachinePrecision], N[(b * a + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -1e238

   1. Initial program 86.3%

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

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 92.5

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

   4. Applied rewrites 92.5%

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

   5. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

      12. *-commutative N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-*.f64 91.1

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

   7. Applied rewrites 91.1%

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

   8. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. *-commutative N/A

         \[\leadsto a \cdot b + t \cdot z \]

      3. +-commutative N/A

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

      4. *-commutative N/A

         \[\leadsto b \cdot a + t \cdot z \]

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 86.3

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

   10. Applied rewrites 86.3%

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

   if -1e238 < (*.f64 a b) < 5.00000000000000031e-10

   1. Initial program 97.6%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 88.8

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

   5. Applied rewrites 88.8%

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

   if 5.00000000000000031e-10 < (*.f64 a b)

   1. Initial program 94.0%

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 81.1

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

   5. Applied rewrites 81.1%

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

Alternative 7: 88.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, a, \mathsf{fma}\left(t, z, y \cdot x\right)\right)\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, t \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma b a (fma t z (* y x)))))
   (if (<= (* x y) -2e-17)
     t_1
     (if (<= (* x y) 1e+32) (fma b a (fma i c (* t z))) 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, fma(t, z, (y * x)));
	double tmp;
	if ((x * y) <= -2e-17) {
		tmp = t_1;
	} else if ((x * y) <= 1e+32) {
		tmp = fma(b, a, fma(i, c, (t * z)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(b, a, fma(t, z, Float64(y * x)))
	tmp = 0.0
	if (Float64(x * y) <= -2e-17)
		tmp = t_1;
	elseif (Float64(x * y) <= 1e+32)
		tmp = fma(b, a, fma(i, c, Float64(t * z)));
	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[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -2e-17], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 1e+32], N[(b * a + N[(i * c + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2.00000000000000014e-17 or 1.00000000000000005e32 < (*.f64 x y)

   1. Initial program 93.8%

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 81.9

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

   5. Applied rewrites 81.9%

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

   if -2.00000000000000014e-17 < (*.f64 x y) < 1.00000000000000005e32

   1. Initial program 97.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 95.0

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

   5. Applied rewrites 95.0%

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

Alternative 8: 86.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, y, t \cdot z\right)\\ \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+188}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+158}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, t \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma x y (* t z))))
   (if (<= (* x y) -5e+188)
     t_1
     (if (<= (* x y) 2e+158) (fma b a (fma i c (* t z))) 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(x, y, (t * z));
	double tmp;
	if ((x * y) <= -5e+188) {
		tmp = t_1;
	} else if ((x * y) <= 2e+158) {
		tmp = fma(b, a, fma(i, c, (t * z)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(x, y, Float64(t * z))
	tmp = 0.0
	if (Float64(x * y) <= -5e+188)
		tmp = t_1;
	elseif (Float64(x * y) <= 2e+158)
		tmp = fma(b, a, fma(i, c, Float64(t * z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x * y + N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -5e+188], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 2e+158], N[(b * a + N[(i * c + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -5.0000000000000001e188 or 1.99999999999999991e158 < (*.f64 x y)

   1. Initial program 90.2%

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

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 94.7

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

   4. Applied rewrites 94.7%

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

   5. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

      12. *-commutative N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-*.f64 88.9

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

   7. Applied rewrites 88.9%

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

   8. Taylor expanded in z around inf

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

      1. lower-*.f64 80.1

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

   10. Applied rewrites 80.1%

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

   if -5.0000000000000001e188 < (*.f64 x y) < 1.99999999999999991e158

   1. Initial program 97.4%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 88.7

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

   5. Applied rewrites 88.7%

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

Alternative 9: 44.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+113}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;z \cdot t \leq 0:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;z \cdot t \leq 10^{+95}:\\ \;\;\;\;i \cdot c\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* z t) -5e+113)
   (* t z)
   (if (<= (* z t) 0.0) (* b a) (if (<= (* z t) 1e+95) (* i c) (* t z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((z * t) <= -5e+113) {
		tmp = t * z;
	} else if ((z * t) <= 0.0) {
		tmp = b * a;
	} else if ((z * t) <= 1e+95) {
		tmp = i * c;
	} 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) <= (-5d+113)) then
        tmp = t * z
    else if ((z * t) <= 0.0d0) then
        tmp = b * a
    else if ((z * t) <= 1d+95) then
        tmp = i * c
    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) <= -5e+113) {
		tmp = t * z;
	} else if ((z * t) <= 0.0) {
		tmp = b * a;
	} else if ((z * t) <= 1e+95) {
		tmp = i * c;
	} else {
		tmp = t * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (z * t) <= -5e+113:
		tmp = t * z
	elif (z * t) <= 0.0:
		tmp = b * a
	elif (z * t) <= 1e+95:
		tmp = i * c
	else:
		tmp = t * z
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(z * t) <= -5e+113)
		tmp = Float64(t * z);
	elseif (Float64(z * t) <= 0.0)
		tmp = Float64(b * a);
	elseif (Float64(z * t) <= 1e+95)
		tmp = Float64(i * c);
	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) <= -5e+113)
		tmp = t * z;
	elseif ((z * t) <= 0.0)
		tmp = b * a;
	elseif ((z * t) <= 1e+95)
		tmp = i * c;
	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], -5e+113], N[(t * z), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 0.0], N[(b * a), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+95], N[(i * c), $MachinePrecision], N[(t * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -5e113 or 1.00000000000000002e95 < (*.f64 z t)

   1. Initial program 91.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 63.0

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

   5. Applied rewrites 63.0%

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

   if -5e113 < (*.f64 z t) < -0.0

   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 a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 34.4

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

   5. Applied rewrites 34.4%

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

   if -0.0 < (*.f64 z t) < 1.00000000000000002e95

   1. Initial program 97.6%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 33.7

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

   5. Applied rewrites 33.7%

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

Alternative 10: 66.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+126}:\\ \;\;\;\;\mathsf{fma}\left(x, y, t \cdot z\right)\\ \mathbf{elif}\;z \cdot t \leq 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(x, y, b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* z t) -2e+126)
   (fma x y (* t z))
   (if (<= (* z t) 1e+95) (fma x y (* b a)) (fma b a (* t z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((z * t) <= -2e+126) {
		tmp = fma(x, y, (t * z));
	} else if ((z * t) <= 1e+95) {
		tmp = fma(x, y, (b * a));
	} else {
		tmp = fma(b, a, (t * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(z * t) <= -2e+126)
		tmp = fma(x, y, Float64(t * z));
	elseif (Float64(z * t) <= 1e+95)
		tmp = fma(x, y, Float64(b * a));
	else
		tmp = fma(b, a, Float64(t * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(z * t), $MachinePrecision], -2e+126], N[(x * y + N[(t * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+95], N[(x * y + N[(b * a), $MachinePrecision]), $MachinePrecision], N[(b * a + N[(t * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -1.99999999999999985e126

   1. Initial program 93.2%

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

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 96.5

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

   4. Applied rewrites 96.5%

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

   5. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

      12. *-commutative N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-*.f64 86.4

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

   7. Applied rewrites 86.4%

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

   8. Taylor expanded in z around inf

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

      1. lower-*.f64 77.7

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

   10. Applied rewrites 77.7%

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

   if -1.99999999999999985e126 < (*.f64 z t) < 1.00000000000000002e95

   1. Initial program 97.6%

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

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 98.8

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

      12. *-commutative N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-*.f64 68.9

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

   7. Applied rewrites 68.9%

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

   8. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 62.2

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

   10. Applied rewrites 62.2%

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

   if 1.00000000000000002e95 < (*.f64 z t)

   1. Initial program 90.3%

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

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 94.6

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

   4. Applied rewrites 94.6%

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

   5. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

      12. *-commutative N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-*.f64 85.2

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

   7. Applied rewrites 85.2%

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

   8. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. *-commutative N/A

         \[\leadsto a \cdot b + t \cdot z \]

      3. +-commutative N/A

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

      4. *-commutative N/A

         \[\leadsto b \cdot a + t \cdot z \]

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 73.5

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

   10. Applied rewrites 73.5%

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

Alternative 11: 66.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, a, t \cdot z\right)\\ \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+126}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(x, y, b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma b a (* t z))))
   (if (<= (* z t) -2e+126)
     t_1
     (if (<= (* z t) 1e+95) (fma x y (* b a)) 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, (t * z));
	double tmp;
	if ((z * t) <= -2e+126) {
		tmp = t_1;
	} else if ((z * t) <= 1e+95) {
		tmp = fma(x, y, (b * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(b, a, Float64(t * z))
	tmp = 0.0
	if (Float64(z * t) <= -2e+126)
		tmp = t_1;
	elseif (Float64(z * t) <= 1e+95)
		tmp = fma(x, y, Float64(b * a));
	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[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -2e+126], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 1e+95], N[(x * y + N[(b * a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.99999999999999985e126 or 1.00000000000000002e95 < (*.f64 z t)

   1. Initial program 91.6%

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

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 95.5

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

   4. Applied rewrites 95.5%

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

   5. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

      12. *-commutative N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-*.f64 85.8

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

   7. Applied rewrites 85.8%

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

   8. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. *-commutative N/A

         \[\leadsto a \cdot b + t \cdot z \]

      3. +-commutative N/A

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

      4. *-commutative N/A

         \[\leadsto b \cdot a + t \cdot z \]

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 74.9

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

   10. Applied rewrites 74.9%

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

   if -1.99999999999999985e126 < (*.f64 z t) < 1.00000000000000002e95

   1. Initial program 97.6%

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

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 98.8

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

      12. *-commutative N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-*.f64 68.9

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

   7. Applied rewrites 68.9%

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

   8. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 62.2

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

   10. Applied rewrites 62.2%

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

Alternative 12: 62.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+206}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \cdot y \leq 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* x y) -5e+206)
   (* y x)
   (if (<= (* x y) 1e+137) (fma b a (* 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 ((x * y) <= -5e+206) {
		tmp = y * x;
	} else if ((x * y) <= 1e+137) {
		tmp = fma(b, a, (t * z));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(x * y) <= -5e+206)
		tmp = Float64(y * x);
	elseif (Float64(x * y) <= 1e+137)
		tmp = fma(b, a, Float64(t * z));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(x * y), $MachinePrecision], -5e+206], N[(y * x), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e+137], N[(b * a + N[(t * z), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -5.0000000000000002e206 or 1e137 < (*.f64 x y)

   1. Initial program 90.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 70.0

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

   5. Applied rewrites 70.0%

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

   if -5.0000000000000002e206 < (*.f64 x y) < 1e137

   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

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

      12. *-commutative N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-*.f64 69.7

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

   7. Applied rewrites 69.7%

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

   8. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. *-commutative N/A

         \[\leadsto a \cdot b + t \cdot z \]

      3. +-commutative N/A

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

      4. *-commutative N/A

         \[\leadsto b \cdot a + t \cdot z \]

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 60.3

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

   10. Applied rewrites 60.3%

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

Alternative 13: 43.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+116}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;a \cdot b \leq 10:\\ \;\;\;\;i \cdot c\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* a b) -2e+116) (* b a) (if (<= (* a b) 10.0) (* i c) (* b a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((a * b) <= -2e+116) {
		tmp = b * a;
	} else if ((a * b) <= 10.0) {
		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 ((a * b) <= (-2d+116)) then
        tmp = b * a
    else if ((a * b) <= 10.0d0) 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 ((a * b) <= -2e+116) {
		tmp = b * a;
	} else if ((a * b) <= 10.0) {
		tmp = i * c;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (a * b) <= -2e+116:
		tmp = b * a
	elif (a * b) <= 10.0:
		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(a * b) <= -2e+116)
		tmp = Float64(b * a);
	elseif (Float64(a * b) <= 10.0)
		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 ((a * b) <= -2e+116)
		tmp = b * a;
	elseif ((a * b) <= 10.0)
		tmp = i * c;
	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], -2e+116], N[(b * a), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 10.0], N[(i * c), $MachinePrecision], N[(b * a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -2.00000000000000003e116 or 10 < (*.f64 a b)

   1. Initial program 92.8%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 56.0

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

   5. Applied rewrites 56.0%

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

   if -2.00000000000000003e116 < (*.f64 a b) < 10

   1. Initial program 97.6%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 34.1

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

   5. Applied rewrites 34.1%

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

Alternative 14: 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 95.6%

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

3. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 27.9

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

5. Applied rewrites 27.9%

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

Reproduce

herbie shell --seed 2025091 
(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)))