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

Percentage Accurate: 95.6% → 97.7%

Time: 4.0s

Alternatives: 14

Speedup: 1.1×

• 40.9% 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: 97.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 95.6%

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

   1. lift-*.f64 N/A

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

   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

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

   4. lift-+.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-+.f64 N/A

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

   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

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

   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 97.7

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

3. Applied rewrites 97.7%

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

Alternative 2: 89.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, y, \mathsf{fma}\left(z, t, a \cdot b\right)\right)\\ \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+118}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 6 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(a, b, \mathsf{fma}\left(z, t, c \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma x y (fma z t (* a b)))))
   (if (<= (* x y) -1e+118)
     t_1
     (if (<= (* x y) 6e+23) (fma a b (fma z t (* c i))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(x, y, fma(z, t, (a * b)));
	double tmp;
	if ((x * y) <= -1e+118) {
		tmp = t_1;
	} else if ((x * y) <= 6e+23) {
		tmp = fma(a, b, fma(z, t, (c * i)));
	} 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(x * y) <= -1e+118)
		tmp = t_1;
	elseif (Float64(x * y) <= 6e+23)
		tmp = fma(a, b, fma(z, t, Float64(c * i)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x * y + N[(z * t + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -1e+118], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 6e+23], N[(a * b + N[(z * t + N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -9.99999999999999967e117 or 6.0000000000000002e23 < (*.f64 x y)

   1. Initial program 92.1%

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

      1. lift-*.f64 N/A

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

      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

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      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

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

      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 95.6

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

   3. Applied rewrites 95.6%

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

   4. Taylor expanded in c around 0

      \[\leadsto \color{blue}{a \cdot b + \left(t \cdot z + x \cdot y\right)} \]
   5. 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 85.0

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

   6. Applied rewrites 85.0%

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

   if -9.99999999999999967e117 < (*.f64 x y) < 6.0000000000000002e23

   1. Initial program 97.7%

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

      1. lift-*.f64 N/A

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

      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

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      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

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

      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 98.9

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

   3. Applied rewrites 98.9%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 92.5

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

   6. Applied rewrites 92.5%

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

Alternative 3: 89.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, z, y \cdot x\right)\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(i, c, t\_1\right)\\ \mathbf{elif}\;x \cdot y \leq 6 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(a, b, \mathsf{fma}\left(z, t, c \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma t z (* y x))))
   (if (<= (* x y) -2e+95)
     (fma i c t_1)
     (if (<= (* x y) 6e+23) (fma a b (fma z t (* c i))) (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 ((x * y) <= -2e+95) {
		tmp = fma(i, c, t_1);
	} else if ((x * y) <= 6e+23) {
		tmp = fma(a, b, fma(z, t, (c * i)));
	} 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(x * y) <= -2e+95)
		tmp = fma(i, c, t_1);
	elseif (Float64(x * y) <= 6e+23)
		tmp = fma(a, b, fma(z, t, Float64(c * i)));
	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[(x * y), $MachinePrecision], -2e+95], N[(i * c + t$95$1), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 6e+23], N[(a * b + N[(z * t + N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * a + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2.00000000000000004e95

   1. Initial program 91.6%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c \cdot i + \left(t \cdot z + x \cdot y\right)} \]
   3. 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

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

      3. +-commutative N/A

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

      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

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

      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 85.3

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

   4. Applied rewrites 85.3%

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

   if -2.00000000000000004e95 < (*.f64 x y) < 6.0000000000000002e23

   1. Initial program 97.7%

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

      1. lift-*.f64 N/A

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

      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

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      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

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

      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 98.9

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

   3. Applied rewrites 98.9%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 93.2

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

   6. Applied rewrites 93.2%

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

   if 6.0000000000000002e23 < (*.f64 x y)

   1. Initial program 93.0%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      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

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

      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 83.1

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

   4. Applied rewrites 83.1%

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

Alternative 4: 89.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, a, \mathsf{fma}\left(t, z, y \cdot x\right)\right)\\ \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+118}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 6 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(a, b, \mathsf{fma}\left(z, t, c \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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) -1e+118)
     t_1
     (if (<= (* x y) 6e+23) (fma a b (fma z t (* c i))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(b, a, fma(t, z, (y * x)));
	double tmp;
	if ((x * y) <= -1e+118) {
		tmp = t_1;
	} else if ((x * y) <= 6e+23) {
		tmp = fma(a, b, fma(z, t, (c * i)));
	} 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) <= -1e+118)
		tmp = t_1;
	elseif (Float64(x * y) <= 6e+23)
		tmp = fma(a, b, fma(z, t, Float64(c * i)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(b * a + N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -1e+118], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 6e+23], N[(a * b + N[(z * t + N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -9.99999999999999967e117 or 6.0000000000000002e23 < (*.f64 x y)

   1. Initial program 92.1%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      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

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

      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 84.2

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

   4. Applied rewrites 84.2%

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

   if -9.99999999999999967e117 < (*.f64 x y) < 6.0000000000000002e23

   1. Initial program 97.7%

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

      1. lift-*.f64 N/A

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

      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

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      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

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

      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 98.9

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

   3. Applied rewrites 98.9%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 92.5

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

   6. Applied rewrites 92.5%

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

Alternative 5: 85.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(x, y, b \cdot a\right)\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+124}:\\ \;\;\;\;\mathsf{fma}\left(a, b, \mathsf{fma}\left(z, t, c \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, t \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* x y) -1e+118)
   (fma x y (* b a))
   (if (<= (* x y) 2e+124) (fma a b (fma z t (* c i))) (fma x y (* t z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((x * y) <= -1e+118) {
		tmp = fma(x, y, (b * a));
	} else if ((x * y) <= 2e+124) {
		tmp = fma(a, b, fma(z, t, (c * i)));
	} else {
		tmp = fma(x, y, (t * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(x * y) <= -1e+118)
		tmp = fma(x, y, Float64(b * a));
	elseif (Float64(x * y) <= 2e+124)
		tmp = fma(a, b, fma(z, t, Float64(c * i)));
	else
		tmp = fma(x, y, Float64(t * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(x * y), $MachinePrecision], -1e+118], N[(x * y + N[(b * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e+124], N[(a * b + N[(z * t + N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y + N[(t * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -9.99999999999999967e117

   1. Initial program 90.8%

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

      1. lift-*.f64 N/A

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

      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

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      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

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

      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 95.0

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

   3. Applied rewrites 95.0%

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

   4. Taylor expanded in c around 0

      \[\leadsto \color{blue}{a \cdot b + \left(t \cdot z + x \cdot y\right)} \]
   5. 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 86.9

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

   6. Applied rewrites 86.9%

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

   7. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 76.3

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

   9. Applied rewrites 76.3%

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

   if -9.99999999999999967e117 < (*.f64 x y) < 1.9999999999999999e124

   1. Initial program 97.7%

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

      1. lift-*.f64 N/A

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

      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

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      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

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

      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

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

   3. Applied rewrites 98.8%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 90.7

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

   6. Applied rewrites 90.7%

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

   if 1.9999999999999999e124 < (*.f64 x y)

   1. Initial program 91.0%

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

      1. lift-*.f64 N/A

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

      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

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      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

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

      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 95.0

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

   3. Applied rewrites 95.0%

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

   4. Taylor expanded in c around 0

      \[\leadsto \color{blue}{a \cdot b + \left(t \cdot z + x \cdot y\right)} \]
   5. 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 86.8

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

   6. Applied rewrites 86.8%

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

   7. Taylor expanded in z around inf

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

      1. lower-*.f64 73.6

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

   9. Applied rewrites 73.6%

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

Alternative 6: 76.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, y, t \cdot z\right)\\ t_2 := x \cdot y + z \cdot t\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+159}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+125}:\\ \;\;\;\;\mathsf{fma}\left(a, b, i \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -1e+159) t_1 (if (<= t_2 5e+125) (fma a b (* i c)) 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 <= -1e+159) {
		tmp = t_1;
	} else if (t_2 <= 5e+125) {
		tmp = fma(a, b, (i * c));
	} 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 <= -1e+159)
		tmp = t_1;
	elseif (t_2 <= 5e+125)
		tmp = fma(a, b, Float64(i * c));
	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, -1e+159], t$95$1, If[LessEqual[t$95$2, 5e+125], N[(a * b + N[(i * c), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x y) (*.f64 z t)) < -9.9999999999999993e158 or 4.99999999999999962e125 < (+.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 91.9%

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

      1. lift-*.f64 N/A

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

      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

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      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

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

      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

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

   3. Applied rewrites 95.7%

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

   4. Taylor expanded in c around 0

      \[\leadsto \color{blue}{a \cdot b + \left(t \cdot z + x \cdot y\right)} \]
   5. 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 87.9

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

   6. Applied rewrites 87.9%

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

   7. Taylor expanded in z around inf

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

      1. lower-*.f64 77.3

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

   9. Applied rewrites 77.3%

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

   if -9.9999999999999993e158 < (+.f64 (*.f64 x y) (*.f64 z t)) < 4.99999999999999962e125

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      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

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      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

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

      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 99.6

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 86.9

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

   6. Applied rewrites 86.9%

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

   7. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 75.0

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

   9. Applied rewrites 75.0%

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

Alternative 7: 67.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, y, b \cdot a\right)\\ \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-216}:\\ \;\;\;\;\mathsf{fma}\left(z, t, b \cdot a\right)\\ \mathbf{elif}\;x \cdot y \leq 6 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(a, b, i \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma x y (* b a))))
   (if (<= (* x y) -1e+85)
     t_1
     (if (<= (* x y) 5e-216)
       (fma z t (* b a))
       (if (<= (* x y) 6e+23) (fma a b (* i c)) 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, (b * a));
	double tmp;
	if ((x * y) <= -1e+85) {
		tmp = t_1;
	} else if ((x * y) <= 5e-216) {
		tmp = fma(z, t, (b * a));
	} else if ((x * y) <= 6e+23) {
		tmp = fma(a, b, (i * c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(x, y, Float64(b * a))
	tmp = 0.0
	if (Float64(x * y) <= -1e+85)
		tmp = t_1;
	elseif (Float64(x * y) <= 5e-216)
		tmp = fma(z, t, Float64(b * a));
	elseif (Float64(x * y) <= 6e+23)
		tmp = fma(a, b, Float64(i * c));
	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[(b * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -1e+85], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 5e-216], N[(z * t + N[(b * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 6e+23], N[(a * b + N[(i * c), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1e85 or 6.0000000000000002e23 < (*.f64 x y)

   1. Initial program 92.4%

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

      1. lift-*.f64 N/A

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

      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

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      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

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

      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

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

   3. Applied rewrites 95.7%

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

   4. Taylor expanded in c around 0

      \[\leadsto \color{blue}{a \cdot b + \left(t \cdot z + x \cdot y\right)} \]
   5. 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 84.5

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

   6. Applied rewrites 84.5%

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

   7. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 71.8

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

   9. Applied rewrites 71.8%

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

   if -1e85 < (*.f64 x y) < 5.00000000000000021e-216

   1. Initial program 97.6%

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

      1. lift-*.f64 N/A

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

      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

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      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

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

      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 98.9

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

   3. Applied rewrites 98.9%

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

   4. Taylor expanded in c around 0

      \[\leadsto \color{blue}{a \cdot b + \left(t \cdot z + x \cdot y\right)} \]
   5. 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 69.4

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

   6. Applied rewrites 69.4%

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

   7. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 64.7

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

   9. Applied rewrites 64.7%

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

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. lift-*.f64 64.8

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

   11. Applied rewrites 64.8%

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

   if 5.00000000000000021e-216 < (*.f64 x y) < 6.0000000000000002e23

   1. Initial program 98.2%

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

      1. lift-*.f64 N/A

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

      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

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      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

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

      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 99.2

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 92.0

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

   6. Applied rewrites 92.0%

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

   7. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 61.4

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

   9. Applied rewrites 61.4%

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

Alternative 8: 63.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, b, i \cdot c\right)\\ \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+159}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-281}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-216}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+124}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma a b (* i c))))
   (if (<= (* x y) -1e+159)
     (* y x)
     (if (<= (* x y) -5e-281)
       t_1
       (if (<= (* x y) 5e-216)
         (fma b a (* t z))
         (if (<= (* x y) 2e+124) t_1 (* y x)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(a, b, (i * c));
	double tmp;
	if ((x * y) <= -1e+159) {
		tmp = y * x;
	} else if ((x * y) <= -5e-281) {
		tmp = t_1;
	} else if ((x * y) <= 5e-216) {
		tmp = fma(b, a, (t * z));
	} else if ((x * y) <= 2e+124) {
		tmp = t_1;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(a, b, Float64(i * c))
	tmp = 0.0
	if (Float64(x * y) <= -1e+159)
		tmp = Float64(y * x);
	elseif (Float64(x * y) <= -5e-281)
		tmp = t_1;
	elseif (Float64(x * y) <= 5e-216)
		tmp = fma(b, a, Float64(t * z));
	elseif (Float64(x * y) <= 2e+124)
		tmp = t_1;
	else
		tmp = Float64(y * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(a * b + N[(i * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -1e+159], N[(y * x), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -5e-281], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 5e-216], N[(b * a + N[(t * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e+124], t$95$1, N[(y * x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -9.9999999999999993e158 or 1.9999999999999999e124 < (*.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. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 67.2

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

   4. Applied rewrites 67.2%

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

   if -9.9999999999999993e158 < (*.f64 x y) < -4.9999999999999998e-281 or 5.00000000000000021e-216 < (*.f64 x y) < 1.9999999999999999e124

   1. Initial program 97.9%

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

      1. lift-*.f64 N/A

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

      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

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      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

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

      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 98.9

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

   3. Applied rewrites 98.9%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 87.0

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

   6. Applied rewrites 87.0%

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

   7. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 59.7

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

   9. Applied rewrites 59.7%

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

   if -4.9999999999999998e-281 < (*.f64 x y) < 5.00000000000000021e-216

   1. Initial program 97.3%

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

      1. lift-*.f64 N/A

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

      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

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      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

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

      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

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

   3. Applied rewrites 98.8%

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

   4. Taylor expanded in c around 0

      \[\leadsto \color{blue}{a \cdot b + \left(t \cdot z + x \cdot y\right)} \]
   5. 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 68.2

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

   6. Applied rewrites 68.2%

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

   7. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 67.8

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

   9. Applied rewrites 67.8%

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

Alternative 9: 63.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+195}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+124}:\\ \;\;\;\;\mathsf{fma}\left(z, t, b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* x y) -5e+195)
   (* y x)
   (if (<= (* x y) 2e+124) (fma z t (* b a)) (* y x))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((x * y) <= -5e+195) {
		tmp = y * x;
	} else if ((x * y) <= 2e+124) {
		tmp = fma(z, t, (b * a));
	} 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+195)
		tmp = Float64(y * x);
	elseif (Float64(x * y) <= 2e+124)
		tmp = fma(z, t, Float64(b * a));
	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+195], N[(y * x), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e+124], N[(z * t + N[(b * a), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -4.9999999999999998e195 or 1.9999999999999999e124 < (*.f64 x y)

   1. Initial program 89.7%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 68.7

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

   4. Applied rewrites 68.7%

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

   if -4.9999999999999998e195 < (*.f64 x y) < 1.9999999999999999e124

   1. Initial program 97.7%

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

      1. lift-*.f64 N/A

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

      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

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      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

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

      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

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

   3. Applied rewrites 98.8%

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

   4. Taylor expanded in c around 0

      \[\leadsto \color{blue}{a \cdot b + \left(t \cdot z + x \cdot y\right)} \]
   5. 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 70.8

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

   6. Applied rewrites 70.8%

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

   7. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 61.2

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

   9. Applied rewrites 61.2%

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

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. lift-*.f64 61.3

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

   11. Applied rewrites 61.3%

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

Alternative 10: 63.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+195}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+124}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* x y) -5e+195)
   (* y x)
   (if (<= (* x y) 2e+124) (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+195) {
		tmp = y * x;
	} else if ((x * y) <= 2e+124) {
		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+195)
		tmp = Float64(y * x);
	elseif (Float64(x * y) <= 2e+124)
		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+195], N[(y * x), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e+124], N[(b * a + N[(t * z), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -4.9999999999999998e195 or 1.9999999999999999e124 < (*.f64 x y)

   1. Initial program 89.7%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 68.7

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

   4. Applied rewrites 68.7%

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

   if -4.9999999999999998e195 < (*.f64 x y) < 1.9999999999999999e124

   1. Initial program 97.7%

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

      1. lift-*.f64 N/A

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

      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

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      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

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

      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

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

   3. Applied rewrites 98.8%

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

   4. Taylor expanded in c around 0

      \[\leadsto \color{blue}{a \cdot b + \left(t \cdot z + x \cdot y\right)} \]
   5. 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 70.8

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

   6. Applied rewrites 70.8%

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

   7. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 61.2

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

   9. Applied rewrites 61.2%

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

Alternative 11: 43.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+85}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-25}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-177}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-281}:\\ \;\;\;\;i \cdot c\\ \mathbf{elif}\;x \cdot y \leq 10^{-196}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+124}:\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* x y) -1e+85)
   (* y x)
   (if (<= (* x y) -5e-25)
     (* t z)
     (if (<= (* x y) -5e-177)
       (* b a)
       (if (<= (* x y) -5e-281)
         (* i c)
         (if (<= (* x y) 1e-196)
           (* t z)
           (if (<= (* x y) 2e+124) (* b a) (* y x))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((x * y) <= -1e+85) {
		tmp = y * x;
	} else if ((x * y) <= -5e-25) {
		tmp = t * z;
	} else if ((x * y) <= -5e-177) {
		tmp = b * a;
	} else if ((x * y) <= -5e-281) {
		tmp = i * c;
	} else if ((x * y) <= 1e-196) {
		tmp = t * z;
	} else if ((x * y) <= 2e+124) {
		tmp = b * a;
	} 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) <= (-1d+85)) then
        tmp = y * x
    else if ((x * y) <= (-5d-25)) then
        tmp = t * z
    else if ((x * y) <= (-5d-177)) then
        tmp = b * a
    else if ((x * y) <= (-5d-281)) then
        tmp = i * c
    else if ((x * y) <= 1d-196) then
        tmp = t * z
    else if ((x * y) <= 2d+124) then
        tmp = b * a
    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) <= -1e+85) {
		tmp = y * x;
	} else if ((x * y) <= -5e-25) {
		tmp = t * z;
	} else if ((x * y) <= -5e-177) {
		tmp = b * a;
	} else if ((x * y) <= -5e-281) {
		tmp = i * c;
	} else if ((x * y) <= 1e-196) {
		tmp = t * z;
	} else if ((x * y) <= 2e+124) {
		tmp = b * a;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (x * y) <= -1e+85:
		tmp = y * x
	elif (x * y) <= -5e-25:
		tmp = t * z
	elif (x * y) <= -5e-177:
		tmp = b * a
	elif (x * y) <= -5e-281:
		tmp = i * c
	elif (x * y) <= 1e-196:
		tmp = t * z
	elif (x * y) <= 2e+124:
		tmp = b * a
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(x * y) <= -1e+85)
		tmp = Float64(y * x);
	elseif (Float64(x * y) <= -5e-25)
		tmp = Float64(t * z);
	elseif (Float64(x * y) <= -5e-177)
		tmp = Float64(b * a);
	elseif (Float64(x * y) <= -5e-281)
		tmp = Float64(i * c);
	elseif (Float64(x * y) <= 1e-196)
		tmp = Float64(t * z);
	elseif (Float64(x * y) <= 2e+124)
		tmp = Float64(b * a);
	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) <= -1e+85)
		tmp = y * x;
	elseif ((x * y) <= -5e-25)
		tmp = t * z;
	elseif ((x * y) <= -5e-177)
		tmp = b * a;
	elseif ((x * y) <= -5e-281)
		tmp = i * c;
	elseif ((x * y) <= 1e-196)
		tmp = t * z;
	elseif ((x * y) <= 2e+124)
		tmp = b * a;
	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], -1e+85], N[(y * x), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -5e-25], N[(t * z), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -5e-177], N[(b * a), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -5e-281], N[(i * c), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e-196], N[(t * z), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e+124], N[(b * a), $MachinePrecision], N[(y * x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -1e85 or 1.9999999999999999e124 < (*.f64 x y)

   1. Initial program 91.4%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 62.0

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

   4. Applied rewrites 62.0%

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

   if -1e85 < (*.f64 x y) < -4.99999999999999962e-25 or -4.9999999999999998e-281 < (*.f64 x y) < 1e-196

   1. Initial program 97.6%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 34.1

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

   4. Applied rewrites 34.1%

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

   if -4.99999999999999962e-25 < (*.f64 x y) < -5e-177 or 1e-196 < (*.f64 x y) < 1.9999999999999999e124

   1. Initial program 97.9%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 30.9

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

   4. Applied rewrites 30.9%

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

   if -5e-177 < (*.f64 x y) < -4.9999999999999998e-281

   1. Initial program 97.2%

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

   2. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 34.0

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

   4. Applied rewrites 34.0%

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

Alternative 12: 43.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+114}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;a \cdot b \leq -5 \cdot 10^{-320}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{-42}:\\ \;\;\;\;i \cdot c\\ \mathbf{elif}\;a \cdot b \leq 10^{+73}:\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* a b) -2e+114)
   (* b a)
   (if (<= (* a b) -5e-320)
     (* t z)
     (if (<= (* a b) 2e-42) (* i c) (if (<= (* a b) 1e+73) (* t z) (* b a))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((a * b) <= -2e+114) {
		tmp = b * a;
	} else if ((a * b) <= -5e-320) {
		tmp = t * z;
	} else if ((a * b) <= 2e-42) {
		tmp = i * c;
	} else if ((a * b) <= 1e+73) {
		tmp = t * z;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if ((a * b) <= (-2d+114)) then
        tmp = b * a
    else if ((a * b) <= (-5d-320)) then
        tmp = t * z
    else if ((a * b) <= 2d-42) then
        tmp = i * c
    else if ((a * b) <= 1d+73) then
        tmp = t * z
    else
        tmp = b * a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((a * b) <= -2e+114) {
		tmp = b * a;
	} else if ((a * b) <= -5e-320) {
		tmp = t * z;
	} else if ((a * b) <= 2e-42) {
		tmp = i * c;
	} else if ((a * b) <= 1e+73) {
		tmp = t * z;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (a * b) <= -2e+114:
		tmp = b * a
	elif (a * b) <= -5e-320:
		tmp = t * z
	elif (a * b) <= 2e-42:
		tmp = i * c
	elif (a * b) <= 1e+73:
		tmp = t * z
	else:
		tmp = b * a
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(a * b) <= -2e+114)
		tmp = Float64(b * a);
	elseif (Float64(a * b) <= -5e-320)
		tmp = Float64(t * z);
	elseif (Float64(a * b) <= 2e-42)
		tmp = Float64(i * c);
	elseif (Float64(a * b) <= 1e+73)
		tmp = Float64(t * z);
	else
		tmp = Float64(b * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((a * b) <= -2e+114)
		tmp = b * a;
	elseif ((a * b) <= -5e-320)
		tmp = t * z;
	elseif ((a * b) <= 2e-42)
		tmp = i * c;
	elseif ((a * b) <= 1e+73)
		tmp = t * z;
	else
		tmp = b * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(a * b), $MachinePrecision], -2e+114], N[(b * a), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], -5e-320], N[(t * z), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 2e-42], N[(i * c), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 1e+73], N[(t * z), $MachinePrecision], N[(b * a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -2e114 or 9.99999999999999983e72 < (*.f64 a b)

   1. Initial program 92.1%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 61.9

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

   4. Applied rewrites 61.9%

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

   if -2e114 < (*.f64 a b) < -4.99994e-320 or 2.00000000000000008e-42 < (*.f64 a b) < 9.99999999999999983e72

   1. Initial program 97.7%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 33.4

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

   4. Applied rewrites 33.4%

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

   if -4.99994e-320 < (*.f64 a b) < 2.00000000000000008e-42

   1. Initial program 97.3%

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

   2. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 33.7

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

   4. Applied rewrites 33.7%

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

Alternative 13: 42.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{+106}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;a \cdot b \leq 10^{+51}:\\ \;\;\;\;i \cdot c\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* a b) -1e+106) (* b a) (if (<= (* a b) 1e+51) (* i c) (* b a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((a * b) <= -1e+106) {
		tmp = b * a;
	} else if ((a * b) <= 1e+51) {
		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) <= (-1d+106)) then
        tmp = b * a
    else if ((a * b) <= 1d+51) 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) <= -1e+106) {
		tmp = b * a;
	} else if ((a * b) <= 1e+51) {
		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) <= -1e+106:
		tmp = b * a
	elif (a * b) <= 1e+51:
		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) <= -1e+106)
		tmp = Float64(b * a);
	elseif (Float64(a * b) <= 1e+51)
		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) <= -1e+106)
		tmp = b * a;
	elseif ((a * b) <= 1e+51)
		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], -1e+106], N[(b * a), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 1e+51], N[(i * c), $MachinePrecision], N[(b * a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -1.00000000000000009e106 or 1e51 < (*.f64 a b)

   1. Initial program 92.4%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 59.6

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

   4. Applied rewrites 59.6%

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

   if -1.00000000000000009e106 < (*.f64 a b) < 1e51

   1. Initial program 97.5%

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

   2. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 32.8

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

   4. Applied rewrites 32.8%

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

Alternative 14: 27.7% accurate, 5.3× 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. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 27.7

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

4. Applied rewrites 27.7%

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

Reproduce

herbie shell --seed 2025117 
(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)))