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

Percentage Accurate: 96.0% → 98.0%

Time: 5.4s

Alternatives: 13

Speedup: 0.5×

• 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: 96.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (+ (+ (+ (* x y) (* z t)) (* a b)) (* c i)) INFINITY)
   (+ (fma y x (fma b a (* t z))) (* c i))
   (fma b a (fma t z (* y x)))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 100.0

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

   3. Applied rewrites 100.0%

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

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

   1. Initial program 0.0%

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

   2. 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 50.8

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

   4. Applied rewrites 50.8%

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

Alternative 2: 90.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(i, c, \mathsf{fma}\left(y, x, t \cdot z\right)\right)\\ \mathbf{if}\;c \cdot i \leq -1 \cdot 10^{+82}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \cdot i \leq 10^{+106}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(t, z, y \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma i c (fma y x (* t z)))))
   (if (<= (* c i) -1e+82)
     t_1
     (if (<= (* c i) 1e+106) (fma b a (fma t z (* y x))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(i, c, fma(y, x, (t * z)));
	double tmp;
	if ((c * i) <= -1e+82) {
		tmp = t_1;
	} else if ((c * i) <= 1e+106) {
		tmp = fma(b, a, fma(t, z, (y * x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(i, c, fma(y, x, Float64(t * z)))
	tmp = 0.0
	if (Float64(c * i) <= -1e+82)
		tmp = t_1;
	elseif (Float64(c * i) <= 1e+106)
		tmp = fma(b, a, fma(t, z, Float64(y * x)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(i * c + N[(y * x + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(c * i), $MachinePrecision], -1e+82], t$95$1, If[LessEqual[N[(c * i), $MachinePrecision], 1e+106], N[(b * a + N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c i) < -9.9999999999999996e81 or 1.00000000000000009e106 < (*.f64 c i)

   1. Initial program 92.2%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 93.1

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

   3. Applied rewrites 93.1%

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

   4. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 83.3

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

   6. Applied rewrites 83.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 85.4

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 85.4

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

   8. Applied rewrites 85.4%

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

   if -9.9999999999999996e81 < (*.f64 c i) < 1.00000000000000009e106

   1. Initial program 98.1%

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

   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 92.5

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

   4. Applied rewrites 92.5%

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

Alternative 3: 85.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -1 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(i, c, t \cdot z\right)\\ \mathbf{elif}\;c \cdot i \leq 10^{+106}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(t, z, y \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -1e+82)
   (fma i c (* t z))
   (if (<= (* c i) 1e+106) (fma b a (fma t z (* y x))) (fma i c (* x y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((c * i) <= -1e+82) {
		tmp = fma(i, c, (t * z));
	} else if ((c * i) <= 1e+106) {
		tmp = fma(b, a, fma(t, z, (y * x)));
	} else {
		tmp = fma(i, c, (x * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(c * i) <= -1e+82)
		tmp = fma(i, c, Float64(t * z));
	elseif (Float64(c * i) <= 1e+106)
		tmp = fma(b, a, fma(t, z, Float64(y * x)));
	else
		tmp = fma(i, c, Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(c * i), $MachinePrecision], -1e+82], N[(i * c + N[(t * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 1e+106], N[(b * a + N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(i * c + N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c i) < -9.9999999999999996e81

   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

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 92.8

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

   3. Applied rewrites 92.8%

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

   4. Taylor expanded in z around inf

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

      1. *-commutative N/A

         \[\leadsto t \cdot z + c \cdot i \]

      2. *-commutative N/A

         \[\leadsto t \cdot z + c \cdot i \]

      3. +-commutative N/A

         \[\leadsto t \cdot z + c \cdot i \]

      4. *-commutative N/A

         \[\leadsto t \cdot z + c \cdot i \]

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 72.1

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

   6. Applied rewrites 72.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 73.5

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

      6. *-commutative 73.5

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

      7. *-commutative 73.5

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

      8. +-commutative 73.5

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

      9. *-commutative 73.5

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

      10. +-commutative 73.5

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-+r+ N/A

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

      14. +-commutative N/A

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

      15. lower-*.f64 73.5

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

   8. Applied rewrites 73.5%

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

   if -9.9999999999999996e81 < (*.f64 c i) < 1.00000000000000009e106

   1. Initial program 98.1%

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

   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 92.5

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

   4. Applied rewrites 92.5%

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

   if 1.00000000000000009e106 < (*.f64 c i)

   1. Initial program 92.5%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{x} + c \cdot i \]

      2. lower-*.f64 73.7

         \[\leadsto y \cdot \color{blue}{x} + c \cdot i \]

   4. Applied rewrites 73.7%

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

      1. lift-*.f64 N/A

         \[\leadsto y \cdot x + \color{blue}{c \cdot i} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{y \cdot x + c \cdot i} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{c \cdot i + y \cdot x} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{i \cdot c} + y \cdot x \]

      5. lower-fma.f64 74.4

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 74.4

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

   6. Applied rewrites 74.4%

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

Alternative 4: 76.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, z, y \cdot x\right)\\ t_2 := x \cdot y + z \cdot t\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+165}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+133}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+207}:\\ \;\;\;\;\mathsf{fma}\left(i, c, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma t z (* y x))) (t_2 (+ (* x y) (* z t))))
   (if (<= t_2 -2e+165)
     t_1
     (if (<= t_2 2e+133)
       (fma i c (* a b))
       (if (<= t_2 2e+207) (fma i c (* x y)) 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 t_2 = (x * y) + (z * t);
	double tmp;
	if (t_2 <= -2e+165) {
		tmp = t_1;
	} else if (t_2 <= 2e+133) {
		tmp = fma(i, c, (a * b));
	} else if (t_2 <= 2e+207) {
		tmp = fma(i, c, (x * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(t, z, Float64(y * x))
	t_2 = Float64(Float64(x * y) + Float64(z * t))
	tmp = 0.0
	if (t_2 <= -2e+165)
		tmp = t_1;
	elseif (t_2 <= 2e+133)
		tmp = fma(i, c, Float64(a * b));
	elseif (t_2 <= 2e+207)
		tmp = fma(i, c, Float64(x * y));
	else
		tmp = 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]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+165], t$95$1, If[LessEqual[t$95$2, 2e+133], N[(i * c + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+207], N[(i * c + N[(x * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 x y) (*.f64 z t)) < -1.9999999999999998e165 or 2.0000000000000001e207 < (+.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. 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 88.7

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

   4. Applied rewrites 88.7%

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

   5. Taylor expanded in a around 0

      \[\leadsto t \cdot z + \color{blue}{x \cdot y} \]
   6. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto t \cdot z + \color{blue}{x} \cdot y \]

      2. +-commutative N/A

         \[\leadsto t \cdot z + \color{blue}{x} \cdot y \]

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 81.3

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

   7. Applied rewrites 81.3%

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

   if -1.9999999999999998e165 < (+.f64 (*.f64 x y) (*.f64 z t)) < 2e133

   1. Initial program 98.9%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 74.3

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

   4. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 74.8

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 74.8

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

   6. Applied rewrites 74.8%

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

   if 2e133 < (+.f64 (*.f64 x y) (*.f64 z t)) < 2.0000000000000001e207

   1. Initial program 99.0%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{x} + c \cdot i \]

      2. lower-*.f64 51.6

         \[\leadsto y \cdot \color{blue}{x} + c \cdot i \]

   4. Applied rewrites 51.6%

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

      1. lift-*.f64 N/A

         \[\leadsto y \cdot x + \color{blue}{c \cdot i} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{y \cdot x + c \cdot i} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{c \cdot i + y \cdot x} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{i \cdot c} + y \cdot x \]

      5. lower-fma.f64 51.6

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 51.6

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

   6. Applied rewrites 51.6%

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

Alternative 5: 75.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, z, y \cdot x\right)\\ t_2 := x \cdot y + z \cdot t\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+165}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+207}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma t z (* y x))) (t_2 (+ (* x y) (* z t))))
   (if (<= t_2 -2e+165) t_1 (if (<= t_2 2e+207) (fma i c (* a b)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(t, z, (y * x));
	double t_2 = (x * y) + (z * t);
	double tmp;
	if (t_2 <= -2e+165) {
		tmp = t_1;
	} else if (t_2 <= 2e+207) {
		tmp = fma(i, c, (a * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(t, z, Float64(y * x))
	t_2 = Float64(Float64(x * y) + Float64(z * t))
	tmp = 0.0
	if (t_2 <= -2e+165)
		tmp = t_1;
	elseif (t_2 <= 2e+207)
		tmp = fma(i, c, Float64(a * b));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+165], t$95$1, If[LessEqual[t$95$2, 2e+207], N[(i * c + N[(a * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x y) (*.f64 z t)) < -1.9999999999999998e165 or 2.0000000000000001e207 < (+.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. 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 88.7

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

   4. Applied rewrites 88.7%

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

   5. Taylor expanded in a around 0

      \[\leadsto t \cdot z + \color{blue}{x \cdot y} \]
   6. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto t \cdot z + \color{blue}{x} \cdot y \]

      2. +-commutative N/A

         \[\leadsto t \cdot z + \color{blue}{x} \cdot y \]

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 81.3

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

   7. Applied rewrites 81.3%

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

   if -1.9999999999999998e165 < (+.f64 (*.f64 x y) (*.f64 z t)) < 2.0000000000000001e207

   1. Initial program 98.9%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 71.1

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

   4. Applied rewrites 71.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 71.6

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 71.6

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

   6. Applied rewrites 71.6%

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

Alternative 6: 66.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, a, y \cdot x\right)\\ t_2 := \mathsf{fma}\left(i, c, t \cdot z\right)\\ \mathbf{if}\;c \cdot i \leq -4 \cdot 10^{+64}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;c \cdot i \leq -2 \cdot 10^{-134}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \cdot i \leq -1 \cdot 10^{-307}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \mathbf{elif}\;c \cdot i \leq 5 \cdot 10^{+182}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma b a (* y x))) (t_2 (fma i c (* t z))))
   (if (<= (* c i) -4e+64)
     t_2
     (if (<= (* c i) -2e-134)
       t_1
       (if (<= (* c i) -1e-307)
         (fma b a (* t z))
         (if (<= (* c i) 5e+182) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(b, a, (y * x));
	double t_2 = fma(i, c, (t * z));
	double tmp;
	if ((c * i) <= -4e+64) {
		tmp = t_2;
	} else if ((c * i) <= -2e-134) {
		tmp = t_1;
	} else if ((c * i) <= -1e-307) {
		tmp = fma(b, a, (t * z));
	} else if ((c * i) <= 5e+182) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(b, a, Float64(y * x))
	t_2 = fma(i, c, Float64(t * z))
	tmp = 0.0
	if (Float64(c * i) <= -4e+64)
		tmp = t_2;
	elseif (Float64(c * i) <= -2e-134)
		tmp = t_1;
	elseif (Float64(c * i) <= -1e-307)
		tmp = fma(b, a, Float64(t * z));
	elseif (Float64(c * i) <= 5e+182)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(b * a + N[(y * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(i * c + N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(c * i), $MachinePrecision], -4e+64], t$95$2, If[LessEqual[N[(c * i), $MachinePrecision], -2e-134], t$95$1, If[LessEqual[N[(c * i), $MachinePrecision], -1e-307], N[(b * a + N[(t * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 5e+182], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c i) < -4.00000000000000009e64 or 4.99999999999999973e182 < (*.f64 c i)

   1. Initial program 91.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 92.6

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

   3. Applied rewrites 92.6%

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

   4. Taylor expanded in z around inf

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

      1. *-commutative N/A

         \[\leadsto t \cdot z + c \cdot i \]

      2. *-commutative N/A

         \[\leadsto t \cdot z + c \cdot i \]

      3. +-commutative N/A

         \[\leadsto t \cdot z + c \cdot i \]

      4. *-commutative N/A

         \[\leadsto t \cdot z + c \cdot i \]

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 74.0

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

   6. Applied rewrites 74.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 75.3

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

      6. *-commutative 75.3

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

      7. *-commutative 75.3

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

      8. +-commutative 75.3

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

      9. *-commutative 75.3

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

      10. +-commutative 75.3

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-+r+ N/A

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

      14. +-commutative N/A

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

      15. lower-*.f64 75.3

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

   8. Applied rewrites 75.3%

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

   if -4.00000000000000009e64 < (*.f64 c i) < -2.00000000000000008e-134 or -9.99999999999999909e-308 < (*.f64 c i) < 4.99999999999999973e182

   1. Initial program 98.0%

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

   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 89.8

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

   4. Applied rewrites 89.8%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 61.2

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

   7. Applied rewrites 61.2%

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

   if -2.00000000000000008e-134 < (*.f64 c i) < -9.99999999999999909e-308

   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 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 97.6

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

   4. Applied rewrites 97.6%

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

   5. Taylor expanded in x around 0

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

      1. lift-*.f64 67.3

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

   7. Applied rewrites 67.3%

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

Alternative 7: 66.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, z, y \cdot x\right)\\ \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+67}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 10^{+28}:\\ \;\;\;\;\mathsf{fma}\left(b, a, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma t z (* y x))))
   (if (<= (* z t) -2e+67) t_1 (if (<= (* z t) 1e+28) (fma b a (* y x)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(t, z, (y * x));
	double tmp;
	if ((z * t) <= -2e+67) {
		tmp = t_1;
	} else if ((z * t) <= 1e+28) {
		tmp = fma(b, a, (y * x));
	} else {
		tmp = 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(z * t) <= -2e+67)
		tmp = t_1;
	elseif (Float64(z * t) <= 1e+28)
		tmp = fma(b, a, Float64(y * x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -2e+67], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 1e+28], N[(b * a + N[(y * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.99999999999999997e67 or 9.99999999999999958e27 < (*.f64 z t)

   1. Initial program 93.6%

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

   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%

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

   5. Taylor expanded in a around 0

      \[\leadsto t \cdot z + \color{blue}{x \cdot y} \]
   6. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto t \cdot z + \color{blue}{x} \cdot y \]

      2. +-commutative N/A

         \[\leadsto t \cdot z + \color{blue}{x} \cdot y \]

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 69.4

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

   7. Applied rewrites 69.4%

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

   if -1.99999999999999997e67 < (*.f64 z t) < 9.99999999999999958e27

   1. Initial program 97.8%

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

   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 69.7

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

   4. Applied rewrites 69.7%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 64.8

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

   7. Applied rewrites 64.8%

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

Alternative 8: 66.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, z, y \cdot x\right)\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+165}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 10^{+130}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma t z (* y x))))
   (if (<= (* x y) -2e+165)
     t_1
     (if (<= (* x y) 1e+130) (fma b a (* t z)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(t, z, (y * x));
	double tmp;
	if ((x * y) <= -2e+165) {
		tmp = t_1;
	} else if ((x * y) <= 1e+130) {
		tmp = fma(b, a, (t * z));
	} else {
		tmp = 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+165)
		tmp = t_1;
	elseif (Float64(x * y) <= 1e+130)
		tmp = fma(b, a, Float64(t * z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -2e+165], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 1e+130], N[(b * a + N[(t * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.9999999999999998e165 or 1.0000000000000001e130 < (*.f64 x y)

   1. Initial program 91.8%

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

   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 87.7

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

   4. Applied rewrites 87.7%

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

   5. Taylor expanded in a around 0

      \[\leadsto t \cdot z + \color{blue}{x \cdot y} \]
   6. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto t \cdot z + \color{blue}{x} \cdot y \]

      2. +-commutative N/A

         \[\leadsto t \cdot z + \color{blue}{x} \cdot y \]

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 79.0

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

   7. Applied rewrites 79.0%

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

   if -1.9999999999999998e165 < (*.f64 x y) < 1.0000000000000001e130

   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 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 70.7

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

   4. Applied rewrites 70.7%

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

   5. Taylor expanded in x around 0

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

      1. lift-*.f64 62.1

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

   7. Applied rewrites 62.1%

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

Alternative 9: 63.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -1 \cdot 10^{+179}:\\ \;\;\;\;i \cdot c\\ \mathbf{elif}\;c \cdot i \leq 10^{+196}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -1e+179)
   (* i c)
   (if (<= (* c i) 1e+196) (fma t z (* y x)) (* i c))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((c * i) <= -1e+179) {
		tmp = i * c;
	} else if ((c * i) <= 1e+196) {
		tmp = fma(t, z, (y * x));
	} else {
		tmp = i * c;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(c * i) <= -1e+179)
		tmp = Float64(i * c);
	elseif (Float64(c * i) <= 1e+196)
		tmp = fma(t, z, Float64(y * x));
	else
		tmp = Float64(i * c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(c * i), $MachinePrecision], -1e+179], N[(i * c), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 1e+196], N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(i * c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c i) < -9.9999999999999998e178 or 9.9999999999999995e195 < (*.f64 c i)

   1. Initial program 89.8%

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

   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 73.8

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

   4. Applied rewrites 73.8%

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

   if -9.9999999999999998e178 < (*.f64 c i) < 9.9999999999999995e195

   1. Initial program 98.0%

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

   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 88.4

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

   4. Applied rewrites 88.4%

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

   5. Taylor expanded in a around 0

      \[\leadsto t \cdot z + \color{blue}{x \cdot y} \]
   6. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto t \cdot z + \color{blue}{x} \cdot y \]

      2. +-commutative N/A

         \[\leadsto t \cdot z + \color{blue}{x} \cdot y \]

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 59.7

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

   7. Applied rewrites 59.7%

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

Alternative 10: 43.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+84}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \cdot y \leq -4 \cdot 10^{-291}:\\ \;\;\;\;i \cdot c\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-115}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;x \cdot y \leq 10^{+130}:\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* x y) -5e+84)
   (* y x)
   (if (<= (* x y) -4e-291)
     (* i c)
     (if (<= (* x y) 5e-115)
       (* t z)
       (if (<= (* x y) 1e+130) (* 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+84) {
		tmp = y * x;
	} else if ((x * y) <= -4e-291) {
		tmp = i * c;
	} else if ((x * y) <= 5e-115) {
		tmp = t * z;
	} else if ((x * y) <= 1e+130) {
		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) <= (-5d+84)) then
        tmp = y * x
    else if ((x * y) <= (-4d-291)) then
        tmp = i * c
    else if ((x * y) <= 5d-115) then
        tmp = t * z
    else if ((x * y) <= 1d+130) 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) <= -5e+84) {
		tmp = y * x;
	} else if ((x * y) <= -4e-291) {
		tmp = i * c;
	} else if ((x * y) <= 5e-115) {
		tmp = t * z;
	} else if ((x * y) <= 1e+130) {
		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) <= -5e+84:
		tmp = y * x
	elif (x * y) <= -4e-291:
		tmp = i * c
	elif (x * y) <= 5e-115:
		tmp = t * z
	elif (x * y) <= 1e+130:
		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) <= -5e+84)
		tmp = Float64(y * x);
	elseif (Float64(x * y) <= -4e-291)
		tmp = Float64(i * c);
	elseif (Float64(x * y) <= 5e-115)
		tmp = Float64(t * z);
	elseif (Float64(x * y) <= 1e+130)
		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) <= -5e+84)
		tmp = y * x;
	elseif ((x * y) <= -4e-291)
		tmp = i * c;
	elseif ((x * y) <= 5e-115)
		tmp = t * z;
	elseif ((x * y) <= 1e+130)
		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], -5e+84], N[(y * x), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -4e-291], N[(i * c), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e-115], N[(t * z), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e+130], N[(b * a), $MachinePrecision], N[(y * x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -5.0000000000000001e84 or 1.0000000000000001e130 < (*.f64 x y)

   1. Initial program 92.7%

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

   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 62.8

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

   4. Applied rewrites 62.8%

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

   if -5.0000000000000001e84 < (*.f64 x y) < -3.99999999999999985e-291

   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 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 31.9

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

   4. Applied rewrites 31.9%

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

   if -3.99999999999999985e-291 < (*.f64 x y) < 5.0000000000000003e-115

   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 z around inf

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

      1. lower-*.f64 34.3

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

   4. Applied rewrites 34.3%

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

   if 5.0000000000000003e-115 < (*.f64 x y) < 1.0000000000000001e130

   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 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 32.3

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

   4. Applied rewrites 32.3%

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

Alternative 11: 43.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -1 \cdot 10^{+82}:\\ \;\;\;\;i \cdot c\\ \mathbf{elif}\;c \cdot i \leq -2 \cdot 10^{-196}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;c \cdot i \leq 10^{+104}:\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;i \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -1e+82)
   (* i c)
   (if (<= (* c i) -2e-196) (* b a) (if (<= (* c i) 1e+104) (* t z) (* i c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((c * i) <= -1e+82) {
		tmp = i * c;
	} else if ((c * i) <= -2e-196) {
		tmp = b * a;
	} else if ((c * i) <= 1e+104) {
		tmp = t * z;
	} else {
		tmp = i * c;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if ((c * i) <= (-1d+82)) then
        tmp = i * c
    else if ((c * i) <= (-2d-196)) then
        tmp = b * a
    else if ((c * i) <= 1d+104) then
        tmp = t * z
    else
        tmp = i * c
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((c * i) <= -1e+82) {
		tmp = i * c;
	} else if ((c * i) <= -2e-196) {
		tmp = b * a;
	} else if ((c * i) <= 1e+104) {
		tmp = t * z;
	} else {
		tmp = i * c;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (c * i) <= -1e+82:
		tmp = i * c
	elif (c * i) <= -2e-196:
		tmp = b * a
	elif (c * i) <= 1e+104:
		tmp = t * z
	else:
		tmp = i * c
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(c * i) <= -1e+82)
		tmp = Float64(i * c);
	elseif (Float64(c * i) <= -2e-196)
		tmp = Float64(b * a);
	elseif (Float64(c * i) <= 1e+104)
		tmp = Float64(t * z);
	else
		tmp = Float64(i * c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((c * i) <= -1e+82)
		tmp = i * c;
	elseif ((c * i) <= -2e-196)
		tmp = b * a;
	elseif ((c * i) <= 1e+104)
		tmp = t * z;
	else
		tmp = i * c;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(c * i), $MachinePrecision], -1e+82], N[(i * c), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], -2e-196], N[(b * a), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 1e+104], N[(t * z), $MachinePrecision], N[(i * c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c i) < -9.9999999999999996e81 or 1e104 < (*.f64 c i)

   1. Initial program 92.2%

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

   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 61.0

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

   4. Applied rewrites 61.0%

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

   if -9.9999999999999996e81 < (*.f64 c i) < -2.0000000000000001e-196

   1. Initial program 98.0%

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

   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 33.6

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

   4. Applied rewrites 33.6%

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

   if -2.0000000000000001e-196 < (*.f64 c i) < 1e104

   1. Initial program 98.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 33.8

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

   4. Applied rewrites 33.8%

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

Alternative 12: 42.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -1 \cdot 10^{+82}:\\ \;\;\;\;i \cdot c\\ \mathbf{elif}\;c \cdot i \leq 10^{+106}:\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;i \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -1e+82) (* i c) (if (<= (* c i) 1e+106) (* b a) (* i c))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((c * i) <= -1e+82) {
		tmp = i * c;
	} else if ((c * i) <= 1e+106) {
		tmp = b * a;
	} else {
		tmp = i * c;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if ((c * i) <= (-1d+82)) then
        tmp = i * c
    else if ((c * i) <= 1d+106) then
        tmp = b * a
    else
        tmp = i * c
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((c * i) <= -1e+82) {
		tmp = i * c;
	} else if ((c * i) <= 1e+106) {
		tmp = b * a;
	} else {
		tmp = i * c;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (c * i) <= -1e+82:
		tmp = i * c
	elif (c * i) <= 1e+106:
		tmp = b * a
	else:
		tmp = i * c
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(c * i) <= -1e+82)
		tmp = Float64(i * c);
	elseif (Float64(c * i) <= 1e+106)
		tmp = Float64(b * a);
	else
		tmp = Float64(i * c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((c * i) <= -1e+82)
		tmp = i * c;
	elseif ((c * i) <= 1e+106)
		tmp = b * a;
	else
		tmp = i * c;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(c * i), $MachinePrecision], -1e+82], N[(i * c), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 1e+106], N[(b * a), $MachinePrecision], N[(i * c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c i) < -9.9999999999999996e81 or 1.00000000000000009e106 < (*.f64 c i)

   1. Initial program 92.2%

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

   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 61.1

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

   4. Applied rewrites 61.1%

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

   if -9.9999999999999996e81 < (*.f64 c i) < 1.00000000000000009e106

   1. Initial program 98.1%

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

   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 34.2

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

   4. Applied rewrites 34.2%

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

Alternative 13: 28.2% 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 96.0%

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

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 28.2

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

4. Applied rewrites 28.2%

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

Reproduce

herbie shell --seed 2025112 
(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)))