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

Percentage Accurate: 95.8% → 97.0%

Time: 4.6s

Alternatives: 14

Speedup: 1.2×

• 41.2% 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: 95.8% 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: 97.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 95.7%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   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

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

   5. associate-+l+ N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   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 97.7

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

4. Applied rewrites 97.7%

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

Alternative 2: 75.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, t, x \cdot y\right)\\ t_2 := x \cdot y + z \cdot t\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+170}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+235}:\\ \;\;\;\;\mathsf{fma}\left(i, c, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma z t (* x y))) (t_2 (+ (* x y) (* z t))))
   (if (<= t_2 -5e+170)
     t_1
     (if (<= t_2 2e+52)
       (fma i c (* a b))
       (if (<= t_2 1e+235) (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(z, t, (x * y));
	double t_2 = (x * y) + (z * t);
	double tmp;
	if (t_2 <= -5e+170) {
		tmp = t_1;
	} else if (t_2 <= 2e+52) {
		tmp = fma(i, c, (a * b));
	} else if (t_2 <= 1e+235) {
		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(z, t, Float64(x * y))
	t_2 = Float64(Float64(x * y) + Float64(z * t))
	tmp = 0.0
	if (t_2 <= -5e+170)
		tmp = t_1;
	elseif (t_2 <= 2e+52)
		tmp = fma(i, c, Float64(a * b));
	elseif (t_2 <= 1e+235)
		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[(z * t + N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+170], t$95$1, If[LessEqual[t$95$2, 2e+52], N[(i * c + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+235], 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)) < -4.99999999999999977e170 or 1.0000000000000001e235 < (+.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 88.6%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      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

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      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 93.8

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

   4. Applied rewrites 93.8%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 90.7

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

   7. Applied rewrites 90.7%

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

   8. Taylor expanded in c around 0

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

      1. *-commutative N/A

         \[\leadsto z \cdot t + x \cdot y \]

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 85.6

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

   10. Applied rewrites 85.6%

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

   if -4.99999999999999977e170 < (+.f64 (*.f64 x y) (*.f64 z t)) < 2e52

   1. Initial program 100.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 81.0

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

   5. Applied rewrites 81.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 81.0

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

      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 81.0

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

   7. Applied rewrites 81.0%

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

   if 2e52 < (+.f64 (*.f64 x y) (*.f64 z t)) < 1.0000000000000001e235

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 76.6

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

   5. Applied rewrites 76.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 76.6

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

      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 76.6

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

   7. Applied rewrites 76.6%

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

Alternative 3: 43.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+128}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-247}:\\ \;\;\;\;i \cdot c\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-208}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;x \cdot y \leq 10^{+31}:\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* x y) -5e+128)
   (* y x)
   (if (<= (* x y) 2e-247)
     (* i c)
     (if (<= (* x y) 5e-208)
       (* t z)
       (if (<= (* x y) 1e+31) (* 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+128) {
		tmp = y * x;
	} else if ((x * y) <= 2e-247) {
		tmp = i * c;
	} else if ((x * y) <= 5e-208) {
		tmp = t * z;
	} else if ((x * y) <= 1e+31) {
		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+128)) then
        tmp = y * x
    else if ((x * y) <= 2d-247) then
        tmp = i * c
    else if ((x * y) <= 5d-208) then
        tmp = t * z
    else if ((x * y) <= 1d+31) 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+128) {
		tmp = y * x;
	} else if ((x * y) <= 2e-247) {
		tmp = i * c;
	} else if ((x * y) <= 5e-208) {
		tmp = t * z;
	} else if ((x * y) <= 1e+31) {
		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+128:
		tmp = y * x
	elif (x * y) <= 2e-247:
		tmp = i * c
	elif (x * y) <= 5e-208:
		tmp = t * z
	elif (x * y) <= 1e+31:
		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+128)
		tmp = Float64(y * x);
	elseif (Float64(x * y) <= 2e-247)
		tmp = Float64(i * c);
	elseif (Float64(x * y) <= 5e-208)
		tmp = Float64(t * z);
	elseif (Float64(x * y) <= 1e+31)
		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+128)
		tmp = y * x;
	elseif ((x * y) <= 2e-247)
		tmp = i * c;
	elseif ((x * y) <= 5e-208)
		tmp = t * z;
	elseif ((x * y) <= 1e+31)
		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+128], N[(y * x), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e-247], N[(i * c), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e-208], N[(t * z), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e+31], N[(b * a), $MachinePrecision], N[(y * x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -5e128 or 9.9999999999999996e30 < (*.f64 x y)

   1. Initial program 91.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 66.7

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

   5. Applied rewrites 66.7%

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

   if -5e128 < (*.f64 x y) < 2e-247

   1. Initial program 99.0%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 41.6

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

   5. Applied rewrites 41.6%

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

   if 2e-247 < (*.f64 x y) < 4.99999999999999963e-208

   1. Initial program 85.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 85.8

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

   5. Applied rewrites 85.8%

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

   if 4.99999999999999963e-208 < (*.f64 x y) < 9.9999999999999996e30

   1. Initial program 97.7%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 48.0

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

   5. Applied rewrites 48.0%

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

Alternative 4: 76.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot y + z \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+170} \lor \neg \left(t\_1 \leq 5 \cdot 10^{+119}\right):\\ \;\;\;\;\mathsf{fma}\left(z, t, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* x y) (* z t))))
   (if (or (<= t_1 -5e+170) (not (<= t_1 5e+119)))
     (fma z t (* x y))
     (fma i c (* a b)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (x * y) + (z * t);
	double tmp;
	if ((t_1 <= -5e+170) || !(t_1 <= 5e+119)) {
		tmp = fma(z, t, (x * y));
	} else {
		tmp = fma(i, c, (a * b));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(x * y) + Float64(z * t))
	tmp = 0.0
	if ((t_1 <= -5e+170) || !(t_1 <= 5e+119))
		tmp = fma(z, t, Float64(x * y));
	else
		tmp = fma(i, c, Float64(a * b));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e+170], N[Not[LessEqual[t$95$1, 5e+119]], $MachinePrecision]], N[(z * t + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(i * c + N[(a * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x y) (*.f64 z t)) < -4.99999999999999977e170 or 4.9999999999999999e119 < (+.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 90.5%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      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

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      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 94.8

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

   4. Applied rewrites 94.8%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 91.4

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

   7. Applied rewrites 91.4%

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

   8. Taylor expanded in c around 0

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

      1. *-commutative N/A

         \[\leadsto z \cdot t + x \cdot y \]

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 80.4

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

   10. Applied rewrites 80.4%

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

   if -4.99999999999999977e170 < (+.f64 (*.f64 x y) (*.f64 z t)) < 4.9999999999999999e119

   1. Initial program 100.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 78.9

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

   5. Applied rewrites 78.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 78.9

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

      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 78.9

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

   7. Applied rewrites 78.9%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y + z \cdot t \leq -5 \cdot 10^{+170} \lor \neg \left(x \cdot y + z \cdot t \leq 5 \cdot 10^{+119}\right):\\ \;\;\;\;\mathsf{fma}\left(z, t, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -1 \cdot 10^{+234}:\\ \;\;\;\;i \cdot c\\ \mathbf{elif}\;c \cdot i \leq 2 \cdot 10^{-301}:\\ \;\;\;\;\mathsf{fma}\left(x, y, b \cdot a\right)\\ \mathbf{elif}\;c \cdot i \leq 10^{+159}:\\ \;\;\;\;\mathsf{fma}\left(z, t, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -1e+234)
   (* i c)
   (if (<= (* c i) 2e-301)
     (fma x y (* b a))
     (if (<= (* c i) 1e+159) (fma z t (* x y)) (* 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+234) {
		tmp = i * c;
	} else if ((c * i) <= 2e-301) {
		tmp = fma(x, y, (b * a));
	} else if ((c * i) <= 1e+159) {
		tmp = fma(z, t, (x * y));
	} 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+234)
		tmp = Float64(i * c);
	elseif (Float64(c * i) <= 2e-301)
		tmp = fma(x, y, Float64(b * a));
	elseif (Float64(c * i) <= 1e+159)
		tmp = fma(z, t, Float64(x * y));
	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+234], N[(i * c), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 2e-301], N[(x * y + N[(b * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 1e+159], N[(z * t + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(i * c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c i) < -1.00000000000000002e234 or 9.9999999999999993e158 < (*.f64 c i)

   1. Initial program 93.3%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 84.9

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

   5. Applied rewrites 84.9%

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

   if -1.00000000000000002e234 < (*.f64 c i) < 2.00000000000000013e-301

   1. Initial program 97.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      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

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      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 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in z around 0

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

      1. lower-*.f64 81.8

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

   7. Applied rewrites 81.8%

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

   8. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      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. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 90.8

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

   10. Applied rewrites 90.8%

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

   11. Taylor expanded in z around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 73.5

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

   13. Applied rewrites 73.5%

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

   if 2.00000000000000013e-301 < (*.f64 c i) < 9.9999999999999993e158

   1. Initial program 94.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      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

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      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 98.7

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 78.8

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

   7. Applied rewrites 78.8%

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

   8. Taylor expanded in c around 0

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

      1. *-commutative N/A

         \[\leadsto z \cdot t + x \cdot y \]

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 67.9

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

   10. Applied rewrites 67.9%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -1 \cdot 10^{+234}:\\ \;\;\;\;i \cdot c\\ \mathbf{elif}\;c \cdot i \leq 2 \cdot 10^{-301}:\\ \;\;\;\;\mathsf{fma}\left(x, y, b \cdot a\right)\\ \mathbf{elif}\;c \cdot i \leq 10^{+159}:\\ \;\;\;\;\mathsf{fma}\left(z, t, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -1 \cdot 10^{+234}:\\ \;\;\;\;i \cdot c\\ \mathbf{elif}\;c \cdot i \leq -1 \cdot 10^{-230}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \mathbf{elif}\;c \cdot i \leq 10^{+159}:\\ \;\;\;\;\mathsf{fma}\left(z, t, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -1e+234)
   (* i c)
   (if (<= (* c i) -1e-230)
     (fma b a (* t z))
     (if (<= (* c i) 1e+159) (fma z t (* x y)) (* 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+234) {
		tmp = i * c;
	} else if ((c * i) <= -1e-230) {
		tmp = fma(b, a, (t * z));
	} else if ((c * i) <= 1e+159) {
		tmp = fma(z, t, (x * y));
	} 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+234)
		tmp = Float64(i * c);
	elseif (Float64(c * i) <= -1e-230)
		tmp = fma(b, a, Float64(t * z));
	elseif (Float64(c * i) <= 1e+159)
		tmp = fma(z, t, Float64(x * y));
	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+234], N[(i * c), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], -1e-230], N[(b * a + N[(t * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 1e+159], N[(z * t + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(i * c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c i) < -1.00000000000000002e234 or 9.9999999999999993e158 < (*.f64 c i)

   1. Initial program 93.3%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 84.9

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

   5. Applied rewrites 84.9%

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

   if -1.00000000000000002e234 < (*.f64 c i) < -1.00000000000000005e-230

   1. Initial program 97.5%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      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

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      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

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in z around 0

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

      1. lower-*.f64 80.9

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

   7. Applied rewrites 80.9%

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

   8. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      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. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 88.0

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

   10. Applied rewrites 88.0%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. associate-+l+ N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 60.0

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

   13. Applied rewrites 60.0%

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

   if -1.00000000000000005e-230 < (*.f64 c i) < 9.9999999999999993e158

   1. Initial program 95.6%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      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

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      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 98.2

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 74.4

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

   7. Applied rewrites 74.4%

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

   8. Taylor expanded in c around 0

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

      1. *-commutative N/A

         \[\leadsto z \cdot t + x \cdot y \]

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 66.9

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

   10. Applied rewrites 66.9%

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

Alternative 7: 87.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+174}:\\ \;\;\;\;\mathsf{fma}\left(i, c, \mathsf{fma}\left(t, z, y \cdot x\right)\right)\\ \mathbf{elif}\;z \cdot t \leq 10^{+235}:\\ \;\;\;\;\mathsf{fma}\left(y, x, a \cdot b\right) + c \cdot i\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, z \cdot t\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* z t) -2e+174)
   (fma i c (fma t z (* y x)))
   (if (<= (* z t) 1e+235)
     (+ (fma y x (* a b)) (* c i))
     (fma x y (fma a b (* z t))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((z * t) <= -2e+174) {
		tmp = fma(i, c, fma(t, z, (y * x)));
	} else if ((z * t) <= 1e+235) {
		tmp = fma(y, x, (a * b)) + (c * i);
	} else {
		tmp = fma(x, y, fma(a, b, (z * t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(z * t) <= -2e+174)
		tmp = fma(i, c, fma(t, z, Float64(y * x)));
	elseif (Float64(z * t) <= 1e+235)
		tmp = Float64(fma(y, x, Float64(a * b)) + Float64(c * i));
	else
		tmp = fma(x, y, fma(a, b, Float64(z * t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(z * t), $MachinePrecision], -2e+174], N[(i * c + N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+235], N[(N[(y * x + N[(a * b), $MachinePrecision]), $MachinePrecision] + N[(c * i), $MachinePrecision]), $MachinePrecision], N[(x * y + N[(a * b + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -2.00000000000000014e174

   1. Initial program 92.3%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 96.2

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

   5. Applied rewrites 96.2%

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

   if -2.00000000000000014e174 < (*.f64 z t) < 1.0000000000000001e235

   1. Initial program 97.6%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      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

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      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 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in z around 0

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

      1. lower-*.f64 91.8

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

   7. Applied rewrites 91.8%

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

   if 1.0000000000000001e235 < (*.f64 z t)

   1. Initial program 78.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      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

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      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 89.5

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

   4. Applied rewrites 89.5%

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

   5. Taylor expanded in z around 0

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

      1. lower-*.f64 16.5

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

   7. Applied rewrites 16.5%

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

   8. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      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. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 94.7

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

   10. Applied rewrites 94.7%

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

Alternative 8: 89.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{+99} \lor \neg \left(c \cdot i \leq 2 \cdot 10^{+93}\right):\\ \;\;\;\;\mathsf{fma}\left(i, c, \mathsf{fma}\left(t, z, y \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, z \cdot t\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* c i) -5e+99) (not (<= (* c i) 2e+93)))
   (fma i c (fma t z (* y x)))
   (fma x y (fma a b (* z t)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((c * i) <= -5e+99) || !((c * i) <= 2e+93)) {
		tmp = fma(i, c, fma(t, z, (y * x)));
	} else {
		tmp = fma(x, y, fma(a, b, (z * t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(c * i) <= -5e+99) || !(Float64(c * i) <= 2e+93))
		tmp = fma(i, c, fma(t, z, Float64(y * x)));
	else
		tmp = fma(x, y, fma(a, b, Float64(z * t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(c * i), $MachinePrecision], -5e+99], N[Not[LessEqual[N[(c * i), $MachinePrecision], 2e+93]], $MachinePrecision]], N[(i * c + N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y + N[(a * b + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c i) < -5.00000000000000008e99 or 2.00000000000000009e93 < (*.f64 c i)

   1. Initial program 93.7%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 88.8

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

   5. Applied rewrites 88.8%

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

   if -5.00000000000000008e99 < (*.f64 c i) < 2.00000000000000009e93

   1. Initial program 96.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      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

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      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 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in z around 0

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

      1. lower-*.f64 75.8

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

   7. Applied rewrites 75.8%

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

   8. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      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. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 94.1

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

   10. Applied rewrites 94.1%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{+99} \lor \neg \left(c \cdot i \leq 2 \cdot 10^{+93}\right):\\ \;\;\;\;\mathsf{fma}\left(i, c, \mathsf{fma}\left(t, z, y \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, z \cdot t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, z, y \cdot x\right)\\ \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{+99} \lor \neg \left(c \cdot i \leq 2 \cdot 10^{+17}\right):\\ \;\;\;\;\mathsf{fma}\left(i, c, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma t z (* y x))))
   (if (or (<= (* c i) -5e+99) (not (<= (* c i) 2e+17)))
     (fma i c t_1)
     (fma b a t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(t, z, (y * x));
	double tmp;
	if (((c * i) <= -5e+99) || !((c * i) <= 2e+17)) {
		tmp = fma(i, c, t_1);
	} else {
		tmp = fma(b, a, t_1);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(t, z, Float64(y * x))
	tmp = 0.0
	if ((Float64(c * i) <= -5e+99) || !(Float64(c * i) <= 2e+17))
		tmp = fma(i, c, t_1);
	else
		tmp = fma(b, a, t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[N[(c * i), $MachinePrecision], -5e+99], N[Not[LessEqual[N[(c * i), $MachinePrecision], 2e+17]], $MachinePrecision]], N[(i * c + t$95$1), $MachinePrecision], N[(b * a + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c i) < -5.00000000000000008e99 or 2e17 < (*.f64 c i)

   1. Initial program 92.7%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 87.5

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

   5. Applied rewrites 87.5%

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

   if -5.00000000000000008e99 < (*.f64 c i) < 2e17

   1. Initial program 97.9%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 94.8

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

   5. Applied rewrites 94.8%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{+99} \lor \neg \left(c \cdot i \leq 2 \cdot 10^{+17}\right):\\ \;\;\;\;\mathsf{fma}\left(i, c, \mathsf{fma}\left(t, z, y \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(t, z, y \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{+207}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \mathbf{elif}\;c \cdot i \leq 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(t, z, y \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -5e+207)
   (fma i c (* a b))
   (if (<= (* c i) 1e+147) (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) <= -5e+207) {
		tmp = fma(i, c, (a * b));
	} else if ((c * i) <= 1e+147) {
		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) <= -5e+207)
		tmp = fma(i, c, Float64(a * b));
	elseif (Float64(c * i) <= 1e+147)
		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], -5e+207], N[(i * c + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 1e+147], 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) < -4.9999999999999999e207

   1. Initial program 87.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 86.7

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

   5. Applied rewrites 86.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 89.9

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

      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 89.9

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

   7. Applied rewrites 89.9%

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

   if -4.9999999999999999e207 < (*.f64 c i) < 9.9999999999999998e146

   1. Initial program 97.3%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 89.8

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

   5. Applied rewrites 89.8%

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

   if 9.9999999999999998e146 < (*.f64 c i)

   1. Initial program 94.4%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 94.6

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

   5. Applied rewrites 94.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 94.6

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

      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 94.6

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

   7. Applied rewrites 94.6%

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

Alternative 11: 43.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -1 \cdot 10^{+99}:\\ \;\;\;\;i \cdot c\\ \mathbf{elif}\;c \cdot i \leq 5 \cdot 10^{-290}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;c \cdot i \leq 5 \cdot 10^{+105}:\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;i \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -1e+99)
   (* i c)
   (if (<= (* c i) 5e-290) (* b a) (if (<= (* c i) 5e+105) (* 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+99) {
		tmp = i * c;
	} else if ((c * i) <= 5e-290) {
		tmp = b * a;
	} else if ((c * i) <= 5e+105) {
		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+99)) then
        tmp = i * c
    else if ((c * i) <= 5d-290) then
        tmp = b * a
    else if ((c * i) <= 5d+105) 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+99) {
		tmp = i * c;
	} else if ((c * i) <= 5e-290) {
		tmp = b * a;
	} else if ((c * i) <= 5e+105) {
		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+99:
		tmp = i * c
	elif (c * i) <= 5e-290:
		tmp = b * a
	elif (c * i) <= 5e+105:
		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+99)
		tmp = Float64(i * c);
	elseif (Float64(c * i) <= 5e-290)
		tmp = Float64(b * a);
	elseif (Float64(c * i) <= 5e+105)
		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+99)
		tmp = i * c;
	elseif ((c * i) <= 5e-290)
		tmp = b * a;
	elseif ((c * i) <= 5e+105)
		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+99], N[(i * c), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 5e-290], N[(b * a), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 5e+105], N[(t * z), $MachinePrecision], N[(i * c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c i) < -9.9999999999999997e98 or 5.00000000000000046e105 < (*.f64 c i)

   1. Initial program 93.4%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 66.8

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

   5. Applied rewrites 66.8%

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

   if -9.9999999999999997e98 < (*.f64 c i) < 5.0000000000000001e-290

   1. Initial program 98.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 44.1

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

   5. Applied rewrites 44.1%

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

   if 5.0000000000000001e-290 < (*.f64 c i) < 5.00000000000000046e105

   1. Initial program 95.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 38.9

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

   5. Applied rewrites 38.9%

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

Alternative 12: 63.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{+207} \lor \neg \left(c \cdot i \leq 10^{+159}\right):\\ \;\;\;\;i \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* c i) -5e+207) (not (<= (* c i) 1e+159)))
   (* i c)
   (fma z t (* 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) <= -5e+207) || !((c * i) <= 1e+159)) {
		tmp = i * c;
	} else {
		tmp = fma(z, t, (x * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(c * i) <= -5e+207) || !(Float64(c * i) <= 1e+159))
		tmp = Float64(i * c);
	else
		tmp = fma(z, t, Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(c * i), $MachinePrecision], -5e+207], N[Not[LessEqual[N[(c * i), $MachinePrecision], 1e+159]], $MachinePrecision]], N[(i * c), $MachinePrecision], N[(z * t + N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c i) < -4.9999999999999999e207 or 9.9999999999999993e158 < (*.f64 c i)

   1. Initial program 92.3%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 81.7

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

   5. Applied rewrites 81.7%

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

   if -4.9999999999999999e207 < (*.f64 c i) < 9.9999999999999993e158

   1. Initial program 96.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      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

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      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 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 69.5

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

   7. Applied rewrites 69.5%

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

   8. Taylor expanded in c around 0

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

      1. *-commutative N/A

         \[\leadsto z \cdot t + x \cdot y \]

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 60.9

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

   10. Applied rewrites 60.9%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{+207} \lor \neg \left(c \cdot i \leq 10^{+159}\right):\\ \;\;\;\;i \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 42.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -1 \cdot 10^{+99} \lor \neg \left(c \cdot i \leq 2 \cdot 10^{+17}\right):\\ \;\;\;\;i \cdot c\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* c i) -1e+99) (not (<= (* c i) 2e+17))) (* i c) (* b a)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((c * i) <= -1e+99) || !((c * i) <= 2e+17)) {
		tmp = i * c;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if (((c * i) <= (-1d+99)) .or. (.not. ((c * i) <= 2d+17))) then
        tmp = i * c
    else
        tmp = b * a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((c * i) <= -1e+99) || !((c * i) <= 2e+17)) {
		tmp = i * c;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if ((c * i) <= -1e+99) or not ((c * i) <= 2e+17):
		tmp = i * c
	else:
		tmp = b * a
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(c * i) <= -1e+99) || !(Float64(c * i) <= 2e+17))
		tmp = Float64(i * c);
	else
		tmp = Float64(b * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (((c * i) <= -1e+99) || ~(((c * i) <= 2e+17)))
		tmp = i * c;
	else
		tmp = b * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(c * i), $MachinePrecision], -1e+99], N[Not[LessEqual[N[(c * i), $MachinePrecision], 2e+17]], $MachinePrecision]], N[(i * c), $MachinePrecision], N[(b * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c i) < -9.9999999999999997e98 or 2e17 < (*.f64 c i)

   1. Initial program 92.8%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 58.7

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

   5. Applied rewrites 58.7%

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

   if -9.9999999999999997e98 < (*.f64 c i) < 2e17

   1. Initial program 97.9%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 36.5

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

   5. Applied rewrites 36.5%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -1 \cdot 10^{+99} \lor \neg \left(c \cdot i \leq 2 \cdot 10^{+17}\right):\\ \;\;\;\;i \cdot c\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 27.6% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.7%

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

3. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 27.7

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

5. Applied rewrites 27.7%

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

Reproduce

herbie shell --seed 2025057 
(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)))