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

Percentage Accurate: 95.4% → 96.5%

Time: 3.2s

Alternatives: 12

Speedup: 1.1×

• 41.6% 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: 95.4% 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: 96.5% accurate, 1.1× 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 95.4%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. associate-+l+ N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-*.f64 96.5

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

3. Applied rewrites 96.5%

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

Alternative 2: 89.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, a, \mathsf{fma}\left(t, z, y \cdot x\right)\right)\\ \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+32}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, y \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma b a (fma t z (* y x)))))
   (if (<= (* z t) -1e+32)
     t_1
     (if (<= (* z t) 1e+122) (fma b a (fma i c (* y x))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(b, a, fma(t, z, (y * x)));
	double tmp;
	if ((z * t) <= -1e+32) {
		tmp = t_1;
	} else if ((z * t) <= 1e+122) {
		tmp = fma(b, a, fma(i, c, (y * x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(b, a, fma(t, z, Float64(y * x)))
	tmp = 0.0
	if (Float64(z * t) <= -1e+32)
		tmp = t_1;
	elseif (Float64(z * t) <= 1e+122)
		tmp = fma(b, a, fma(i, c, Float64(y * x)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(b * a + N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -1e+32], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 1e+122], N[(b * a + N[(i * c + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.00000000000000005e32 or 1.00000000000000001e122 < (*.f64 z t)

   1. Initial program 92.1%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 84.6

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

   4. Applied rewrites 84.6%

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

   if -1.00000000000000005e32 < (*.f64 z t) < 1.00000000000000001e122

   1. Initial program 97.4%

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

   2. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 92.2

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

   4. Applied rewrites 92.2%

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

Alternative 3: 85.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+143}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{+145}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, y \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, t \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* z t) -5e+143)
   (fma b a (* t z))
   (if (<= (* z t) 4e+145) (fma b a (fma i c (* y x))) (fma x y (* t z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((z * t) <= -5e+143) {
		tmp = fma(b, a, (t * z));
	} else if ((z * t) <= 4e+145) {
		tmp = fma(b, a, fma(i, c, (y * x)));
	} else {
		tmp = fma(x, y, (t * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(z * t) <= -5e+143)
		tmp = fma(b, a, Float64(t * z));
	elseif (Float64(z * t) <= 4e+145)
		tmp = fma(b, a, fma(i, c, Float64(y * x)));
	else
		tmp = fma(x, y, Float64(t * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(z * t), $MachinePrecision], -5e+143], N[(b * a + N[(t * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 4e+145], N[(b * a + N[(i * c + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * y + N[(t * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -5.00000000000000012e143

   1. Initial program 89.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 92.8

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

   3. Applied rewrites 92.8%

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

   4. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      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

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

      7. +-commutative N/A

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

      8. +-commutative N/A

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

      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

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-*.f64 84.5

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

   6. Applied rewrites 84.5%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 75.3

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

   9. Applied rewrites 75.3%

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

   if -5.00000000000000012e143 < (*.f64 z t) < 4e145

   1. Initial program 97.4%

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

   2. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 89.3

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

   4. Applied rewrites 89.3%

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

   if 4e145 < (*.f64 z t)

   1. Initial program 91.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 94.1

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

   3. Applied rewrites 94.1%

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

   4. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      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

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

      7. +-commutative N/A

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

      8. +-commutative N/A

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

      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

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-*.f64 87.3

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

   6. Applied rewrites 87.3%

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

   7. Taylor expanded in z around inf

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

      1. lower-*.f64 77.4

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

   9. Applied rewrites 77.4%

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

Alternative 4: 74.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, y, t \cdot z\right)\\ t_2 := x \cdot y + z \cdot t\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+143}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(b, a, i \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma x y (* t z))) (t_2 (+ (* x y) (* z t))))
   (if (<= t_2 -5e+143) t_1 (if (<= t_2 5e+151) (fma b a (* i c)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(x, y, (t * z));
	double t_2 = (x * y) + (z * t);
	double tmp;
	if (t_2 <= -5e+143) {
		tmp = t_1;
	} else if (t_2 <= 5e+151) {
		tmp = fma(b, a, (i * c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(x, y, Float64(t * z))
	t_2 = Float64(Float64(x * y) + Float64(z * t))
	tmp = 0.0
	if (t_2 <= -5e+143)
		tmp = t_1;
	elseif (t_2 <= 5e+151)
		tmp = fma(b, a, Float64(i * c));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x * y + N[(t * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+143], t$95$1, If[LessEqual[t$95$2, 5e+151], N[(b * a + N[(i * c), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x y) (*.f64 z t)) < -5.00000000000000012e143 or 5.0000000000000002e151 < (+.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 91.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 94.1

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

   3. Applied rewrites 94.1%

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

   4. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      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

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

      7. +-commutative N/A

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

      8. +-commutative N/A

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

      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

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-*.f64 86.5

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

   6. Applied rewrites 86.5%

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

   7. Taylor expanded in z around inf

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

      1. lower-*.f64 76.2

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

   9. Applied rewrites 76.2%

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

   if -5.00000000000000012e143 < (+.f64 (*.f64 x y) (*.f64 z t)) < 5.0000000000000002e151

   1. Initial program 98.9%

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

   2. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 86.1

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

   4. Applied rewrites 86.1%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 73.2

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

   7. Applied rewrites 73.2%

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

Alternative 5: 65.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, a, i \cdot c\right)\\ t_2 := \mathsf{fma}\left(b, a, t \cdot z\right)\\ \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+32}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \cdot t \leq -5 \cdot 10^{-140}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{-319}:\\ \;\;\;\;\mathsf{fma}\left(x, y, b \cdot a\right)\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-128}:\\ \;\;\;\;\mathsf{fma}\left(i, c, x \cdot y\right)\\ \mathbf{elif}\;z \cdot t \leq 10^{+111}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma b a (* i c))) (t_2 (fma b a (* t z))))
   (if (<= (* z t) -1e+32)
     t_2
     (if (<= (* z t) -5e-140)
       t_1
       (if (<= (* z t) -1e-319)
         (fma x y (* b a))
         (if (<= (* z t) 2e-128)
           (fma i c (* x y))
           (if (<= (* z t) 1e+111) t_1 t_2)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(b, a, (i * c));
	double t_2 = fma(b, a, (t * z));
	double tmp;
	if ((z * t) <= -1e+32) {
		tmp = t_2;
	} else if ((z * t) <= -5e-140) {
		tmp = t_1;
	} else if ((z * t) <= -1e-319) {
		tmp = fma(x, y, (b * a));
	} else if ((z * t) <= 2e-128) {
		tmp = fma(i, c, (x * y));
	} else if ((z * t) <= 1e+111) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(b, a, Float64(i * c))
	t_2 = fma(b, a, Float64(t * z))
	tmp = 0.0
	if (Float64(z * t) <= -1e+32)
		tmp = t_2;
	elseif (Float64(z * t) <= -5e-140)
		tmp = t_1;
	elseif (Float64(z * t) <= -1e-319)
		tmp = fma(x, y, Float64(b * a));
	elseif (Float64(z * t) <= 2e-128)
		tmp = fma(i, c, Float64(x * y));
	elseif (Float64(z * t) <= 1e+111)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(b * a + N[(i * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b * a + N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -1e+32], t$95$2, If[LessEqual[N[(z * t), $MachinePrecision], -5e-140], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], -1e-319], N[(x * y + N[(b * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e-128], N[(i * c + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+111], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 z t) < -1.00000000000000005e32 or 9.99999999999999957e110 < (*.f64 z t)

   1. Initial program 92.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 94.6

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

   3. Applied rewrites 94.6%

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

   4. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      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

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

      7. +-commutative N/A

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

      8. +-commutative N/A

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

      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

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-*.f64 83.9

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

   6. Applied rewrites 83.9%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 71.2

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

   9. Applied rewrites 71.2%

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

   if -1.00000000000000005e32 < (*.f64 z t) < -5.00000000000000015e-140 or 2.00000000000000011e-128 < (*.f64 z t) < 9.99999999999999957e110

   1. Initial program 97.0%

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

   2. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 86.9

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

   4. Applied rewrites 86.9%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 58.8

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

   7. Applied rewrites 58.8%

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

   if -5.00000000000000015e-140 < (*.f64 z t) < -9.99989e-320

   1. Initial program 97.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 98.0

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

   3. Applied rewrites 98.0%

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

   4. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      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

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

      7. +-commutative N/A

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

      8. +-commutative N/A

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

      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

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-*.f64 68.0

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

   6. Applied rewrites 68.0%

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

   7. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 66.9

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

   9. Applied rewrites 66.9%

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

   if -9.99989e-320 < (*.f64 z t) < 2.00000000000000011e-128

   1. Initial program 97.9%

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

   2. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 98.0

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

         \[\leadsto i \cdot c + x \cdot y \]

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 65.2

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

   7. Applied rewrites 65.2%

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

Alternative 6: 65.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, a, i \cdot c\right)\\ t_2 := \mathsf{fma}\left(b, a, t \cdot z\right)\\ \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+32}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{-200}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{-319}:\\ \;\;\;\;\mathsf{fma}\left(b, a, x \cdot y\right)\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-128}:\\ \;\;\;\;\mathsf{fma}\left(i, c, x \cdot y\right)\\ \mathbf{elif}\;z \cdot t \leq 10^{+111}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma b a (* i c))) (t_2 (fma b a (* t z))))
   (if (<= (* z t) -1e+32)
     t_2
     (if (<= (* z t) -1e-200)
       t_1
       (if (<= (* z t) -1e-319)
         (fma b a (* x y))
         (if (<= (* z t) 2e-128)
           (fma i c (* x y))
           (if (<= (* z t) 1e+111) t_1 t_2)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(b, a, (i * c));
	double t_2 = fma(b, a, (t * z));
	double tmp;
	if ((z * t) <= -1e+32) {
		tmp = t_2;
	} else if ((z * t) <= -1e-200) {
		tmp = t_1;
	} else if ((z * t) <= -1e-319) {
		tmp = fma(b, a, (x * y));
	} else if ((z * t) <= 2e-128) {
		tmp = fma(i, c, (x * y));
	} else if ((z * t) <= 1e+111) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(b, a, Float64(i * c))
	t_2 = fma(b, a, Float64(t * z))
	tmp = 0.0
	if (Float64(z * t) <= -1e+32)
		tmp = t_2;
	elseif (Float64(z * t) <= -1e-200)
		tmp = t_1;
	elseif (Float64(z * t) <= -1e-319)
		tmp = fma(b, a, Float64(x * y));
	elseif (Float64(z * t) <= 2e-128)
		tmp = fma(i, c, Float64(x * y));
	elseif (Float64(z * t) <= 1e+111)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(b * a + N[(i * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b * a + N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -1e+32], t$95$2, If[LessEqual[N[(z * t), $MachinePrecision], -1e-200], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], -1e-319], N[(b * a + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e-128], N[(i * c + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+111], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 z t) < -1.00000000000000005e32 or 9.99999999999999957e110 < (*.f64 z t)

   1. Initial program 92.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 94.6

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

   3. Applied rewrites 94.6%

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

   4. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      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

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

      7. +-commutative N/A

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

      8. +-commutative N/A

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

      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

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-*.f64 83.9

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

   6. Applied rewrites 83.9%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 71.2

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

   9. Applied rewrites 71.2%

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

   if -1.00000000000000005e32 < (*.f64 z t) < -9.9999999999999998e-201 or 2.00000000000000011e-128 < (*.f64 z t) < 9.99999999999999957e110

   1. Initial program 97.0%

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

   2. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 87.8

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

   4. Applied rewrites 87.8%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 59.6

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

   7. Applied rewrites 59.6%

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

   if -9.9999999999999998e-201 < (*.f64 z t) < -9.99989e-320

   1. Initial program 97.8%

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

   2. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 98.1

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

   4. Applied rewrites 98.1%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 64.4

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

   7. Applied rewrites 64.4%

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

   if -9.99989e-320 < (*.f64 z t) < 2.00000000000000011e-128

   1. Initial program 97.9%

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

   2. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 98.0

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

         \[\leadsto i \cdot c + x \cdot y \]

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 65.2

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

   7. Applied rewrites 65.2%

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

Alternative 7: 65.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, a, i \cdot c\right)\\ t_2 := \mathsf{fma}\left(b, a, t \cdot z\right)\\ \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+32}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{-319}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-128}:\\ \;\;\;\;\mathsf{fma}\left(i, c, x \cdot y\right)\\ \mathbf{elif}\;z \cdot t \leq 10^{+111}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma b a (* i c))) (t_2 (fma b a (* t z))))
   (if (<= (* z t) -1e+32)
     t_2
     (if (<= (* z t) -1e-319)
       t_1
       (if (<= (* z t) 2e-128)
         (fma i c (* x y))
         (if (<= (* z t) 1e+111) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(b, a, (i * c));
	double t_2 = fma(b, a, (t * z));
	double tmp;
	if ((z * t) <= -1e+32) {
		tmp = t_2;
	} else if ((z * t) <= -1e-319) {
		tmp = t_1;
	} else if ((z * t) <= 2e-128) {
		tmp = fma(i, c, (x * y));
	} else if ((z * t) <= 1e+111) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(b, a, Float64(i * c))
	t_2 = fma(b, a, Float64(t * z))
	tmp = 0.0
	if (Float64(z * t) <= -1e+32)
		tmp = t_2;
	elseif (Float64(z * t) <= -1e-319)
		tmp = t_1;
	elseif (Float64(z * t) <= 2e-128)
		tmp = fma(i, c, Float64(x * y));
	elseif (Float64(z * t) <= 1e+111)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(b * a + N[(i * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b * a + N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -1e+32], t$95$2, If[LessEqual[N[(z * t), $MachinePrecision], -1e-319], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 2e-128], N[(i * c + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+111], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -1.00000000000000005e32 or 9.99999999999999957e110 < (*.f64 z t)

   1. Initial program 92.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 94.6

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

   3. Applied rewrites 94.6%

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

   4. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      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

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

      7. +-commutative N/A

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

      8. +-commutative N/A

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

      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

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-*.f64 83.9

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

   6. Applied rewrites 83.9%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 71.2

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

   9. Applied rewrites 71.2%

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

   if -1.00000000000000005e32 < (*.f64 z t) < -9.99989e-320 or 2.00000000000000011e-128 < (*.f64 z t) < 9.99999999999999957e110

   1. Initial program 97.1%

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

   2. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 89.2

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

   4. Applied rewrites 89.2%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 60.3

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

   7. Applied rewrites 60.3%

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

   if -9.99989e-320 < (*.f64 z t) < 2.00000000000000011e-128

   1. Initial program 97.9%

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

   2. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 98.0

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

         \[\leadsto i \cdot c + x \cdot y \]

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 65.2

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

   7. Applied rewrites 65.2%

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

Alternative 8: 65.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, a, t \cdot z\right)\\ \mathbf{if}\;z \cdot t \leq -20000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{+117}:\\ \;\;\;\;\mathsf{fma}\left(i, c, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma b a (* t z))))
   (if (<= (* z t) -20000000.0)
     t_1
     (if (<= (* z t) 4e+117) (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(b, a, (t * z));
	double tmp;
	if ((z * t) <= -20000000.0) {
		tmp = t_1;
	} else if ((z * t) <= 4e+117) {
		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(b, a, Float64(t * z))
	tmp = 0.0
	if (Float64(z * t) <= -20000000.0)
		tmp = t_1;
	elseif (Float64(z * t) <= 4e+117)
		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[(b * a + N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -20000000.0], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 4e+117], N[(i * c + N[(x * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2e7 or 4.0000000000000002e117 < (*.f64 z t)

   1. Initial program 92.4%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 94.7

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

   3. Applied rewrites 94.7%

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

   4. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      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

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

      7. +-commutative N/A

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

      8. +-commutative N/A

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

      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

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-*.f64 83.5

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

   6. Applied rewrites 83.5%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 70.3

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

   9. Applied rewrites 70.3%

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

   if -2e7 < (*.f64 z t) < 4.0000000000000002e117

   1. Initial program 97.4%

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

   2. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 92.5

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

   4. Applied rewrites 92.5%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

         \[\leadsto i \cdot c + x \cdot y \]

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 62.6

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

   7. Applied rewrites 62.6%

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

Alternative 9: 61.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+104}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;z \cdot t \leq 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(i, c, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* z t) -5e+104)
   (* t z)
   (if (<= (* z t) 1e+122) (fma i c (* x y)) (* t z))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((z * t) <= -5e+104) {
		tmp = t * z;
	} else if ((z * t) <= 1e+122) {
		tmp = fma(i, c, (x * y));
	} else {
		tmp = t * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(z * t) <= -5e+104)
		tmp = Float64(t * z);
	elseif (Float64(z * t) <= 1e+122)
		tmp = fma(i, c, Float64(x * y));
	else
		tmp = Float64(t * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(z * t), $MachinePrecision], -5e+104], N[(t * z), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+122], N[(i * c + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(t * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -4.9999999999999997e104 or 1.00000000000000001e122 < (*.f64 z t)

   1. Initial program 91.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 61.7

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

   4. Applied rewrites 61.7%

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

   if -4.9999999999999997e104 < (*.f64 z t) < 1.00000000000000001e122

   1. Initial program 97.3%

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

   2. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 90.8

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

   4. Applied rewrites 90.8%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

         \[\leadsto i \cdot c + x \cdot y \]

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 61.4

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

   7. Applied rewrites 61.4%

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

Alternative 10: 42.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+62}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{-319}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;z \cdot t \leq 10^{+111}:\\ \;\;\;\;i \cdot c\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* z t) -1e+62)
   (* t z)
   (if (<= (* z t) -1e-319) (* b a) (if (<= (* z t) 1e+111) (* i c) (* t z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((z * t) <= -1e+62) {
		tmp = t * z;
	} else if ((z * t) <= -1e-319) {
		tmp = b * a;
	} else if ((z * t) <= 1e+111) {
		tmp = i * c;
	} else {
		tmp = t * z;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if ((z * t) <= (-1d+62)) then
        tmp = t * z
    else if ((z * t) <= (-1d-319)) then
        tmp = b * a
    else if ((z * t) <= 1d+111) then
        tmp = i * c
    else
        tmp = t * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((z * t) <= -1e+62) {
		tmp = t * z;
	} else if ((z * t) <= -1e-319) {
		tmp = b * a;
	} else if ((z * t) <= 1e+111) {
		tmp = i * c;
	} else {
		tmp = t * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (z * t) <= -1e+62:
		tmp = t * z
	elif (z * t) <= -1e-319:
		tmp = b * a
	elif (z * t) <= 1e+111:
		tmp = i * c
	else:
		tmp = t * z
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(z * t) <= -1e+62)
		tmp = Float64(t * z);
	elseif (Float64(z * t) <= -1e-319)
		tmp = Float64(b * a);
	elseif (Float64(z * t) <= 1e+111)
		tmp = Float64(i * c);
	else
		tmp = Float64(t * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((z * t) <= -1e+62)
		tmp = t * z;
	elseif ((z * t) <= -1e-319)
		tmp = b * a;
	elseif ((z * t) <= 1e+111)
		tmp = i * c;
	else
		tmp = t * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(z * t), $MachinePrecision], -1e+62], N[(t * z), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], -1e-319], N[(b * a), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+111], N[(i * c), $MachinePrecision], N[(t * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -1.00000000000000004e62 or 9.99999999999999957e110 < (*.f64 z t)

   1. Initial program 92.0%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 58.3

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

   4. Applied rewrites 58.3%

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

   if -1.00000000000000004e62 < (*.f64 z t) < -9.99989e-320

   1. Initial program 97.3%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 33.4

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

   4. Applied rewrites 33.4%

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

   if -9.99989e-320 < (*.f64 z t) < 9.99999999999999957e110

   1. Initial program 97.4%

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

   2. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 31.3

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

   4. Applied rewrites 31.3%

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

Alternative 11: 41.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+179}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{+153}:\\ \;\;\;\;i \cdot c\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* a b) -2e+179) (* b a) (if (<= (* a b) 5e+153) (* i c) (* b a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((a * b) <= -2e+179) {
		tmp = b * a;
	} else if ((a * b) <= 5e+153) {
		tmp = i * c;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if ((a * b) <= (-2d+179)) then
        tmp = b * a
    else if ((a * b) <= 5d+153) then
        tmp = i * c
    else
        tmp = b * a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((a * b) <= -2e+179) {
		tmp = b * a;
	} else if ((a * b) <= 5e+153) {
		tmp = i * c;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (a * b) <= -2e+179:
		tmp = b * a
	elif (a * b) <= 5e+153:
		tmp = i * c
	else:
		tmp = b * a
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(a * b) <= -2e+179)
		tmp = Float64(b * a);
	elseif (Float64(a * b) <= 5e+153)
		tmp = Float64(i * c);
	else
		tmp = Float64(b * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((a * b) <= -2e+179)
		tmp = b * a;
	elseif ((a * b) <= 5e+153)
		tmp = i * c;
	else
		tmp = b * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(a * b), $MachinePrecision], -2e+179], N[(b * a), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 5e+153], N[(i * c), $MachinePrecision], N[(b * a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -1.99999999999999996e179 or 5.00000000000000018e153 < (*.f64 a b)

   1. Initial program 89.9%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 71.5

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

   4. Applied rewrites 71.5%

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

   if -1.99999999999999996e179 < (*.f64 a b) < 5.00000000000000018e153

   1. Initial program 97.4%

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

   2. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 31.0

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

   4. Applied rewrites 31.0%

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

Alternative 12: 28.3% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

2. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 28.3

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

4. Applied rewrites 28.3%

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

Reproduce

herbie shell --seed 2025120 
(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)))