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

Percentage Accurate: 95.6% → 97.8%

Time: 3.1s

Alternatives: 10

Speedup: 1.1×

• 42.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 95.6%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-+l+ N/A

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

   7. lift-*.f64 N/A

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

   8. lower-fma.f64 N/A

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

   9. +-commutative N/A

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

   10. +-commutative N/A

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

   11. associate-+l+ N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. add-flip N/A

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

   16. sub-flip N/A

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. remove-double-neg N/A

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

   20. lower-fma.f64 97.8

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

   21. lift-*.f64 N/A

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

   22. *-commutative N/A

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

   23. lower-*.f64 97.8

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

3. Applied rewrites 97.8%

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

Alternative 2: 89.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, b, \mathsf{fma}\left(c, i, x \cdot y\right)\right)\\ \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+200}:\\ \;\;\;\;\mathsf{fma}\left(a, b, t \cdot z\right)\\ \mathbf{elif}\;a \cdot b \leq -2 \cdot 10^{+68}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+28}:\\ \;\;\;\;\mathsf{fma}\left(c, i, \mathsf{fma}\left(t, z, x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma a b (fma c i (* x y)))))
   (if (<= (* a b) -2e+200)
     (fma a b (* t z))
     (if (<= (* a b) -2e+68)
       t_1
       (if (<= (* a b) 2e+28) (fma c i (fma t z (* 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(a, b, fma(c, i, (x * y)));
	double tmp;
	if ((a * b) <= -2e+200) {
		tmp = fma(a, b, (t * z));
	} else if ((a * b) <= -2e+68) {
		tmp = t_1;
	} else if ((a * b) <= 2e+28) {
		tmp = fma(c, i, fma(t, z, (x * y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(a, b, fma(c, i, Float64(x * y)))
	tmp = 0.0
	if (Float64(a * b) <= -2e+200)
		tmp = fma(a, b, Float64(t * z));
	elseif (Float64(a * b) <= -2e+68)
		tmp = t_1;
	elseif (Float64(a * b) <= 2e+28)
		tmp = fma(c, i, fma(t, z, 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[(a * b + N[(c * i + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -2e+200], N[(a * b + N[(t * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], -2e+68], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], 2e+28], N[(c * i + N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -1.9999999999999999e200

   1. Initial program 95.6%

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

   2. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 76.2

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

   4. Applied rewrites 76.2%

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

   5. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 52.5

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

   7. Applied rewrites 52.5%

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

   if -1.9999999999999999e200 < (*.f64 a b) < -1.99999999999999991e68 or 1.99999999999999992e28 < (*.f64 a b)

   1. Initial program 95.6%

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

   2. Taylor expanded in z around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 74.9

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

   4. Applied rewrites 74.9%

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

   if -1.99999999999999991e68 < (*.f64 a b) < 1.99999999999999992e28

   1. Initial program 95.6%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 74.5

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

   4. Applied rewrites 74.5%

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

Alternative 3: 88.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, b, \mathsf{fma}\left(c, i, x \cdot y\right)\right)\\ \mathbf{if}\;x \cdot y \leq -2.05 \cdot 10^{+55}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 3.1 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(a, b, \mathsf{fma}\left(c, i, t \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma a b (fma c i (* x y)))))
   (if (<= (* x y) -2.05e+55)
     t_1
     (if (<= (* x y) 3.1e+144) (fma a b (fma c i (* t z))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(a, b, fma(c, i, (x * y)));
	double tmp;
	if ((x * y) <= -2.05e+55) {
		tmp = t_1;
	} else if ((x * y) <= 3.1e+144) {
		tmp = fma(a, b, fma(c, i, (t * z)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(a, b, fma(c, i, Float64(x * y)))
	tmp = 0.0
	if (Float64(x * y) <= -2.05e+55)
		tmp = t_1;
	elseif (Float64(x * y) <= 3.1e+144)
		tmp = fma(a, b, fma(c, i, Float64(t * z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(a * b + N[(c * i + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -2.05e+55], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 3.1e+144], N[(a * b + N[(c * i + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2.04999999999999991e55 or 3.1000000000000002e144 < (*.f64 x y)

   1. Initial program 95.6%

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

   2. Taylor expanded in z around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 74.9

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

   4. Applied rewrites 74.9%

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

   if -2.04999999999999991e55 < (*.f64 x y) < 3.1000000000000002e144

   1. Initial program 95.6%

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

   2. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 76.2

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

   4. Applied rewrites 76.2%

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

Alternative 4: 85.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1.25 \cdot 10^{+241}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 1.65 \cdot 10^{+180}:\\ \;\;\;\;\mathsf{fma}\left(a, b, \mathsf{fma}\left(c, i, t \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* x y) -1.25e+241)
   (* x y)
   (if (<= (* x y) 1.65e+180) (fma a b (fma c i (* t z))) (fma a b (* x y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((x * y) <= -1.25e+241) {
		tmp = x * y;
	} else if ((x * y) <= 1.65e+180) {
		tmp = fma(a, b, fma(c, i, (t * z)));
	} else {
		tmp = fma(a, b, (x * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(x * y) <= -1.25e+241)
		tmp = Float64(x * y);
	elseif (Float64(x * y) <= 1.65e+180)
		tmp = fma(a, b, fma(c, i, Float64(t * z)));
	else
		tmp = fma(a, b, Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(x * y), $MachinePrecision], -1.25e+241], N[(x * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1.65e+180], N[(a * b + N[(c * i + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * b + N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1.25000000000000006e241

   1. Initial program 95.6%

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

   2. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 76.2

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

   4. Applied rewrites 76.2%

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

   5. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 52.5

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

   7. Applied rewrites 52.5%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 26.7

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

   10. Applied rewrites 26.7%

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

   if -1.25000000000000006e241 < (*.f64 x y) < 1.64999999999999995e180

   1. Initial program 95.6%

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

   2. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 76.2

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

   4. Applied rewrites 76.2%

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

   if 1.64999999999999995e180 < (*.f64 x y)

   1. Initial program 95.6%

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

   2. Taylor expanded in z around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 74.9

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

   4. Applied rewrites 74.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 52.6

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

   7. Applied rewrites 52.6%

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

   8. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 51.4

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

   10. Applied rewrites 51.4%

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

Alternative 5: 67.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+192}:\\ \;\;\;\;\mathsf{fma}\left(a, b, t \cdot z\right)\\ \mathbf{elif}\;a \cdot b \leq -2 \cdot 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(a, b, c \cdot i\right)\\ \mathbf{elif}\;a \cdot b \leq 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(c, i, t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* a b) -2e+192)
   (fma a b (* t z))
   (if (<= (* a b) -2e+68)
     (fma a b (* c i))
     (if (<= (* a b) 1e+30) (fma c i (* t z)) (fma a b (* x y))))))

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+192) {
		tmp = fma(a, b, (t * z));
	} else if ((a * b) <= -2e+68) {
		tmp = fma(a, b, (c * i));
	} else if ((a * b) <= 1e+30) {
		tmp = fma(c, i, (t * z));
	} else {
		tmp = fma(a, b, (x * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(a * b) <= -2e+192)
		tmp = fma(a, b, Float64(t * z));
	elseif (Float64(a * b) <= -2e+68)
		tmp = fma(a, b, Float64(c * i));
	elseif (Float64(a * b) <= 1e+30)
		tmp = fma(c, i, Float64(t * z));
	else
		tmp = fma(a, b, Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(a * b), $MachinePrecision], -2e+192], N[(a * b + N[(t * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], -2e+68], N[(a * b + N[(c * i), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 1e+30], N[(c * i + N[(t * z), $MachinePrecision]), $MachinePrecision], N[(a * b + N[(x * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 a b) < -2.00000000000000008e192

   1. Initial program 95.6%

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

   2. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 76.2

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

   4. Applied rewrites 76.2%

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

   5. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 52.5

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

   7. Applied rewrites 52.5%

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

   if -2.00000000000000008e192 < (*.f64 a b) < -1.99999999999999991e68

   1. Initial program 95.6%

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

   2. Taylor expanded in z around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 74.9

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

   4. Applied rewrites 74.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 52.6

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

   7. Applied rewrites 52.6%

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

   if -1.99999999999999991e68 < (*.f64 a b) < 1e30

   1. Initial program 95.6%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 74.5

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

   4. Applied rewrites 74.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 52.0

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

   7. Applied rewrites 52.0%

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

   if 1e30 < (*.f64 a b)

   1. Initial program 95.6%

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

   2. Taylor expanded in z around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 74.9

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

   4. Applied rewrites 74.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 52.6

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

   7. Applied rewrites 52.6%

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

   8. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 51.4

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

   10. Applied rewrites 51.4%

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

Alternative 6: 67.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, b, c \cdot i\right)\\ \mathbf{if}\;c \cdot i \leq -2.5 \cdot 10^{+98}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \cdot i \leq 1.16 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(a, b, t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma a b (* c i))))
   (if (<= (* c i) -2.5e+98)
     t_1
     (if (<= (* c i) 1.16e+41) (fma a b (* t z)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(a, b, (c * i));
	double tmp;
	if ((c * i) <= -2.5e+98) {
		tmp = t_1;
	} else if ((c * i) <= 1.16e+41) {
		tmp = fma(a, b, (t * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(a, b, Float64(c * i))
	tmp = 0.0
	if (Float64(c * i) <= -2.5e+98)
		tmp = t_1;
	elseif (Float64(c * i) <= 1.16e+41)
		tmp = fma(a, b, Float64(t * z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(a * b + N[(c * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(c * i), $MachinePrecision], -2.5e+98], t$95$1, If[LessEqual[N[(c * i), $MachinePrecision], 1.16e+41], N[(a * b + N[(t * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c i) < -2.4999999999999999e98 or 1.16000000000000007e41 < (*.f64 c i)

   1. Initial program 95.6%

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

   2. Taylor expanded in z around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 74.9

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

   4. Applied rewrites 74.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 52.6

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

   7. Applied rewrites 52.6%

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

   if -2.4999999999999999e98 < (*.f64 c i) < 1.16000000000000007e41

   1. Initial program 95.6%

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

   2. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 76.2

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

   4. Applied rewrites 76.2%

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

   5. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 52.5

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

   7. Applied rewrites 52.5%

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

Alternative 7: 62.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -4.5 \cdot 10^{+106}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq 3.5 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(a, b, t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -4.5e+106)
   (* c i)
   (if (<= (* c i) 3.5e+51) (fma a b (* t z)) (* c i))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((c * i) <= -4.5e+106) {
		tmp = c * i;
	} else if ((c * i) <= 3.5e+51) {
		tmp = fma(a, b, (t * z));
	} else {
		tmp = c * i;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(c * i) <= -4.5e+106)
		tmp = Float64(c * i);
	elseif (Float64(c * i) <= 3.5e+51)
		tmp = fma(a, b, Float64(t * z));
	else
		tmp = Float64(c * i);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(c * i), $MachinePrecision], -4.5e+106], N[(c * i), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 3.5e+51], N[(a * b + N[(t * z), $MachinePrecision]), $MachinePrecision], N[(c * i), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c i) < -4.4999999999999997e106 or 3.5e51 < (*.f64 c i)

   1. Initial program 95.6%

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

   2. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 76.2

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

   4. Applied rewrites 76.2%

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

   5. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 52.5

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

   7. Applied rewrites 52.5%

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

   8. Taylor expanded in z around 0

      \[\leadsto a \cdot b \]
   9. Step-by-step derivation

      1. lower-*.f64 28.2

         \[\leadsto a \cdot b \]

   10. Applied rewrites 28.2%

       \[\leadsto a \cdot b \]

   11. Taylor expanded in c around inf

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

       1. lower-*.f64 27.7

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

   13. Applied rewrites 27.7%

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

   if -4.4999999999999997e106 < (*.f64 c i) < 3.5e51

   1. Initial program 95.6%

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

   2. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 76.2

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

   4. Applied rewrites 76.2%

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

   5. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 52.5

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

   7. Applied rewrites 52.5%

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

Alternative 8: 43.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -2.5 \cdot 10^{+98}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;c \cdot i \leq 5 \cdot 10^{-199}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;c \cdot i \leq 5.6 \cdot 10^{-111}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;c \cdot i \leq 1.16 \cdot 10^{+41}:\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;c \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -2.5e+98)
   (* c i)
   (if (<= (* c i) 5e-199)
     (* t z)
     (if (<= (* c i) 5.6e-111)
       (* a b)
       (if (<= (* c i) 1.16e+41) (* t z) (* c i))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((c * i) <= -2.5e+98) {
		tmp = c * i;
	} else if ((c * i) <= 5e-199) {
		tmp = t * z;
	} else if ((c * i) <= 5.6e-111) {
		tmp = a * b;
	} else if ((c * i) <= 1.16e+41) {
		tmp = t * z;
	} else {
		tmp = c * i;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if ((c * i) <= (-2.5d+98)) then
        tmp = c * i
    else if ((c * i) <= 5d-199) then
        tmp = t * z
    else if ((c * i) <= 5.6d-111) then
        tmp = a * b
    else if ((c * i) <= 1.16d+41) then
        tmp = t * z
    else
        tmp = c * i
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((c * i) <= -2.5e+98) {
		tmp = c * i;
	} else if ((c * i) <= 5e-199) {
		tmp = t * z;
	} else if ((c * i) <= 5.6e-111) {
		tmp = a * b;
	} else if ((c * i) <= 1.16e+41) {
		tmp = t * z;
	} else {
		tmp = c * i;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (c * i) <= -2.5e+98:
		tmp = c * i
	elif (c * i) <= 5e-199:
		tmp = t * z
	elif (c * i) <= 5.6e-111:
		tmp = a * b
	elif (c * i) <= 1.16e+41:
		tmp = t * z
	else:
		tmp = c * i
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(c * i) <= -2.5e+98)
		tmp = Float64(c * i);
	elseif (Float64(c * i) <= 5e-199)
		tmp = Float64(t * z);
	elseif (Float64(c * i) <= 5.6e-111)
		tmp = Float64(a * b);
	elseif (Float64(c * i) <= 1.16e+41)
		tmp = Float64(t * z);
	else
		tmp = Float64(c * i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((c * i) <= -2.5e+98)
		tmp = c * i;
	elseif ((c * i) <= 5e-199)
		tmp = t * z;
	elseif ((c * i) <= 5.6e-111)
		tmp = a * b;
	elseif ((c * i) <= 1.16e+41)
		tmp = t * z;
	else
		tmp = c * i;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(c * i), $MachinePrecision], -2.5e+98], N[(c * i), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 5e-199], N[(t * z), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 5.6e-111], N[(a * b), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 1.16e+41], N[(t * z), $MachinePrecision], N[(c * i), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c i) < -2.4999999999999999e98 or 1.16000000000000007e41 < (*.f64 c i)

   1. Initial program 95.6%

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

   2. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 76.2

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

   4. Applied rewrites 76.2%

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

   5. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 52.5

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

   7. Applied rewrites 52.5%

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

   8. Taylor expanded in z around 0

      \[\leadsto a \cdot b \]
   9. Step-by-step derivation

      1. lower-*.f64 28.2

         \[\leadsto a \cdot b \]

   10. Applied rewrites 28.2%

       \[\leadsto a \cdot b \]

   11. Taylor expanded in c around inf

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

       1. lower-*.f64 27.7

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

   13. Applied rewrites 27.7%

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

   if -2.4999999999999999e98 < (*.f64 c i) < 4.9999999999999996e-199 or 5.5999999999999999e-111 < (*.f64 c i) < 1.16000000000000007e41

   1. Initial program 95.6%

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

   2. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 76.2

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

   4. Applied rewrites 76.2%

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

   5. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 52.5

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

   7. Applied rewrites 52.5%

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

   8. Taylor expanded in z around 0

      \[\leadsto a \cdot b \]
   9. Step-by-step derivation

      1. lower-*.f64 28.2

         \[\leadsto a \cdot b \]

   10. Applied rewrites 28.2%

       \[\leadsto a \cdot b \]

   11. Taylor expanded in z around inf

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

       1. lower-*.f64 27.6

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

   13. Applied rewrites 27.6%

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

   if 4.9999999999999996e-199 < (*.f64 c i) < 5.5999999999999999e-111

   1. Initial program 95.6%

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

   2. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 76.2

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

   4. Applied rewrites 76.2%

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

   5. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 52.5

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

   7. Applied rewrites 52.5%

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

   8. Taylor expanded in z around 0

      \[\leadsto a \cdot b \]
   9. Step-by-step derivation

      1. lower-*.f64 28.2

         \[\leadsto a \cdot b \]

   10. Applied rewrites 28.2%

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

Alternative 9: 43.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1.7 \cdot 10^{+175}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 2.5 \cdot 10^{+43}:\\ \;\;\;\;c \cdot i\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* a b) -1.7e+175) (* a b) (if (<= (* a b) 2.5e+43) (* c i) (* a b))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((a * b) <= -1.7e+175) {
		tmp = a * b;
	} else if ((a * b) <= 2.5e+43) {
		tmp = c * i;
	} else {
		tmp = a * b;
	}
	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) <= (-1.7d+175)) then
        tmp = a * b
    else if ((a * b) <= 2.5d+43) then
        tmp = c * i
    else
        tmp = a * b
    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) <= -1.7e+175) {
		tmp = a * b;
	} else if ((a * b) <= 2.5e+43) {
		tmp = c * i;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (a * b) <= -1.7e+175:
		tmp = a * b
	elif (a * b) <= 2.5e+43:
		tmp = c * i
	else:
		tmp = a * b
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(a * b) <= -1.7e+175)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= 2.5e+43)
		tmp = Float64(c * i);
	else
		tmp = Float64(a * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((a * b) <= -1.7e+175)
		tmp = a * b;
	elseif ((a * b) <= 2.5e+43)
		tmp = c * i;
	else
		tmp = a * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(a * b), $MachinePrecision], -1.7e+175], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 2.5e+43], N[(c * i), $MachinePrecision], N[(a * b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -1.70000000000000014e175 or 2.5000000000000002e43 < (*.f64 a b)

   1. Initial program 95.6%

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

   2. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 76.2

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

   4. Applied rewrites 76.2%

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

   5. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 52.5

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

   7. Applied rewrites 52.5%

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

   8. Taylor expanded in z around 0

      \[\leadsto a \cdot b \]
   9. Step-by-step derivation

      1. lower-*.f64 28.2

         \[\leadsto a \cdot b \]

   10. Applied rewrites 28.2%

       \[\leadsto a \cdot b \]

   if -1.70000000000000014e175 < (*.f64 a b) < 2.5000000000000002e43

   1. Initial program 95.6%

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

   2. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 76.2

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

   4. Applied rewrites 76.2%

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

   5. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 52.5

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

   7. Applied rewrites 52.5%

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

   8. Taylor expanded in z around 0

      \[\leadsto a \cdot b \]
   9. Step-by-step derivation

      1. lower-*.f64 28.2

         \[\leadsto a \cdot b \]

   10. Applied rewrites 28.2%

       \[\leadsto a \cdot b \]

   11. Taylor expanded in c around inf

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

       1. lower-*.f64 27.7

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

   13. Applied rewrites 27.7%

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

Alternative 10: 28.2% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

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 = a * b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.6%

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

2. Taylor expanded in x around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 76.2

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

4. Applied rewrites 76.2%

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

5. Taylor expanded in c around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-*.f64 52.5

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

7. Applied rewrites 52.5%

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

8. Taylor expanded in z around 0

   \[\leadsto a \cdot b \]
9. Step-by-step derivation

   1. lower-*.f64 28.2

      \[\leadsto a \cdot b \]

10. Applied rewrites 28.2%

    \[\leadsto a \cdot b \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025155 
(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)))