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

Percentage Accurate: 95.9% → 97.9%

Time: 5.9s

Alternatives: 10

Speedup: 1.3×

• 41.8% 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: 95.9% 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: 97.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.5%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. fp-cancel-sign-sub-inv N/A

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

   4. lift-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. associate-+l+ N/A

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

   7. associate--l+ N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. fp-cancel-sign-sub-inv N/A

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

   11. lift-*.f64 N/A

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

   12. lower-fma.f64 N/A

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

   13. associate-+r+ N/A

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

   14. lift-*.f64 N/A

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

   15. remove-double-neg N/A

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

   16. remove-double-neg N/A

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

   17. *-commutative N/A

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

   18. lower-fma.f64 N/A

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

   19. +-commutative N/A

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

4. Applied rewrites 98.8%

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

Alternative 2: 66.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(i, c, z \cdot t\right)\\ \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{+100}:\\ \;\;\;\;\mathsf{fma}\left(y, x, b \cdot a\right)\\ \mathbf{elif}\;a \cdot b \leq -2 \cdot 10^{-67}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \cdot b \leq 0:\\ \;\;\;\;\mathsf{fma}\left(i, c, y \cdot x\right)\\ \mathbf{elif}\;a \cdot b \leq 5000000000:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, c \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma i c (* z t))))
   (if (<= (* a b) -1e+100)
     (fma y x (* b a))
     (if (<= (* a b) -2e-67)
       t_1
       (if (<= (* a b) 0.0)
         (fma i c (* y x))
         (if (<= (* a b) 5000000000.0) t_1 (fma b a (* c i))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(i, c, (z * t));
	double tmp;
	if ((a * b) <= -1e+100) {
		tmp = fma(y, x, (b * a));
	} else if ((a * b) <= -2e-67) {
		tmp = t_1;
	} else if ((a * b) <= 0.0) {
		tmp = fma(i, c, (y * x));
	} else if ((a * b) <= 5000000000.0) {
		tmp = t_1;
	} else {
		tmp = fma(b, a, (c * i));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(i, c, Float64(z * t))
	tmp = 0.0
	if (Float64(a * b) <= -1e+100)
		tmp = fma(y, x, Float64(b * a));
	elseif (Float64(a * b) <= -2e-67)
		tmp = t_1;
	elseif (Float64(a * b) <= 0.0)
		tmp = fma(i, c, Float64(y * x));
	elseif (Float64(a * b) <= 5000000000.0)
		tmp = t_1;
	else
		tmp = fma(b, a, Float64(c * i));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(i * c + N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -1e+100], N[(y * x + N[(b * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], -2e-67], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], 0.0], N[(i * c + N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 5000000000.0], t$95$1, N[(b * a + N[(c * i), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 a b) < -1.00000000000000002e100

   1. Initial program 92.0%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. distribute-lft-neg-out N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \color{blue}{-1 \cdot \left(x \cdot y\right)}\right) \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{c \cdot i + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{i \cdot c} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(b, a, i \cdot c + \color{blue}{1} \cdot \left(x \cdot y\right)\right) \]

      11. *-lft-identity N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 90.2

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 82.4%

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

   10. Applied rewrites 84.4%

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

   if -1.00000000000000002e100 < (*.f64 a b) < -1.99999999999999989e-67 or -0.0 < (*.f64 a b) < 5e9

   1. Initial program 97.7%

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

   3. Taylor expanded in a around 0

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

      1. *-lft-identity N/A

         \[\leadsto c \cdot i + \color{blue}{1 \cdot \left(t \cdot z + x \cdot y\right)} \]

      2. metadata-eval N/A

         \[\leadsto c \cdot i + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \left(t \cdot z + x \cdot y\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{i \cdot c} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(t \cdot z + x \cdot y\right) \]

      4. metadata-eval N/A

         \[\leadsto i \cdot c + \color{blue}{1} \cdot \left(t \cdot z + x \cdot y\right) \]

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 89.8

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 72.6%

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

   if -1.99999999999999989e-67 < (*.f64 a b) < -0.0

   1. Initial program 96.6%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. distribute-lft-neg-out N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \color{blue}{-1 \cdot \left(x \cdot y\right)}\right) \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{c \cdot i + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{i \cdot c} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(b, a, i \cdot c + \color{blue}{1} \cdot \left(x \cdot y\right)\right) \]

      11. *-lft-identity N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 74.4

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

   5. Applied rewrites 74.4%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 44.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 3.8%

       \[\leadsto b \cdot a \]

   12. Taylor expanded in a around 0

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

   14. Applied rewrites 72.8%

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

   if 5e9 < (*.f64 a b)

   1. Initial program 98.3%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. distribute-lft-neg-out N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \color{blue}{-1 \cdot \left(x \cdot y\right)}\right) \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{c \cdot i + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{i \cdot c} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(b, a, i \cdot c + \color{blue}{1} \cdot \left(x \cdot y\right)\right) \]

      11. *-lft-identity N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 95.0

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

   5. Applied rewrites 95.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 84.3%

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

Alternative 3: 89.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+80} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+130}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, t \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, y \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* z t) -2e+80) (not (<= (* z t) 2e+130)))
   (fma b a (fma i c (* t z)))
   (fma b a (fma i c (* y x)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((z * t) <= -2e+80) || !((z * t) <= 2e+130)) {
		tmp = fma(b, a, fma(i, c, (t * z)));
	} else {
		tmp = fma(b, a, fma(i, c, (y * x)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(z * t) <= -2e+80) || !(Float64(z * t) <= 2e+130))
		tmp = fma(b, a, fma(i, c, Float64(t * z)));
	else
		tmp = fma(b, a, fma(i, c, Float64(y * x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -2e+80], N[Not[LessEqual[N[(z * t), $MachinePrecision], 2e+130]], $MachinePrecision]], N[(b * a + N[(i * c + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * a + N[(i * c + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2e80 or 2.0000000000000001e130 < (*.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. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 91.2

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

   5. Applied rewrites 91.2%

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

   if -2e80 < (*.f64 z t) < 2.0000000000000001e130

   1. Initial program 98.8%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. distribute-lft-neg-out N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \color{blue}{-1 \cdot \left(x \cdot y\right)}\right) \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{c \cdot i + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{i \cdot c} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(b, a, i \cdot c + \color{blue}{1} \cdot \left(x \cdot y\right)\right) \]

      11. *-lft-identity N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 94.1

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

   5. Applied rewrites 94.1%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+80} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+130}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, t \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, y \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -1 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, t \cdot z\right)\right)\\ \mathbf{elif}\;c \cdot i \leq 2 \cdot 10^{+89}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(a, b, z \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, y \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -1e+47)
   (fma b a (fma i c (* t z)))
   (if (<= (* c i) 2e+89)
     (fma y x (fma a b (* z t)))
     (fma b a (fma i c (* y x))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((c * i) <= -1e+47) {
		tmp = fma(b, a, fma(i, c, (t * z)));
	} else if ((c * i) <= 2e+89) {
		tmp = fma(y, x, fma(a, b, (z * t)));
	} else {
		tmp = fma(b, a, fma(i, c, (y * x)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(c * i) <= -1e+47)
		tmp = fma(b, a, fma(i, c, Float64(t * z)));
	elseif (Float64(c * i) <= 2e+89)
		tmp = fma(y, x, fma(a, b, Float64(z * t)));
	else
		tmp = fma(b, a, fma(i, c, Float64(y * x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(c * i), $MachinePrecision], -1e+47], N[(b * a + N[(i * c + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 2e+89], N[(y * x + N[(a * b + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * a + N[(i * c + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c i) < -1e47

   1. Initial program 93.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 88.8

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

   5. Applied rewrites 88.8%

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

   if -1e47 < (*.f64 c i) < 1.99999999999999999e89

   1. Initial program 97.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. associate--l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. associate-+r+ N/A

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

      14. lift-*.f64 N/A

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

      15. remove-double-neg N/A

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

      16. remove-double-neg N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. +-commutative N/A

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 96.1

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

   7. Applied rewrites 96.1%

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

   if 1.99999999999999999e89 < (*.f64 c i)

   1. Initial program 97.7%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. distribute-lft-neg-out N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \color{blue}{-1 \cdot \left(x \cdot y\right)}\right) \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{c \cdot i + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{i \cdot c} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(b, a, i \cdot c + \color{blue}{1} \cdot \left(x \cdot y\right)\right) \]

      11. *-lft-identity N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 93.3

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

   5. Applied rewrites 93.3%

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

Alternative 5: 84.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+295}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \cdot y \leq 10^{+136}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, t \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, b \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* x y) -4e+295)
   (* y x)
   (if (<= (* x y) 1e+136) (fma b a (fma i c (* t z))) (fma y x (* b a)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((x * y) <= -4e+295) {
		tmp = y * x;
	} else if ((x * y) <= 1e+136) {
		tmp = fma(b, a, fma(i, c, (t * z)));
	} else {
		tmp = fma(y, x, (b * a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(x * y) <= -4e+295)
		tmp = Float64(y * x);
	elseif (Float64(x * y) <= 1e+136)
		tmp = fma(b, a, fma(i, c, Float64(t * z)));
	else
		tmp = fma(y, x, Float64(b * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(x * y), $MachinePrecision], -4e+295], N[(y * x), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e+136], N[(b * a + N[(i * c + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x + N[(b * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -3.9999999999999999e295

   1. Initial program 94.4%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. distribute-lft-neg-out N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \color{blue}{-1 \cdot \left(x \cdot y\right)}\right) \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{c \cdot i + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{i \cdot c} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(b, a, i \cdot c + \color{blue}{1} \cdot \left(x \cdot y\right)\right) \]

      11. *-lft-identity N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 94.4

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

   5. Applied rewrites 94.4%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 94.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 7.7%

       \[\leadsto b \cdot a \]

   12. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 94.4

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

   14. Applied rewrites 94.4%

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

   if -3.9999999999999999e295 < (*.f64 x y) < 1.00000000000000006e136

   1. Initial program 98.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 89.4

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

   5. Applied rewrites 89.4%

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

   if 1.00000000000000006e136 < (*.f64 x y)

   1. Initial program 89.7%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. distribute-lft-neg-out N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \color{blue}{-1 \cdot \left(x \cdot y\right)}\right) \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{c \cdot i + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{i \cdot c} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(b, a, i \cdot c + \color{blue}{1} \cdot \left(x \cdot y\right)\right) \]

      11. *-lft-identity N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 85.1

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 77.5%

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

   10. Applied rewrites 80.1%

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

Alternative 6: 62.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -1 \cdot 10^{+95} \lor \neg \left(c \cdot i \leq 4 \cdot 10^{+223}\right):\\ \;\;\;\;i \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* c i) -1e+95) (not (<= (* c i) 4e+223)))
   (* i c)
   (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 (((c * i) <= -1e+95) || !((c * i) <= 4e+223)) {
		tmp = i * c;
	} 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(c * i) <= -1e+95) || !(Float64(c * i) <= 4e+223))
		tmp = Float64(i * c);
	else
		tmp = fma(a, b, Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(c * i), $MachinePrecision], -1e+95], N[Not[LessEqual[N[(c * i), $MachinePrecision], 4e+223]], $MachinePrecision]], N[(i * c), $MachinePrecision], N[(a * b + N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c i) < -1.00000000000000002e95 or 4.00000000000000019e223 < (*.f64 c i)

   1. Initial program 94.2%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 81.5

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

   5. Applied rewrites 81.5%

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

   if -1.00000000000000002e95 < (*.f64 c i) < 4.00000000000000019e223

   1. Initial program 97.3%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. distribute-lft-neg-out N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \color{blue}{-1 \cdot \left(x \cdot y\right)}\right) \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{c \cdot i + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{i \cdot c} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(b, a, i \cdot c + \color{blue}{1} \cdot \left(x \cdot y\right)\right) \]

      11. *-lft-identity N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 73.2

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

   5. Applied rewrites 73.2%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 65.1%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -1 \cdot 10^{+95} \lor \neg \left(c \cdot i \leq 4 \cdot 10^{+223}\right):\\ \;\;\;\;i \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -1 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(b, a, c \cdot i\right)\\ \mathbf{elif}\;c \cdot i \leq 4 \cdot 10^{+223}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -1e+95)
   (fma b a (* c i))
   (if (<= (* c i) 4e+223) (fma a b (* x y)) (* i c))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((c * i) <= -1e+95) {
		tmp = fma(b, a, (c * i));
	} else if ((c * i) <= 4e+223) {
		tmp = fma(a, b, (x * y));
	} else {
		tmp = i * c;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(c * i) <= -1e+95)
		tmp = fma(b, a, Float64(c * i));
	elseif (Float64(c * i) <= 4e+223)
		tmp = fma(a, b, Float64(x * y));
	else
		tmp = Float64(i * c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(c * i), $MachinePrecision], -1e+95], N[(b * a + N[(c * i), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 4e+223], N[(a * b + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(i * c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c i) < -1.00000000000000002e95

   1. Initial program 92.1%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. distribute-lft-neg-out N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \color{blue}{-1 \cdot \left(x \cdot y\right)}\right) \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{c \cdot i + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{i \cdot c} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(b, a, i \cdot c + \color{blue}{1} \cdot \left(x \cdot y\right)\right) \]

      11. *-lft-identity N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 86.4

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

   5. Applied rewrites 86.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 86.1%

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

   if -1.00000000000000002e95 < (*.f64 c i) < 4.00000000000000019e223

   1. Initial program 97.3%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. distribute-lft-neg-out N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \color{blue}{-1 \cdot \left(x \cdot y\right)}\right) \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{c \cdot i + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{i \cdot c} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(b, a, i \cdot c + \color{blue}{1} \cdot \left(x \cdot y\right)\right) \]

      11. *-lft-identity N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 73.2

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

   5. Applied rewrites 73.2%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 65.1%

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

   if 4.00000000000000019e223 < (*.f64 c i)

   1. Initial program 99.9%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 95.5

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

   5. Applied rewrites 95.5%

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

Alternative 8: 64.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -1 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(i, c, y \cdot x\right)\\ \mathbf{elif}\;c \cdot i \leq 4 \cdot 10^{+223}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -1e+47)
   (fma i c (* y x))
   (if (<= (* c i) 4e+223) (fma a b (* x y)) (* i c))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((c * i) <= -1e+47) {
		tmp = fma(i, c, (y * x));
	} else if ((c * i) <= 4e+223) {
		tmp = fma(a, b, (x * y));
	} else {
		tmp = i * c;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(c * i) <= -1e+47)
		tmp = fma(i, c, Float64(y * x));
	elseif (Float64(c * i) <= 4e+223)
		tmp = fma(a, b, Float64(x * y));
	else
		tmp = Float64(i * c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(c * i), $MachinePrecision], -1e+47], N[(i * c + N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 4e+223], N[(a * b + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(i * c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c i) < -1e47

   1. Initial program 93.5%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. distribute-lft-neg-out N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \color{blue}{-1 \cdot \left(x \cdot y\right)}\right) \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{c \cdot i + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{i \cdot c} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(b, a, i \cdot c + \color{blue}{1} \cdot \left(x \cdot y\right)\right) \]

      11. *-lft-identity N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 85.7

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

   5. Applied rewrites 85.7%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 26.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 14.8%

       \[\leadsto b \cdot a \]

   12. Taylor expanded in a around 0

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

   14. Applied rewrites 77.8%

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

   if -1e47 < (*.f64 c i) < 4.00000000000000019e223

   1. Initial program 97.1%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. distribute-lft-neg-out N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \color{blue}{-1 \cdot \left(x \cdot y\right)}\right) \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{c \cdot i + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{i \cdot c} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(b, a, i \cdot c + \color{blue}{1} \cdot \left(x \cdot y\right)\right) \]

      11. *-lft-identity N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 72.6

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

   5. Applied rewrites 72.6%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 66.2%

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

   if 4.00000000000000019e223 < (*.f64 c i)

   1. Initial program 99.9%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 95.5

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

   5. Applied rewrites 95.5%

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

Alternative 9: 41.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -2 \cdot 10^{+41} \lor \neg \left(c \cdot i \leq 4 \cdot 10^{+223}\right):\\ \;\;\;\;i \cdot c\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* c i) -2e+41) (not (<= (* c i) 4e+223))) (* i c) (* b a)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((c * i) <= -2e+41) || !((c * i) <= 4e+223)) {
		tmp = i * c;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if (((c * i) <= (-2d+41)) .or. (.not. ((c * i) <= 4d+223))) then
        tmp = i * c
    else
        tmp = b * a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((c * i) <= -2e+41) || !((c * i) <= 4e+223)) {
		tmp = i * c;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if ((c * i) <= -2e+41) or not ((c * i) <= 4e+223):
		tmp = i * c
	else:
		tmp = b * a
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(c * i) <= -2e+41) || !(Float64(c * i) <= 4e+223))
		tmp = Float64(i * c);
	else
		tmp = Float64(b * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (((c * i) <= -2e+41) || ~(((c * i) <= 4e+223)))
		tmp = i * c;
	else
		tmp = b * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(c * i), $MachinePrecision], -2e+41], N[Not[LessEqual[N[(c * i), $MachinePrecision], 4e+223]], $MachinePrecision]], N[(i * c), $MachinePrecision], N[(b * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c i) < -2.00000000000000001e41 or 4.00000000000000019e223 < (*.f64 c i)

   1. Initial program 95.1%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 73.7

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

   5. Applied rewrites 73.7%

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

   if -2.00000000000000001e41 < (*.f64 c i) < 4.00000000000000019e223

   1. Initial program 97.1%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. distribute-lft-neg-out N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \color{blue}{-1 \cdot \left(x \cdot y\right)}\right) \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{c \cdot i + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{i \cdot c} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(b, a, i \cdot c + \color{blue}{1} \cdot \left(x \cdot y\right)\right) \]

      11. *-lft-identity N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 72.3

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

   5. Applied rewrites 72.3%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 65.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 38.5%

       \[\leadsto b \cdot a \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -2 \cdot 10^{+41} \lor \neg \left(c \cdot i \leq 4 \cdot 10^{+223}\right):\\ \;\;\;\;i \cdot c\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 28.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.5%

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

3. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. *-commutative N/A

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

   4. fp-cancel-sign-sub-inv N/A

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

   5. distribute-lft-neg-out N/A

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

   6. *-commutative N/A

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

   7. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(b, a, c \cdot i - \color{blue}{-1 \cdot \left(x \cdot y\right)}\right) \]

   8. fp-cancel-sub-sign-inv N/A

      \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{c \cdot i + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)}\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(b, a, \color{blue}{i \cdot c} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(x \cdot y\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(b, a, i \cdot c + \color{blue}{1} \cdot \left(x \cdot y\right)\right) \]

   11. *-lft-identity N/A

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

   12. lower-fma.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 77.4

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

5. Applied rewrites 77.4%

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

6. Taylor expanded in c around 0

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

8. Applied rewrites 52.8%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 29.9%

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

Reproduce

herbie shell --seed 2024346 
(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)))