Linear.V3:$cdot from linear-1.19.1.3, B

Percentage Accurate: 97.9% → 99.0%

Time: 7.6s

Alternatives: 8

Speedup: 1.1×

• 32.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((x * y) + (z * t)) + (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)
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
    code = ((x * y) + (z * t)) + (a * b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((x * y) + (z * t)) + (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)
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
    code = ((x * y) + (z * t)) + (a * b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.9%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. +-commutative N/A

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

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

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

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

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

   13. mul-1-neg N/A

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

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

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

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

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

   16. mul-1-neg N/A

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

   17. lower-fma.f64 N/A

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

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

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

   19. mul-1-neg N/A

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

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

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

   21. mul-1-neg N/A

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

   22. remove-double-neg N/A

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

   23. *-commutative N/A

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

   24. lower-*.f64 99.0

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

3. Applied rewrites 99.0%

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

Alternative 2: 85.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, z, y \cdot x\right)\\ \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(b, a, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma t z (* y x))))
   (if (<= (* z t) -5e+33) t_1 (if (<= (* z t) 2e+39) (fma b a (* x y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(t, z, (y * x));
	double tmp;
	if ((z * t) <= -5e+33) {
		tmp = t_1;
	} else if ((z * t) <= 2e+39) {
		tmp = fma(b, a, (x * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(t, z, Float64(y * x))
	tmp = 0.0
	if (Float64(z * t) <= -5e+33)
		tmp = t_1;
	elseif (Float64(z * t) <= 2e+39)
		tmp = fma(b, a, Float64(x * y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -5e+33], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 2e+39], N[(b * a + N[(x * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -4.99999999999999973e33 or 1.99999999999999988e39 < (*.f64 z t)

   1. Initial program 96.1%

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

   2. Taylor expanded in a around 0

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

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

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

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

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

      3. mul-1-neg N/A

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

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

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

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

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

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, z, \mathsf{neg}\left(\left(-1 \cdot x\right) \cdot y\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(t, z, \left(\mathsf{neg}\left(-1 \cdot x\right)\right) \cdot y\right) \]

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 81.7

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

   4. Applied rewrites 81.7%

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

   if -4.99999999999999973e33 < (*.f64 z t) < 1.99999999999999988e39

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

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

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

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

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

      13. mul-1-neg N/A

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

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

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

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

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

      16. mul-1-neg N/A

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

      17. lower-fma.f64 N/A

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

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

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

      19. mul-1-neg N/A

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

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

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

      21. mul-1-neg N/A

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

      22. remove-double-neg N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 89.4

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

   6. Applied rewrites 89.4%

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

Alternative 3: 85.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, z, y \cdot x\right)\\ \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(y, x, b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma t z (* y x))))
   (if (<= (* z t) -5e+33) t_1 (if (<= (* z t) 2e+39) (fma y x (* b a)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(t, z, (y * x));
	double tmp;
	if ((z * t) <= -5e+33) {
		tmp = t_1;
	} else if ((z * t) <= 2e+39) {
		tmp = fma(y, x, (b * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(t, z, Float64(y * x))
	tmp = 0.0
	if (Float64(z * t) <= -5e+33)
		tmp = t_1;
	elseif (Float64(z * t) <= 2e+39)
		tmp = fma(y, x, Float64(b * a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -5e+33], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 2e+39], N[(y * x + N[(b * a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -4.99999999999999973e33 or 1.99999999999999988e39 < (*.f64 z t)

   1. Initial program 96.1%

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

   2. Taylor expanded in a around 0

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

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

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

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

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

      3. mul-1-neg N/A

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

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

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

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

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

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, z, \mathsf{neg}\left(\left(-1 \cdot x\right) \cdot y\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(t, z, \left(\mathsf{neg}\left(-1 \cdot x\right)\right) \cdot y\right) \]

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 81.7

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

   4. Applied rewrites 81.7%

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

   if -4.99999999999999973e33 < (*.f64 z t) < 1.99999999999999988e39

   1. Initial program 99.4%

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

   2. Taylor expanded in z around 0

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

      1. /-rgt-identity N/A

         \[\leadsto \frac{a \cdot b + x \cdot y}{\color{blue}{1}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{a \cdot b + x \cdot y}{\mathsf{neg}\left(-1\right)} \]

      3. +-commutative N/A

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

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

         \[\leadsto \frac{x \cdot y - \left(\mathsf{neg}\left(a\right)\right) \cdot b}{\mathsf{neg}\left(\color{blue}{-1}\right)} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{x \cdot y - \left(-1 \cdot a\right) \cdot b}{\mathsf{neg}\left(-1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{x \cdot y - -1 \cdot \left(a \cdot b\right)}{\mathsf{neg}\left(-1\right)} \]

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

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

      8. div-add N/A

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

      9. frac-2neg-rev N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

          \[\leadsto \frac{\mathsf{neg}\left(x \cdot y\right)}{-1} + \frac{1 \cdot \left(a \cdot b\right)}{\mathsf{neg}\left(-1\right)} \]

      13. *-lft-identity N/A

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

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{neg}\left(x \cdot y\right)}{-1} + \frac{a \cdot b}{1} \]

      15. /-rgt-identity N/A

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

      16. remove-double-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(x \cdot y\right)}{-1} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot b\right)\right)\right)\right) \]

   4. Applied rewrites 89.4%

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

Alternative 4: 83.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, z, b \cdot a\right)\\ \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+139}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \cdot b \leq 10^{-87}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma t z (* b a))))
   (if (<= (* a b) -5e+139)
     t_1
     (if (<= (* a b) 1e-87) (fma t z (* y x)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(t, z, (b * a));
	double tmp;
	if ((a * b) <= -5e+139) {
		tmp = t_1;
	} else if ((a * b) <= 1e-87) {
		tmp = fma(t, z, (y * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(t, z, Float64(b * a))
	tmp = 0.0
	if (Float64(a * b) <= -5e+139)
		tmp = t_1;
	elseif (Float64(a * b) <= 1e-87)
		tmp = fma(t, z, Float64(y * x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t * z + N[(b * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -5e+139], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], 1e-87], N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -5.0000000000000003e139 or 1.00000000000000002e-87 < (*.f64 a b)

   1. Initial program 96.4%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 79.2

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

   4. Applied rewrites 79.2%

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

   if -5.0000000000000003e139 < (*.f64 a b) < 1.00000000000000002e-87

   1. Initial program 99.2%

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

   2. Taylor expanded in a around 0

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

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

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

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

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

      3. mul-1-neg N/A

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

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

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

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

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

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, z, \mathsf{neg}\left(\left(-1 \cdot x\right) \cdot y\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(t, z, \left(\mathsf{neg}\left(-1 \cdot x\right)\right) \cdot y\right) \]

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 87.9

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

   4. Applied rewrites 87.9%

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

Alternative 5: 81.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+139}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+203}:\\ \;\;\;\;\mathsf{fma}\left(t, z, b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* x y) -2e+139)
   (* y x)
   (if (<= (* x y) 5e+203) (fma t z (* b a)) (* y x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x * y) <= -2e+139) {
		tmp = y * x;
	} else if ((x * y) <= 5e+203) {
		tmp = fma(t, z, (b * a));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x * y) <= -2e+139)
		tmp = Float64(y * x);
	elseif (Float64(x * y) <= 5e+203)
		tmp = fma(t, z, Float64(b * a));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e+139], N[(y * x), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+203], N[(t * z + N[(b * a), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2.00000000000000007e139 or 4.99999999999999994e203 < (*.f64 x y)

   1. Initial program 94.3%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 79.0

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

   4. Applied rewrites 79.0%

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

   if -2.00000000000000007e139 < (*.f64 x y) < 4.99999999999999994e203

   1. Initial program 99.2%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 82.8

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

   4. Applied rewrites 82.8%

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

Alternative 6: 55.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+139}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \cdot y \leq 5000:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+68}:\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* x y) -2e+139)
   (* y x)
   (if (<= (* x y) 5000.0) (* t z) (if (<= (* x y) 4e+68) (* b a) (* y x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x * y) <= -2e+139) {
		tmp = y * x;
	} else if ((x * y) <= 5000.0) {
		tmp = t * z;
	} else if ((x * y) <= 4e+68) {
		tmp = b * a;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
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) :: tmp
    if ((x * y) <= (-2d+139)) then
        tmp = y * x
    else if ((x * y) <= 5000.0d0) then
        tmp = t * z
    else if ((x * y) <= 4d+68) then
        tmp = b * a
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x * y) <= -2e+139) {
		tmp = y * x;
	} else if ((x * y) <= 5000.0) {
		tmp = t * z;
	} else if ((x * y) <= 4e+68) {
		tmp = b * a;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (x * y) <= -2e+139:
		tmp = y * x
	elif (x * y) <= 5000.0:
		tmp = t * z
	elif (x * y) <= 4e+68:
		tmp = b * a
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x * y) <= -2e+139)
		tmp = Float64(y * x);
	elseif (Float64(x * y) <= 5000.0)
		tmp = Float64(t * z);
	elseif (Float64(x * y) <= 4e+68)
		tmp = Float64(b * a);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((x * y) <= -2e+139)
		tmp = y * x;
	elseif ((x * y) <= 5000.0)
		tmp = t * z;
	elseif ((x * y) <= 4e+68)
		tmp = b * a;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e+139], N[(y * x), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5000.0], N[(t * z), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 4e+68], N[(b * a), $MachinePrecision], N[(y * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2.00000000000000007e139 or 3.99999999999999981e68 < (*.f64 x y)

   1. Initial program 95.4%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 71.4

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

   4. Applied rewrites 71.4%

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

   if -2.00000000000000007e139 < (*.f64 x y) < 5e3

   1. Initial program 99.2%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 45.9

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

   4. Applied rewrites 45.9%

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

   if 5e3 < (*.f64 x y) < 3.99999999999999981e68

   1. Initial program 99.3%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 35.6

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

   4. Applied rewrites 35.6%

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

Alternative 7: 54.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+33}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+78}:\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* z t) -5e+33) (* t z) (if (<= (* z t) 5e+78) (* b a) (* t z))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z * t) <= -5e+33) {
		tmp = t * z;
	} else if ((z * t) <= 5e+78) {
		tmp = b * a;
	} else {
		tmp = t * z;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
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) :: tmp
    if ((z * t) <= (-5d+33)) then
        tmp = t * z
    else if ((z * t) <= 5d+78) then
        tmp = b * a
    else
        tmp = t * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z * t) <= -5e+33) {
		tmp = t * z;
	} else if ((z * t) <= 5e+78) {
		tmp = b * a;
	} else {
		tmp = t * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z * t) <= -5e+33:
		tmp = t * z
	elif (z * t) <= 5e+78:
		tmp = b * a
	else:
		tmp = t * z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(z * t) <= -5e+33)
		tmp = Float64(t * z);
	elseif (Float64(z * t) <= 5e+78)
		tmp = Float64(b * a);
	else
		tmp = Float64(t * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((z * t) <= -5e+33)
		tmp = t * z;
	elseif ((z * t) <= 5e+78)
		tmp = b * a;
	else
		tmp = t * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(z * t), $MachinePrecision], -5e+33], N[(t * z), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+78], N[(b * a), $MachinePrecision], N[(t * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -4.99999999999999973e33 or 4.99999999999999984e78 < (*.f64 z t)

   1. Initial program 95.8%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 66.8

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

   4. Applied rewrites 66.8%

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

   if -4.99999999999999973e33 < (*.f64 z t) < 4.99999999999999984e78

   1. Initial program 99.4%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 46.4

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

   4. Applied rewrites 46.4%

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

Alternative 8: 35.4% accurate, 3.9× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	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)
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
    code = b * a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 35.4

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

4. Applied rewrites 35.4%

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

Reproduce

herbie shell --seed 2025130 
(FPCore (x y z t a b)
  :name "Linear.V3:$cdot from linear-1.19.1.3, B"
  :precision binary64
  (+ (+ (* x y) (* z t)) (* a b)))