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

Percentage Accurate: 95.7% → 97.9%

Time: 4.1s

Alternatives: 13

Speedup: 0.5×

• 41.9% 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: 95.7% 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, 0.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((x * y) + (z * t)) + (a * b)) + (c * i)) <= ((double) INFINITY)) {
		tmp = fma(y, x, fma(b, a, (t * z))) + (c * i);
	} else {
		tmp = fma(b, a, fma(t, z, (y * x)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * y) + Float64(z * t)) + Float64(a * b)) + Float64(c * i)) <= Inf)
		tmp = Float64(fma(y, x, fma(b, a, Float64(t * z))) + Float64(c * i));
	else
		tmp = fma(b, a, fma(t, z, Float64(y * x)));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (*.f64 x y) (*.f64 z t)) (*.f64 a b)) (*.f64 c i)) < +inf.0

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   if +inf.0 < (+.f64 (+.f64 (+.f64 (*.f64 x y) (*.f64 z t)) (*.f64 a b)) (*.f64 c i))

   1. Initial program 0.0%

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

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 76.2%

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

Alternative 2: 66.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, a, y \cdot x\right)\\ t_2 := \mathsf{fma}\left(i, c, t \cdot z\right)\\ \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+158}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \cdot t \leq -500000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq -2 \cdot 10^{-236}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \mathbf{elif}\;z \cdot t \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-145}:\\ \;\;\;\;\mathsf{fma}\left(i, c, x \cdot y\right)\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+17}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma b a (* y x))) (t_2 (fma i c (* t z))))
   (if (<= (* z t) -2e+158)
     t_2
     (if (<= (* z t) -500000000.0)
       t_1
       (if (<= (* z t) -2e-236)
         (fma i c (* a b))
         (if (<= (* z t) 0.0)
           t_1
           (if (<= (* z t) 2e-145)
             (fma i c (* x y))
             (if (<= (* z t) 5e+17) t_1 t_2))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(b, a, (y * x));
	double t_2 = fma(i, c, (t * z));
	double tmp;
	if ((z * t) <= -2e+158) {
		tmp = t_2;
	} else if ((z * t) <= -500000000.0) {
		tmp = t_1;
	} else if ((z * t) <= -2e-236) {
		tmp = fma(i, c, (a * b));
	} else if ((z * t) <= 0.0) {
		tmp = t_1;
	} else if ((z * t) <= 2e-145) {
		tmp = fma(i, c, (x * y));
	} else if ((z * t) <= 5e+17) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(b, a, Float64(y * x))
	t_2 = fma(i, c, Float64(t * z))
	tmp = 0.0
	if (Float64(z * t) <= -2e+158)
		tmp = t_2;
	elseif (Float64(z * t) <= -500000000.0)
		tmp = t_1;
	elseif (Float64(z * t) <= -2e-236)
		tmp = fma(i, c, Float64(a * b));
	elseif (Float64(z * t) <= 0.0)
		tmp = t_1;
	elseif (Float64(z * t) <= 2e-145)
		tmp = fma(i, c, Float64(x * y));
	elseif (Float64(z * t) <= 5e+17)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(b * a + N[(y * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(i * c + N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -2e+158], t$95$2, If[LessEqual[N[(z * t), $MachinePrecision], -500000000.0], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], -2e-236], N[(i * c + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 0.0], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 2e-145], N[(i * c + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+17], t$95$1, t$95$2]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 z t) < -1.99999999999999991e158 or 5e17 < (*.f64 z t)

   1. Initial program 86.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 91.2

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

   4. Applied rewrites 91.2%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 75.4%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{z \cdot t + c \cdot i} \]

      2. lift-*.f64 N/A

         \[\leadsto z \cdot t + \color{blue}{c \cdot i} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{c \cdot i + z \cdot t} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{i \cdot c} + z \cdot t \]

      5. lower-fma.f64 77.6

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

   9. Applied rewrites 77.6%

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

   if -1.99999999999999991e158 < (*.f64 z t) < -5e8 or -2.0000000000000001e-236 < (*.f64 z t) < -0.0 or 1.99999999999999983e-145 < (*.f64 z t) < 5e17

   1. Initial program 93.3%

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

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 87.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 81.2%

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

   if -5e8 < (*.f64 z t) < -2.0000000000000001e-236

   1. Initial program 95.5%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 71.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 75.8

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

   7. Applied rewrites 75.8%

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

   if -0.0 < (*.f64 z t) < 1.99999999999999983e-145

   1. Initial program 96.8%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 84.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 84.0

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

   7. Applied rewrites 84.0%

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

Alternative 3: 43.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+159}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;z \cdot t \leq -2 \cdot 10^{-7}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;z \cdot t \leq 0:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-145}:\\ \;\;\;\;i \cdot c\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+93}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* z t) -2e+159)
   (* t z)
   (if (<= (* z t) -2e-7)
     (* y x)
     (if (<= (* z t) 0.0)
       (* b a)
       (if (<= (* z t) 2e-145)
         (* i c)
         (if (<= (* z t) 2e+93) (* y x) (* t z)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((z * t) <= -2e+159) {
		tmp = t * z;
	} else if ((z * t) <= -2e-7) {
		tmp = y * x;
	} else if ((z * t) <= 0.0) {
		tmp = b * a;
	} else if ((z * t) <= 2e-145) {
		tmp = i * c;
	} else if ((z * t) <= 2e+93) {
		tmp = y * x;
	} else {
		tmp = t * z;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if ((z * t) <= (-2d+159)) then
        tmp = t * z
    else if ((z * t) <= (-2d-7)) then
        tmp = y * x
    else if ((z * t) <= 0.0d0) then
        tmp = b * a
    else if ((z * t) <= 2d-145) then
        tmp = i * c
    else if ((z * t) <= 2d+93) then
        tmp = y * x
    else
        tmp = t * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((z * t) <= -2e+159) {
		tmp = t * z;
	} else if ((z * t) <= -2e-7) {
		tmp = y * x;
	} else if ((z * t) <= 0.0) {
		tmp = b * a;
	} else if ((z * t) <= 2e-145) {
		tmp = i * c;
	} else if ((z * t) <= 2e+93) {
		tmp = y * x;
	} else {
		tmp = t * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (z * t) <= -2e+159:
		tmp = t * z
	elif (z * t) <= -2e-7:
		tmp = y * x
	elif (z * t) <= 0.0:
		tmp = b * a
	elif (z * t) <= 2e-145:
		tmp = i * c
	elif (z * t) <= 2e+93:
		tmp = y * x
	else:
		tmp = t * z
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(z * t) <= -2e+159)
		tmp = Float64(t * z);
	elseif (Float64(z * t) <= -2e-7)
		tmp = Float64(y * x);
	elseif (Float64(z * t) <= 0.0)
		tmp = Float64(b * a);
	elseif (Float64(z * t) <= 2e-145)
		tmp = Float64(i * c);
	elseif (Float64(z * t) <= 2e+93)
		tmp = Float64(y * x);
	else
		tmp = Float64(t * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((z * t) <= -2e+159)
		tmp = t * z;
	elseif ((z * t) <= -2e-7)
		tmp = y * x;
	elseif ((z * t) <= 0.0)
		tmp = b * a;
	elseif ((z * t) <= 2e-145)
		tmp = i * c;
	elseif ((z * t) <= 2e+93)
		tmp = y * x;
	else
		tmp = t * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(z * t), $MachinePrecision], -2e+159], N[(t * z), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], -2e-7], N[(y * x), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 0.0], N[(b * a), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e-145], N[(i * c), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+93], N[(y * x), $MachinePrecision], N[(t * z), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 z t) < -1.9999999999999999e159 or 2.00000000000000009e93 < (*.f64 z t)

   1. Initial program 85.3%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 68.2%

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

   if -1.9999999999999999e159 < (*.f64 z t) < -1.9999999999999999e-7 or 1.99999999999999983e-145 < (*.f64 z t) < 2.00000000000000009e93

   1. Initial program 90.6%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 51.0%

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

   if -1.9999999999999999e-7 < (*.f64 z t) < -0.0

   1. Initial program 97.5%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 46.2%

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

   if -0.0 < (*.f64 z t) < 1.99999999999999983e-145

   1. Initial program 96.8%

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

   3. Taylor expanded in c around inf

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

   5. Applied rewrites 59.4%

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

Alternative 4: 66.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, a, y \cdot x\right)\\ t_2 := \mathsf{fma}\left(i, c, t \cdot z\right)\\ \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+158}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \cdot t \leq -500000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-115}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+17}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma b a (* y x))) (t_2 (fma i c (* t z))))
   (if (<= (* z t) -2e+158)
     t_2
     (if (<= (* z t) -500000000.0)
       t_1
       (if (<= (* z t) 5e-115)
         (fma i c (* a b))
         (if (<= (* z t) 5e+17) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(b, a, (y * x));
	double t_2 = fma(i, c, (t * z));
	double tmp;
	if ((z * t) <= -2e+158) {
		tmp = t_2;
	} else if ((z * t) <= -500000000.0) {
		tmp = t_1;
	} else if ((z * t) <= 5e-115) {
		tmp = fma(i, c, (a * b));
	} else if ((z * t) <= 5e+17) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(b, a, Float64(y * x))
	t_2 = fma(i, c, Float64(t * z))
	tmp = 0.0
	if (Float64(z * t) <= -2e+158)
		tmp = t_2;
	elseif (Float64(z * t) <= -500000000.0)
		tmp = t_1;
	elseif (Float64(z * t) <= 5e-115)
		tmp = fma(i, c, Float64(a * b));
	elseif (Float64(z * t) <= 5e+17)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(b * a + N[(y * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(i * c + N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -2e+158], t$95$2, If[LessEqual[N[(z * t), $MachinePrecision], -500000000.0], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 5e-115], N[(i * c + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+17], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -1.99999999999999991e158 or 5e17 < (*.f64 z t)

   1. Initial program 86.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 91.2

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

   4. Applied rewrites 91.2%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 75.4%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{z \cdot t + c \cdot i} \]

      2. lift-*.f64 N/A

         \[\leadsto z \cdot t + \color{blue}{c \cdot i} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{c \cdot i + z \cdot t} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{i \cdot c} + z \cdot t \]

      5. lower-fma.f64 77.6

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

   9. Applied rewrites 77.6%

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

   if -1.99999999999999991e158 < (*.f64 z t) < -5e8 or 5.0000000000000003e-115 < (*.f64 z t) < 5e17

   1. Initial program 87.9%

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

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 79.8%

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

   if -5e8 < (*.f64 z t) < 5.0000000000000003e-115

   1. Initial program 97.4%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 70.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 72.3

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

   7. Applied rewrites 72.3%

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

Alternative 5: 67.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, a, y \cdot x\right)\\ t_2 := \mathsf{fma}\left(b, a, t \cdot z\right)\\ \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+159}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \cdot t \leq -500000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-115}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+93}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma b a (* y x))) (t_2 (fma b a (* t z))))
   (if (<= (* z t) -2e+159)
     t_2
     (if (<= (* z t) -500000000.0)
       t_1
       (if (<= (* z t) 5e-115)
         (fma i c (* a b))
         (if (<= (* z t) 2e+93) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(b, a, (y * x));
	double t_2 = fma(b, a, (t * z));
	double tmp;
	if ((z * t) <= -2e+159) {
		tmp = t_2;
	} else if ((z * t) <= -500000000.0) {
		tmp = t_1;
	} else if ((z * t) <= 5e-115) {
		tmp = fma(i, c, (a * b));
	} else if ((z * t) <= 2e+93) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(b, a, Float64(y * x))
	t_2 = fma(b, a, Float64(t * z))
	tmp = 0.0
	if (Float64(z * t) <= -2e+159)
		tmp = t_2;
	elseif (Float64(z * t) <= -500000000.0)
		tmp = t_1;
	elseif (Float64(z * t) <= 5e-115)
		tmp = fma(i, c, Float64(a * b));
	elseif (Float64(z * t) <= 2e+93)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(b * a + N[(y * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b * a + N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -2e+159], t$95$2, If[LessEqual[N[(z * t), $MachinePrecision], -500000000.0], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 5e-115], N[(i * c + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+93], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -1.9999999999999999e159 or 2.00000000000000009e93 < (*.f64 z t)

   1. Initial program 85.3%

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

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 77.5%

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

   if -1.9999999999999999e159 < (*.f64 z t) < -5e8 or 5.0000000000000003e-115 < (*.f64 z t) < 2.00000000000000009e93

   1. Initial program 89.8%

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

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 73.2%

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

   if -5e8 < (*.f64 z t) < 5.0000000000000003e-115

   1. Initial program 97.4%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 70.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 72.3

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

   7. Applied rewrites 72.3%

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

Alternative 6: 89.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+71} \lor \neg \left(z \cdot t \leq 10^{+56}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(t, z, y \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, \mathsf{fma}\left(y, x, b \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* z t) -5e+71) (not (<= (* z t) 1e+56)))
   (fma b a (fma t z (* y x)))
   (fma i c (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 (((z * t) <= -5e+71) || !((z * t) <= 1e+56)) {
		tmp = fma(b, a, fma(t, z, (y * x)));
	} else {
		tmp = fma(i, c, fma(y, x, (b * a)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(z * t) <= -5e+71) || !(Float64(z * t) <= 1e+56))
		tmp = fma(b, a, fma(t, z, Float64(y * x)));
	else
		tmp = fma(i, c, fma(y, x, Float64(b * a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -5e+71], N[Not[LessEqual[N[(z * t), $MachinePrecision], 1e+56]], $MachinePrecision]], N[(b * a + N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(i * c + N[(y * x + N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -4.99999999999999972e71 or 1.00000000000000009e56 < (*.f64 z t)

   1. Initial program 86.0%

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

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 85.3%

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

   if -4.99999999999999972e71 < (*.f64 z t) < 1.00000000000000009e56

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 95.1

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

   4. Applied rewrites 95.1%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 90.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 94.9

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

   9. Applied rewrites 94.9%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+71} \lor \neg \left(z \cdot t \leq 10^{+56}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(t, z, y \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, \mathsf{fma}\left(y, x, b \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+71} \lor \neg \left(z \cdot t \leq 10^{+56}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(t, z, y \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, y \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* z t) -5e+71) (not (<= (* z t) 1e+56)))
   (fma b a (fma t z (* y x)))
   (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) <= -5e+71) || !((z * t) <= 1e+56)) {
		tmp = fma(b, a, fma(t, z, (y * x)));
	} 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) <= -5e+71) || !(Float64(z * t) <= 1e+56))
		tmp = fma(b, a, fma(t, z, Float64(y * x)));
	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], -5e+71], N[Not[LessEqual[N[(z * t), $MachinePrecision], 1e+56]], $MachinePrecision]], N[(b * a + N[(t * z + N[(y * x), $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) < -4.99999999999999972e71 or 1.00000000000000009e56 < (*.f64 z t)

   1. Initial program 86.0%

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

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 85.3%

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

   if -4.99999999999999972e71 < (*.f64 z t) < 1.00000000000000009e56

   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 z around 0

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

   5. Applied rewrites 92.5%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+71} \lor \neg \left(z \cdot t \leq 10^{+56}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(t, z, y \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, y \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+159} \lor \neg \left(z \cdot t \leq 5 \cdot 10^{+101}\right):\\ \;\;\;\;\mathsf{fma}\left(i, c, t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, y \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* z t) -2e+159) (not (<= (* z t) 5e+101)))
   (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+159) || !((z * t) <= 5e+101)) {
		tmp = 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+159) || !(Float64(z * t) <= 5e+101))
		tmp = 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+159], N[Not[LessEqual[N[(z * t), $MachinePrecision], 5e+101]], $MachinePrecision]], N[(i * c + N[(t * z), $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) < -1.9999999999999999e159 or 4.99999999999999989e101 < (*.f64 z t)

   1. Initial program 84.6%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 89.7

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

   4. Applied rewrites 89.7%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 77.8%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{z \cdot t + c \cdot i} \]

      2. lift-*.f64 N/A

         \[\leadsto z \cdot t + \color{blue}{c \cdot i} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{c \cdot i + z \cdot t} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{i \cdot c} + z \cdot t \]

      5. lower-fma.f64 80.4

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

   9. Applied rewrites 80.4%

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

   if -1.9999999999999999e159 < (*.f64 z t) < 4.99999999999999989e101

   1. Initial program 94.9%

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

   3. 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

   5. Applied rewrites 91.0%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+159} \lor \neg \left(z \cdot t \leq 5 \cdot 10^{+101}\right):\\ \;\;\;\;\mathsf{fma}\left(i, c, t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(i, c, y \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+24}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;z \cdot t \leq 0:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;z \cdot t \leq 10^{+56}:\\ \;\;\;\;i \cdot c\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* z t) -1e+24)
   (* t z)
   (if (<= (* z t) 0.0) (* b a) (if (<= (* z t) 1e+56) (* i c) (* t z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((z * t) <= -1e+24) {
		tmp = t * z;
	} else if ((z * t) <= 0.0) {
		tmp = b * a;
	} else if ((z * t) <= 1e+56) {
		tmp = i * c;
	} else {
		tmp = t * z;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if ((z * t) <= (-1d+24)) then
        tmp = t * z
    else if ((z * t) <= 0.0d0) then
        tmp = b * a
    else if ((z * t) <= 1d+56) then
        tmp = i * c
    else
        tmp = t * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((z * t) <= -1e+24) {
		tmp = t * z;
	} else if ((z * t) <= 0.0) {
		tmp = b * a;
	} else if ((z * t) <= 1e+56) {
		tmp = i * c;
	} else {
		tmp = t * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (z * t) <= -1e+24:
		tmp = t * z
	elif (z * t) <= 0.0:
		tmp = b * a
	elif (z * t) <= 1e+56:
		tmp = i * c
	else:
		tmp = t * z
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(z * t) <= -1e+24)
		tmp = Float64(t * z);
	elseif (Float64(z * t) <= 0.0)
		tmp = Float64(b * a);
	elseif (Float64(z * t) <= 1e+56)
		tmp = Float64(i * c);
	else
		tmp = Float64(t * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((z * t) <= -1e+24)
		tmp = t * z;
	elseif ((z * t) <= 0.0)
		tmp = b * a;
	elseif ((z * t) <= 1e+56)
		tmp = i * c;
	else
		tmp = t * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(z * t), $MachinePrecision], -1e+24], N[(t * z), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 0.0], N[(b * a), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+56], N[(i * c), $MachinePrecision], N[(t * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -9.9999999999999998e23 or 1.00000000000000009e56 < (*.f64 z t)

   1. Initial program 86.4%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 59.0%

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

   if -9.9999999999999998e23 < (*.f64 z t) < -0.0

   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 a around inf

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

   5. Applied rewrites 45.4%

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

   if -0.0 < (*.f64 z t) < 1.00000000000000009e56

   1. Initial program 94.1%

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

   3. Taylor expanded in c around inf

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

   5. Applied rewrites 38.0%

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

Alternative 10: 66.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+159} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+93}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* z t) -2e+159) (not (<= (* z t) 2e+93)))
   (fma b a (* t z))
   (fma b a (* y x))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((z * t) <= -2e+159) || !((z * t) <= 2e+93)) {
		tmp = fma(b, a, (t * z));
	} else {
		tmp = fma(b, a, (y * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(z * t) <= -2e+159) || !(Float64(z * t) <= 2e+93))
		tmp = fma(b, a, Float64(t * z));
	else
		tmp = fma(b, a, 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+159], N[Not[LessEqual[N[(z * t), $MachinePrecision], 2e+93]], $MachinePrecision]], N[(b * a + N[(t * z), $MachinePrecision]), $MachinePrecision], N[(b * a + N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.9999999999999999e159 or 2.00000000000000009e93 < (*.f64 z t)

   1. Initial program 85.3%

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

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 77.5%

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

   if -1.9999999999999999e159 < (*.f64 z t) < 2.00000000000000009e93

   1. Initial program 94.8%

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

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 72.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 66.6%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+159} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+93}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+179} \lor \neg \left(x \cdot y \leq 2 \cdot 10^{+102}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* x y) -4e+179) (not (<= (* x y) 2e+102)))
   (* y x)
   (fma b a (* t z))))

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+179) || !((x * y) <= 2e+102)) {
		tmp = y * x;
	} else {
		tmp = fma(b, a, (t * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(x * y) <= -4e+179) || !(Float64(x * y) <= 2e+102))
		tmp = Float64(y * x);
	else
		tmp = fma(b, a, Float64(t * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -4e+179], N[Not[LessEqual[N[(x * y), $MachinePrecision], 2e+102]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(b * a + N[(t * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -3.99999999999999992e179 or 1.99999999999999995e102 < (*.f64 x y)

   1. Initial program 85.7%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 68.1%

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

   if -3.99999999999999992e179 < (*.f64 x y) < 1.99999999999999995e102

   1. Initial program 94.7%

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

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 70.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 62.4%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+179} \lor \neg \left(x \cdot y \leq 2 \cdot 10^{+102}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -4 \cdot 10^{+29} \lor \neg \left(a \cdot b \leq 10^{+152}\right):\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;i \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* a b) -4e+29) (not (<= (* a b) 1e+152))) (* b a) (* i c)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((a * b) <= -4e+29) || !((a * b) <= 1e+152)) {
		tmp = b * a;
	} else {
		tmp = i * c;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if (((a * b) <= (-4d+29)) .or. (.not. ((a * b) <= 1d+152))) then
        tmp = b * a
    else
        tmp = i * c
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((a * b) <= -4e+29) || !((a * b) <= 1e+152)) {
		tmp = b * a;
	} else {
		tmp = i * c;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if ((a * b) <= -4e+29) or not ((a * b) <= 1e+152):
		tmp = b * a
	else:
		tmp = i * c
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(a * b) <= -4e+29) || !(Float64(a * b) <= 1e+152))
		tmp = Float64(b * a);
	else
		tmp = Float64(i * c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (((a * b) <= -4e+29) || ~(((a * b) <= 1e+152)))
		tmp = b * a;
	else
		tmp = i * c;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(a * b), $MachinePrecision], -4e+29], N[Not[LessEqual[N[(a * b), $MachinePrecision], 1e+152]], $MachinePrecision]], N[(b * a), $MachinePrecision], N[(i * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -3.99999999999999966e29 or 1e152 < (*.f64 a b)

   1. Initial program 87.2%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 50.5%

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

   if -3.99999999999999966e29 < (*.f64 a b) < 1e152

   1. Initial program 94.8%

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

   3. Taylor expanded in c around inf

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

   5. Applied rewrites 34.8%

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

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -4 \cdot 10^{+29} \lor \neg \left(a \cdot b \leq 10^{+152}\right):\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;i \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 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 91.8%

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

3. Taylor expanded in a around inf

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

5. Applied rewrites 25.4%

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

Reproduce

herbie shell --seed 2025026 
(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)))