AI.Clustering.Hierarchical.Internal:ward from clustering-0.2.1

Percentage Accurate: 60.6% → 98.5%

Time: 10.1s

Alternatives: 22

Speedup: 1.2×

Specification

\[\mathsf{TRUE}\left(\right)\]

Math

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

FPCore

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

C

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

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 + y) * a)) - (y * b)) / ((x + t) + y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 60.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 + y) * a)) - (y * b)) / ((x + t) + y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t + \left(x + y\right)\\ t_2 := \left(y + x\right) + t\\ t_3 := \mathsf{fma}\left(\frac{y + x}{t\_2}, z, a \cdot \mathsf{fma}\left(-1, \frac{b}{a} \cdot \frac{y}{t\_1}, \frac{t + y}{t\_1}\right)\right)\\ \mathbf{if}\;a \leq -5.5 \cdot 10^{-18}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;a \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x + y}{t\_1}, \frac{a \cdot \left(t + y\right)}{t\_1}\right) - b \cdot \frac{y}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ t (+ x y)))
        (t_2 (+ (+ y x) t))
        (t_3
         (fma
          (/ (+ y x) t_2)
          z
          (* a (fma -1.0 (* (/ b a) (/ y t_1)) (/ (+ t y) t_1))))))
   (if (<= a -5.5e-18)
     t_3
     (if (<= a 5e-46)
       (- (fma z (/ (+ x y) t_1) (/ (* a (+ t y)) t_1)) (* b (/ y t_2)))
       t_3))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t + (x + y);
	double t_2 = (y + x) + t;
	double t_3 = fma(((y + x) / t_2), z, (a * fma(-1.0, ((b / a) * (y / t_1)), ((t + y) / t_1))));
	double tmp;
	if (a <= -5.5e-18) {
		tmp = t_3;
	} else if (a <= 5e-46) {
		tmp = fma(z, ((x + y) / t_1), ((a * (t + y)) / t_1)) - (b * (y / t_2));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t + Float64(x + y))
	t_2 = Float64(Float64(y + x) + t)
	t_3 = fma(Float64(Float64(y + x) / t_2), z, Float64(a * fma(-1.0, Float64(Float64(b / a) * Float64(y / t_1)), Float64(Float64(t + y) / t_1))))
	tmp = 0.0
	if (a <= -5.5e-18)
		tmp = t_3;
	elseif (a <= 5e-46)
		tmp = Float64(fma(z, Float64(Float64(x + y) / t_1), Float64(Float64(a * Float64(t + y)) / t_1)) - Float64(b * Float64(y / t_2)));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(y + x), $MachinePrecision] / t$95$2), $MachinePrecision] * z + N[(a * N[(-1.0 * N[(N[(b / a), $MachinePrecision] * N[(y / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -5.5e-18], t$95$3, If[LessEqual[a, 5e-46], N[(N[(z * N[(N[(x + y), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(N[(a * N[(t + y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] - N[(b * N[(y / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if a < -5.5e-18 or 4.99999999999999992e-46 < a

   1. Initial program 56.3%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   4. Applied rewrites 66.4%

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

   5. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, \color{blue}{a \cdot \left(-1 \cdot \frac{b \cdot y}{a \cdot \left(t + \left(x + y\right)\right)} + \left(\frac{t}{t + \left(x + y\right)} + \frac{y}{t + \left(x + y\right)}\right)\right)}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. div-add-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lift-+.f64 98.5

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

   7. Applied rewrites 98.5%

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

   if -5.5e-18 < a < 4.99999999999999992e-46

   1. Initial program 77.8%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   4. Applied rewrites 80.0%

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

   5. Taylor expanded in z around 0

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

      1. lower-fma.f64 N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 96.4

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

   7. Applied rewrites 96.4%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.5 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, a \cdot \mathsf{fma}\left(-1, \frac{b}{a} \cdot \frac{y}{t + \left(x + y\right)}, \frac{t + y}{t + \left(x + y\right)}\right)\right)\\ \mathbf{elif}\;a \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x + y}{t + \left(x + y\right)}, \frac{a \cdot \left(t + y\right)}{t + \left(x + y\right)}\right) - b \cdot \frac{y}{\left(y + x\right) + t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, a \cdot \mathsf{fma}\left(-1, \frac{b}{a} \cdot \frac{y}{t + \left(x + y\right)}, \frac{t + y}{t + \left(x + y\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ t_3 := \left(y + x\right) + t\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+170}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{t\_3}, z, a\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+35}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t + y, a, \left(y + x\right) \cdot z\right)}{t\_1}\\ \mathbf{elif}\;t\_2 \leq 1.3 \cdot 10^{+258}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \frac{\mathsf{fma}\left(t + y, a, \left(-b\right) \cdot y\right)}{t\_3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ x t) y))
        (t_2 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) t_1))
        (t_3 (+ (+ y x) t)))
   (if (<= t_2 -2e+170)
     (fma (/ (+ y x) t_3) z a)
     (if (<= t_2 2e+35)
       (/ (fma (+ t y) a (* (+ y x) z)) t_1)
       (if (<= t_2 1.3e+258)
         (fma 1.0 z (/ (fma (+ t y) a (* (- b) y)) t_3))
         (- (+ a z) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + t) + y;
	double t_2 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / t_1;
	double t_3 = (y + x) + t;
	double tmp;
	if (t_2 <= -2e+170) {
		tmp = fma(((y + x) / t_3), z, a);
	} else if (t_2 <= 2e+35) {
		tmp = fma((t + y), a, ((y + x) * z)) / t_1;
	} else if (t_2 <= 1.3e+258) {
		tmp = fma(1.0, z, (fma((t + y), a, (-b * y)) / t_3));
	} else {
		tmp = (a + z) - b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + t) + y)
	t_2 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / t_1)
	t_3 = Float64(Float64(y + x) + t)
	tmp = 0.0
	if (t_2 <= -2e+170)
		tmp = fma(Float64(Float64(y + x) / t_3), z, a);
	elseif (t_2 <= 2e+35)
		tmp = Float64(fma(Float64(t + y), a, Float64(Float64(y + x) * z)) / t_1);
	elseif (t_2 <= 1.3e+258)
		tmp = fma(1.0, z, Float64(fma(Float64(t + y), a, Float64(Float64(-b) * y)) / t_3));
	else
		tmp = Float64(Float64(a + z) - b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+170], N[(N[(N[(y + x), $MachinePrecision] / t$95$3), $MachinePrecision] * z + a), $MachinePrecision], If[LessEqual[t$95$2, 2e+35], N[(N[(N[(t + y), $MachinePrecision] * a + N[(N[(y + x), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 1.3e+258], N[(1.0 * z + N[(N[(N[(t + y), $MachinePrecision] * a + N[((-b) * y), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -2.00000000000000007e170

   1. Initial program 27.4%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   4. Applied rewrites 63.6%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 78.4%

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

   if -2.00000000000000007e170 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1.9999999999999999e35

   1. Initial program 99.6%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 80.8

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

   5. Applied rewrites 80.8%

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

   if 1.9999999999999999e35 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1.30000000000000005e258

   1. Initial program 99.7%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, z, \frac{\mathsf{fma}\left(t + y, a, \left(-b\right) \cdot y\right)}{\left(y + x\right) + t}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 93.7%

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

   if 1.30000000000000005e258 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 17.8%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

         \[\leadsto \left(a + z\right) - \color{blue}{b} \]

      2. lower-+.f64 72.6

         \[\leadsto \left(a + z\right) - b \]

   5. Applied rewrites 72.6%

      \[\leadsto \color{blue}{\left(a + z\right) - b} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq -2 \cdot 10^{+170}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, a\right)\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq 2 \cdot 10^{+35}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t + y, a, \left(y + x\right) \cdot z\right)}{\left(x + t\right) + y}\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq 1.3 \cdot 10^{+258}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \frac{\mathsf{fma}\left(t + y, a, \left(-b\right) \cdot y\right)}{\left(y + x\right) + t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t + \left(x + y\right)\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ t_3 := \left(y + x\right) + t\\ t_4 := \frac{y + x}{t\_3}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(t\_4, z, a \cdot \frac{t + y}{t\_1}\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\mathsf{fma}\left(t\_4, z, \frac{\mathsf{fma}\left(t + y, a, \left(-b\right) \cdot y\right)}{t\_3}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_4, z, a \cdot \mathsf{fma}\left(-1, \frac{b}{a} \cdot \frac{y}{t\_1}, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ t (+ x y)))
        (t_2 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))
        (t_3 (+ (+ y x) t))
        (t_4 (/ (+ y x) t_3)))
   (if (<= t_2 (- INFINITY))
     (fma t_4 z (* a (/ (+ t y) t_1)))
     (if (<= t_2 5e+292)
       (fma t_4 z (/ (fma (+ t y) a (* (- b) y)) t_3))
       (fma t_4 z (* a (fma -1.0 (* (/ b a) (/ y t_1)) 1.0)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t + (x + y);
	double t_2 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
	double t_3 = (y + x) + t;
	double t_4 = (y + x) / t_3;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma(t_4, z, (a * ((t + y) / t_1)));
	} else if (t_2 <= 5e+292) {
		tmp = fma(t_4, z, (fma((t + y), a, (-b * y)) / t_3));
	} else {
		tmp = fma(t_4, z, (a * fma(-1.0, ((b / a) * (y / t_1)), 1.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t + Float64(x + y))
	t_2 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / Float64(Float64(x + t) + y))
	t_3 = Float64(Float64(y + x) + t)
	t_4 = Float64(Float64(y + x) / t_3)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = fma(t_4, z, Float64(a * Float64(Float64(t + y) / t_1)));
	elseif (t_2 <= 5e+292)
		tmp = fma(t_4, z, Float64(fma(Float64(t + y), a, Float64(Float64(-b) * y)) / t_3));
	else
		tmp = fma(t_4, z, Float64(a * fma(-1.0, Float64(Float64(b / a) * Float64(y / t_1)), 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$4 = N[(N[(y + x), $MachinePrecision] / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(t$95$4 * z + N[(a * N[(N[(t + y), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+292], N[(t$95$4 * z + N[(N[(N[(t + y), $MachinePrecision] * a + N[((-b) * y), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision], N[(t$95$4 * z + N[(a * N[(-1.0 * N[(N[(b / a), $MachinePrecision] * N[(y / t$95$1), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 6.2%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   4. Applied rewrites 52.9%

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

   5. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, \color{blue}{-1 \cdot \frac{b \cdot y}{t + \left(x + y\right)} + a \cdot \left(\frac{t}{t + \left(x + y\right)} + \frac{y}{t + \left(x + y\right)}\right)}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lift-+.f64 88.8

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

   7. Applied rewrites 88.8%

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

   8. Taylor expanded in a around inf

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

      1. div-add-rev N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 88.9

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

   10. Applied rewrites 88.9%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4.9999999999999996e292

   1. Initial program 99.7%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   4. Applied rewrites 99.7%

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

   if 4.9999999999999996e292 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 7.9%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   4. Applied rewrites 29.9%

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

   5. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, \color{blue}{a \cdot \left(-1 \cdot \frac{b \cdot y}{a \cdot \left(t + \left(x + y\right)\right)} + \left(\frac{t}{t + \left(x + y\right)} + \frac{y}{t + \left(x + y\right)}\right)\right)}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. div-add-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lift-+.f64 86.8

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

   7. Applied rewrites 86.8%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, a \cdot \mathsf{fma}\left(-1, \frac{b}{a} \cdot \frac{y}{t + \left(x + y\right)}, 1\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 78.2%

       \[\leadsto \mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, a \cdot \mathsf{fma}\left(-1, \frac{b}{a} \cdot \frac{y}{t + \left(x + y\right)}, 1\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, a \cdot \frac{t + y}{t + \left(x + y\right)}\right)\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, \frac{\mathsf{fma}\left(t + y, a, \left(-b\right) \cdot y\right)}{\left(y + x\right) + t}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, a \cdot \mathsf{fma}\left(-1, \frac{b}{a} \cdot \frac{y}{t + \left(x + y\right)}, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ t_2 := \left(y + x\right) + t\\ t_3 := \frac{y + x}{t\_2}\\ t_4 := \mathsf{fma}\left(t\_3, z, a \cdot \frac{t + y}{t + \left(x + y\right)}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\mathsf{fma}\left(t\_3, z, \frac{\mathsf{fma}\left(t + y, a, \left(-b\right) \cdot y\right)}{t\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))
        (t_2 (+ (+ y x) t))
        (t_3 (/ (+ y x) t_2))
        (t_4 (fma t_3 z (* a (/ (+ t y) (+ t (+ x y)))))))
   (if (<= t_1 (- INFINITY))
     t_4
     (if (<= t_1 5e+292)
       (fma t_3 z (/ (fma (+ t y) a (* (- b) y)) t_2))
       t_4))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
	double t_2 = (y + x) + t;
	double t_3 = (y + x) / t_2;
	double t_4 = fma(t_3, z, (a * ((t + y) / (t + (x + y)))));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_1 <= 5e+292) {
		tmp = fma(t_3, z, (fma((t + y), a, (-b * y)) / t_2));
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / Float64(Float64(x + t) + y))
	t_2 = Float64(Float64(y + x) + t)
	t_3 = Float64(Float64(y + x) / t_2)
	t_4 = fma(t_3, z, Float64(a * Float64(Float64(t + y) / Float64(t + Float64(x + y)))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_1 <= 5e+292)
		tmp = fma(t_3, z, Float64(fma(Float64(t + y), a, Float64(Float64(-b) * y)) / t_2));
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y + x), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * z + N[(a * N[(N[(t + y), $MachinePrecision] / N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$4, If[LessEqual[t$95$1, 5e+292], N[(t$95$3 * z + N[(N[(N[(t + y), $MachinePrecision] * a + N[((-b) * y), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -inf.0 or 4.9999999999999996e292 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 7.3%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   4. Applied rewrites 38.4%

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

   5. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, \color{blue}{-1 \cdot \frac{b \cdot y}{t + \left(x + y\right)} + a \cdot \left(\frac{t}{t + \left(x + y\right)} + \frac{y}{t + \left(x + y\right)}\right)}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lift-+.f64 77.3

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

   7. Applied rewrites 77.3%

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

   8. Taylor expanded in a around inf

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

      1. div-add-rev N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 80.9

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

   10. Applied rewrites 80.9%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4.9999999999999996e292

   1. Initial program 99.7%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   4. Applied rewrites 99.7%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, a \cdot \frac{t + y}{t + \left(x + y\right)}\right)\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, \frac{\mathsf{fma}\left(t + y, a, \left(-b\right) \cdot y\right)}{\left(y + x\right) + t}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, a \cdot \frac{t + y}{t + \left(x + y\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ t_3 := \mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, a \cdot \frac{t + y}{t + \left(x + y\right)}\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y + x, z, \mathsf{fma}\left(t + y, a, \left(-b\right) \cdot y\right)\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ x t) y))
        (t_2 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) t_1))
        (t_3
         (fma (/ (+ y x) (+ (+ y x) t)) z (* a (/ (+ t y) (+ t (+ x y)))))))
   (if (<= t_2 (- INFINITY))
     t_3
     (if (<= t_2 5e+292)
       (/ (fma (+ y x) z (fma (+ t y) a (* (- b) y))) t_1)
       t_3))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + t) + y;
	double t_2 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / t_1;
	double t_3 = fma(((y + x) / ((y + x) + t)), z, (a * ((t + y) / (t + (x + y)))));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= 5e+292) {
		tmp = fma((y + x), z, fma((t + y), a, (-b * y))) / t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + t) + y)
	t_2 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / t_1)
	t_3 = fma(Float64(Float64(y + x) / Float64(Float64(y + x) + t)), z, Float64(a * Float64(Float64(t + y) / Float64(t + Float64(x + y)))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= 5e+292)
		tmp = Float64(fma(Float64(y + x), z, fma(Float64(t + y), a, Float64(Float64(-b) * y))) / t_1);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(y + x), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] * z + N[(a * N[(N[(t + y), $MachinePrecision] / N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, 5e+292], N[(N[(N[(y + x), $MachinePrecision] * z + N[(N[(t + y), $MachinePrecision] * a + N[((-b) * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -inf.0 or 4.9999999999999996e292 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 7.3%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   4. Applied rewrites 38.4%

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

   5. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, \color{blue}{-1 \cdot \frac{b \cdot y}{t + \left(x + y\right)} + a \cdot \left(\frac{t}{t + \left(x + y\right)} + \frac{y}{t + \left(x + y\right)}\right)}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lift-+.f64 77.3

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

   7. Applied rewrites 77.3%

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

   8. Taylor expanded in a around inf

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

      1. div-add-rev N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 80.9

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

   10. Applied rewrites 80.9%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4.9999999999999996e292

   1. Initial program 99.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate--l+ N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

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

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

      15. *-commutative N/A

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

      16. mul-1-neg N/A

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

      17. associate-*r* N/A

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

      18. mul-1-neg N/A

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

      19. lower-fma.f64 N/A

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

      20. lift-+.f64 N/A

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

      21. mul-1-neg N/A

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

      22. associate-*r* N/A

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

      23. lower-*.f64 N/A

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

      24. mul-1-neg N/A

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

      25. lower-neg.f64 99.7

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

   4. Applied rewrites 99.7%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, a \cdot \frac{t + y}{t + \left(x + y\right)}\right)\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y + x, z, \mathsf{fma}\left(t + y, a, \left(-b\right) \cdot y\right)\right)}{\left(x + t\right) + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, a \cdot \frac{t + y}{t + \left(x + y\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, a \cdot \frac{t}{t + x}\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+263}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y + x, z, \mathsf{fma}\left(t + y, a, \left(-b\right) \cdot y\right)\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ x t) y))
        (t_2 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) t_1)))
   (if (<= t_2 (- INFINITY))
     (fma (/ (+ y x) (+ (+ y x) t)) z (* a (/ t (+ t x))))
     (if (<= t_2 1e+263)
       (/ (fma (+ y x) z (fma (+ t y) a (* (- b) y))) t_1)
       (- (+ a z) b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + t) + y;
	double t_2 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma(((y + x) / ((y + x) + t)), z, (a * (t / (t + x))));
	} else if (t_2 <= 1e+263) {
		tmp = fma((y + x), z, fma((t + y), a, (-b * y))) / t_1;
	} else {
		tmp = (a + z) - b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + t) + y)
	t_2 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = fma(Float64(Float64(y + x) / Float64(Float64(y + x) + t)), z, Float64(a * Float64(t / Float64(t + x))));
	elseif (t_2 <= 1e+263)
		tmp = Float64(fma(Float64(y + x), z, fma(Float64(t + y), a, Float64(Float64(-b) * y))) / t_1);
	else
		tmp = Float64(Float64(a + z) - b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[(y + x), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] * z + N[(a * N[(t / N[(t + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+263], N[(N[(N[(y + x), $MachinePrecision] * z + N[(N[(t + y), $MachinePrecision] * a + N[((-b) * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 6.2%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   4. Applied rewrites 52.9%

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

   5. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, \color{blue}{a \cdot \left(-1 \cdot \frac{b \cdot y}{a \cdot \left(t + \left(x + y\right)\right)} + \left(\frac{t}{t + \left(x + y\right)} + \frac{y}{t + \left(x + y\right)}\right)\right)}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. div-add-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lift-+.f64 91.3

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

   7. Applied rewrites 91.3%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lift-+.f64 83.1

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

   10. Applied rewrites 83.1%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1.00000000000000002e263

   1. Initial program 99.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate--l+ N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

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

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

      15. *-commutative N/A

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

      16. mul-1-neg N/A

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

      17. associate-*r* N/A

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

      18. mul-1-neg N/A

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

      19. lower-fma.f64 N/A

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

      20. lift-+.f64 N/A

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

      21. mul-1-neg N/A

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

      22. associate-*r* N/A

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

      23. lower-*.f64 N/A

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

      24. mul-1-neg N/A

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

      25. lower-neg.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if 1.00000000000000002e263 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 13.9%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

         \[\leadsto \left(a + z\right) - \color{blue}{b} \]

      2. lower-+.f64 74.1

         \[\leadsto \left(a + z\right) - b \]

   5. Applied rewrites 74.1%

      \[\leadsto \color{blue}{\left(a + z\right) - b} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, a \cdot \frac{t}{t + x}\right)\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq 10^{+263}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y + x, z, \mathsf{fma}\left(t + y, a, \left(-b\right) \cdot y\right)\right)}{\left(x + t\right) + y}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, a\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+263}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y + x, z, \mathsf{fma}\left(t + y, a, \left(-b\right) \cdot y\right)\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ x t) y))
        (t_2 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) t_1)))
   (if (<= t_2 (- INFINITY))
     (fma (/ (+ y x) (+ (+ y x) t)) z a)
     (if (<= t_2 1e+263)
       (/ (fma (+ y x) z (fma (+ t y) a (* (- b) y))) t_1)
       (- (+ a z) b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + t) + y;
	double t_2 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma(((y + x) / ((y + x) + t)), z, a);
	} else if (t_2 <= 1e+263) {
		tmp = fma((y + x), z, fma((t + y), a, (-b * y))) / t_1;
	} else {
		tmp = (a + z) - b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + t) + y)
	t_2 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = fma(Float64(Float64(y + x) / Float64(Float64(y + x) + t)), z, a);
	elseif (t_2 <= 1e+263)
		tmp = Float64(fma(Float64(y + x), z, fma(Float64(t + y), a, Float64(Float64(-b) * y))) / t_1);
	else
		tmp = Float64(Float64(a + z) - b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[(y + x), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] * z + a), $MachinePrecision], If[LessEqual[t$95$2, 1e+263], N[(N[(N[(y + x), $MachinePrecision] * z + N[(N[(t + y), $MachinePrecision] * a + N[((-b) * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 6.2%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   4. Applied rewrites 52.9%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 80.2%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1.00000000000000002e263

   1. Initial program 99.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate--l+ N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

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

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

      15. *-commutative N/A

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

      16. mul-1-neg N/A

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

      17. associate-*r* N/A

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

      18. mul-1-neg N/A

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

      19. lower-fma.f64 N/A

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

      20. lift-+.f64 N/A

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

      21. mul-1-neg N/A

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

      22. associate-*r* N/A

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

      23. lower-*.f64 N/A

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

      24. mul-1-neg N/A

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

      25. lower-neg.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if 1.00000000000000002e263 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 13.9%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

         \[\leadsto \left(a + z\right) - \color{blue}{b} \]

      2. lower-+.f64 74.1

         \[\leadsto \left(a + z\right) - b \]

   5. Applied rewrites 74.1%

      \[\leadsto \color{blue}{\left(a + z\right) - b} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, a\right)\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq 10^{+263}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y + x, z, \mathsf{fma}\left(t + y, a, \left(-b\right) \cdot y\right)\right)}{\left(x + t\right) + y}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t + \left(x + y\right)\\ t_2 := \frac{y + x}{\left(y + x\right) + t}\\ \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq 4 \cdot 10^{+299}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, z, \mathsf{fma}\left(-1, \frac{b \cdot y}{t\_1}, a \cdot \frac{t + y}{t\_1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, z, a \cdot \left(b \cdot \mathsf{fma}\left(-1, \frac{y}{a \cdot t\_1}, \frac{t + y}{b \cdot t\_1}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ t (+ x y))) (t_2 (/ (+ y x) (+ (+ y x) t))))
   (if (<=
        (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y))
        4e+299)
     (fma t_2 z (fma -1.0 (/ (* b y) t_1) (* a (/ (+ t y) t_1))))
     (fma
      t_2
      z
      (* a (* b (fma -1.0 (/ y (* a t_1)) (/ (+ t y) (* b t_1)))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t + (x + y);
	double t_2 = (y + x) / ((y + x) + t);
	double tmp;
	if ((((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y)) <= 4e+299) {
		tmp = fma(t_2, z, fma(-1.0, ((b * y) / t_1), (a * ((t + y) / t_1))));
	} else {
		tmp = fma(t_2, z, (a * (b * fma(-1.0, (y / (a * t_1)), ((t + y) / (b * t_1))))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t + Float64(x + y))
	t_2 = Float64(Float64(y + x) / Float64(Float64(y + x) + t))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / Float64(Float64(x + t) + y)) <= 4e+299)
		tmp = fma(t_2, z, fma(-1.0, Float64(Float64(b * y) / t_1), Float64(a * Float64(Float64(t + y) / t_1))));
	else
		tmp = fma(t_2, z, Float64(a * Float64(b * fma(-1.0, Float64(y / Float64(a * t_1)), Float64(Float64(t + y) / Float64(b * t_1))))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y + x), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], 4e+299], N[(t$95$2 * z + N[(-1.0 * N[(N[(b * y), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(a * N[(N[(t + y), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * z + N[(a * N[(b * N[(-1.0 * N[(y / N[(a * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] / N[(b * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4.0000000000000002e299

   1. Initial program 83.8%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   4. Applied rewrites 91.8%

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

   5. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, \color{blue}{-1 \cdot \frac{b \cdot y}{t + \left(x + y\right)} + a \cdot \left(\frac{t}{t + \left(x + y\right)} + \frac{y}{t + \left(x + y\right)}\right)}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lift-+.f64 97.0

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

   7. Applied rewrites 97.0%

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

   if 4.0000000000000002e299 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 4.6%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   4. Applied rewrites 27.4%

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

   5. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, \color{blue}{a \cdot \left(-1 \cdot \frac{b \cdot y}{a \cdot \left(t + \left(x + y\right)\right)} + \left(\frac{t}{t + \left(x + y\right)} + \frac{y}{t + \left(x + y\right)}\right)\right)}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. div-add-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lift-+.f64 86.4

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

   7. Applied rewrites 86.4%

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

   8. Taylor expanded in b around inf

      \[\leadsto \mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, a \cdot \left(b \cdot \color{blue}{\left(-1 \cdot \frac{y}{a \cdot \left(t + \left(x + y\right)\right)} + \left(\frac{t}{b \cdot \left(t + \left(x + y\right)\right)} + \frac{y}{b \cdot \left(t + \left(x + y\right)\right)}\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 84.1

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

   10. Applied rewrites 84.1%

       \[\leadsto \mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, a \cdot \left(b \cdot \color{blue}{\mathsf{fma}\left(-1, \frac{y}{a \cdot \left(t + \left(x + y\right)\right)}, \frac{t + y}{b \cdot \left(t + \left(x + y\right)\right)}\right)}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq 4 \cdot 10^{+299}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, \mathsf{fma}\left(-1, \frac{b \cdot y}{t + \left(x + y\right)}, a \cdot \frac{t + y}{t + \left(x + y\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, a \cdot \left(b \cdot \mathsf{fma}\left(-1, \frac{y}{a \cdot \left(t + \left(x + y\right)\right)}, \frac{t + y}{b \cdot \left(t + \left(x + y\right)\right)}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ t_2 := \left(a + z\right) - b\\ \mathbf{if}\;t\_1 \leq -8 \cdot 10^{-36}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-58}:\\ \;\;\;\;\frac{\left(y + x\right) \cdot z}{t + x}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))
        (t_2 (- (+ a z) b)))
   (if (<= t_1 -8e-36) t_2 (if (<= t_1 4e-58) (/ (* (+ y x) z) (+ t x)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
	double t_2 = (a + z) - b;
	double tmp;
	if (t_1 <= -8e-36) {
		tmp = t_2;
	} else if (t_1 <= 4e-58) {
		tmp = ((y + x) * z) / (t + x);
	} else {
		tmp = t_2;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y)
    t_2 = (a + z) - b
    if (t_1 <= (-8d-36)) then
        tmp = t_2
    else if (t_1 <= 4d-58) then
        tmp = ((y + x) * z) / (t + x)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
	double t_2 = (a + z) - b;
	double tmp;
	if (t_1 <= -8e-36) {
		tmp = t_2;
	} else if (t_1 <= 4e-58) {
		tmp = ((y + x) * z) / (t + x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y)
	t_2 = (a + z) - b
	tmp = 0
	if t_1 <= -8e-36:
		tmp = t_2
	elif t_1 <= 4e-58:
		tmp = ((y + x) * z) / (t + x)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / Float64(Float64(x + t) + y))
	t_2 = Float64(Float64(a + z) - b)
	tmp = 0.0
	if (t_1 <= -8e-36)
		tmp = t_2;
	elseif (t_1 <= 4e-58)
		tmp = Float64(Float64(Float64(y + x) * z) / Float64(t + x));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
	t_2 = (a + z) - b;
	tmp = 0.0;
	if (t_1 <= -8e-36)
		tmp = t_2;
	elseif (t_1 <= 4e-58)
		tmp = ((y + x) * z) / (t + x);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[t$95$1, -8e-36], t$95$2, If[LessEqual[t$95$1, 4e-58], N[(N[(N[(y + x), $MachinePrecision] * z), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -7.9999999999999995e-36 or 4.0000000000000001e-58 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 59.0%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

         \[\leadsto \left(a + z\right) - \color{blue}{b} \]

      2. lower-+.f64 64.1

         \[\leadsto \left(a + z\right) - b \]

   5. Applied rewrites 64.1%

      \[\leadsto \color{blue}{\left(a + z\right) - b} \]

   if -7.9999999999999995e-36 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4.0000000000000001e-58

   1. Initial program 99.6%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 54.3

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

   5. Applied rewrites 54.3%

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

   6. Taylor expanded in y around inf

      \[\leadsto \frac{\left(y + x\right) \cdot z}{\color{blue}{y}} \]
   7. Step-by-step derivation

   8. Applied rewrites 7.6%

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

   9. Taylor expanded in y around 0

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

       1. lift-+.f64 50.0

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

   11. Applied rewrites 50.0%

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

Alternative 10: 92.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t + \left(x + y\right)\\ t_2 := \left(y + x\right) + t\\ t_3 := \mathsf{fma}\left(\frac{y + x}{t\_2}, z, \mathsf{fma}\left(-1, \frac{b \cdot y}{t\_1}, a \cdot \frac{t + y}{t\_1}\right)\right)\\ \mathbf{if}\;a \leq -2.4 \cdot 10^{+55}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;a \leq 180000000:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x + y}{t\_1}, \frac{a \cdot \left(t + y\right)}{t\_1}\right) - b \cdot \frac{y}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ t (+ x y)))
        (t_2 (+ (+ y x) t))
        (t_3
         (fma
          (/ (+ y x) t_2)
          z
          (fma -1.0 (/ (* b y) t_1) (* a (/ (+ t y) t_1))))))
   (if (<= a -2.4e+55)
     t_3
     (if (<= a 180000000.0)
       (- (fma z (/ (+ x y) t_1) (/ (* a (+ t y)) t_1)) (* b (/ y t_2)))
       t_3))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t + (x + y);
	double t_2 = (y + x) + t;
	double t_3 = fma(((y + x) / t_2), z, fma(-1.0, ((b * y) / t_1), (a * ((t + y) / t_1))));
	double tmp;
	if (a <= -2.4e+55) {
		tmp = t_3;
	} else if (a <= 180000000.0) {
		tmp = fma(z, ((x + y) / t_1), ((a * (t + y)) / t_1)) - (b * (y / t_2));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t + Float64(x + y))
	t_2 = Float64(Float64(y + x) + t)
	t_3 = fma(Float64(Float64(y + x) / t_2), z, fma(-1.0, Float64(Float64(b * y) / t_1), Float64(a * Float64(Float64(t + y) / t_1))))
	tmp = 0.0
	if (a <= -2.4e+55)
		tmp = t_3;
	elseif (a <= 180000000.0)
		tmp = Float64(fma(z, Float64(Float64(x + y) / t_1), Float64(Float64(a * Float64(t + y)) / t_1)) - Float64(b * Float64(y / t_2)));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(y + x), $MachinePrecision] / t$95$2), $MachinePrecision] * z + N[(-1.0 * N[(N[(b * y), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(a * N[(N[(t + y), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2.4e+55], t$95$3, If[LessEqual[a, 180000000.0], N[(N[(z * N[(N[(x + y), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(N[(a * N[(t + y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] - N[(b * N[(y / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.3999999999999999e55 or 1.8e8 < a

   1. Initial program 53.3%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   4. Applied rewrites 62.4%

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

   5. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, \color{blue}{-1 \cdot \frac{b \cdot y}{t + \left(x + y\right)} + a \cdot \left(\frac{t}{t + \left(x + y\right)} + \frac{y}{t + \left(x + y\right)}\right)}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lift-+.f64 93.7

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

   7. Applied rewrites 93.7%

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

   if -2.3999999999999999e55 < a < 1.8e8

   1. Initial program 76.0%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   4. Applied rewrites 80.4%

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

   5. Taylor expanded in z around 0

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

      1. lower-fma.f64 N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 96.4

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

   7. Applied rewrites 96.4%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, \mathsf{fma}\left(-1, \frac{b \cdot y}{t + \left(x + y\right)}, a \cdot \frac{t + y}{t + \left(x + y\right)}\right)\right)\\ \mathbf{elif}\;a \leq 180000000:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x + y}{t + \left(x + y\right)}, \frac{a \cdot \left(t + y\right)}{t + \left(x + y\right)}\right) - b \cdot \frac{y}{\left(y + x\right) + t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, \mathsf{fma}\left(-1, \frac{b \cdot y}{t + \left(x + y\right)}, a \cdot \frac{t + y}{t + \left(x + y\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 93.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t + \left(x + y\right)\\ t_2 := \left(y + x\right) + t\\ t_3 := \mathsf{fma}\left(\frac{y + x}{t\_2}, z, a \cdot \frac{t + y}{t\_1}\right)\\ \mathbf{if}\;a \leq -2.4 \cdot 10^{+55}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x + y}{t\_1}, \frac{a \cdot \left(t + y\right)}{t\_1}\right) - b \cdot \frac{y}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ t (+ x y)))
        (t_2 (+ (+ y x) t))
        (t_3 (fma (/ (+ y x) t_2) z (* a (/ (+ t y) t_1)))))
   (if (<= a -2.4e+55)
     t_3
     (if (<= a 1.35e+41)
       (- (fma z (/ (+ x y) t_1) (/ (* a (+ t y)) t_1)) (* b (/ y t_2)))
       t_3))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t + (x + y);
	double t_2 = (y + x) + t;
	double t_3 = fma(((y + x) / t_2), z, (a * ((t + y) / t_1)));
	double tmp;
	if (a <= -2.4e+55) {
		tmp = t_3;
	} else if (a <= 1.35e+41) {
		tmp = fma(z, ((x + y) / t_1), ((a * (t + y)) / t_1)) - (b * (y / t_2));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t + Float64(x + y))
	t_2 = Float64(Float64(y + x) + t)
	t_3 = fma(Float64(Float64(y + x) / t_2), z, Float64(a * Float64(Float64(t + y) / t_1)))
	tmp = 0.0
	if (a <= -2.4e+55)
		tmp = t_3;
	elseif (a <= 1.35e+41)
		tmp = Float64(fma(z, Float64(Float64(x + y) / t_1), Float64(Float64(a * Float64(t + y)) / t_1)) - Float64(b * Float64(y / t_2)));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(y + x), $MachinePrecision] / t$95$2), $MachinePrecision] * z + N[(a * N[(N[(t + y), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2.4e+55], t$95$3, If[LessEqual[a, 1.35e+41], N[(N[(z * N[(N[(x + y), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(N[(a * N[(t + y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] - N[(b * N[(y / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.3999999999999999e55 or 1.35e41 < a

   1. Initial program 53.4%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   4. Applied rewrites 61.3%

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

   5. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, \color{blue}{-1 \cdot \frac{b \cdot y}{t + \left(x + y\right)} + a \cdot \left(\frac{t}{t + \left(x + y\right)} + \frac{y}{t + \left(x + y\right)}\right)}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lift-+.f64 94.2

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

   7. Applied rewrites 94.2%

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

   8. Taylor expanded in a around inf

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

      1. div-add-rev N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 90.4

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

   10. Applied rewrites 90.4%

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

   if -2.3999999999999999e55 < a < 1.35e41

   1. Initial program 74.8%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   4. Applied rewrites 78.9%

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

   5. Taylor expanded in z around 0

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

      1. lower-fma.f64 N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 95.9

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

   7. Applied rewrites 95.9%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, a \cdot \frac{t + y}{t + \left(x + y\right)}\right)\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x + y}{t + \left(x + y\right)}, \frac{a \cdot \left(t + y\right)}{t + \left(x + y\right)}\right) - b \cdot \frac{y}{\left(y + x\right) + t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, a \cdot \frac{t + y}{t + \left(x + y\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t + \left(x + y\right)\\ t_2 := \left(a + z\right) - b\\ \mathbf{if}\;a \leq -3 \cdot 10^{-60}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;a \leq 2 \cdot 10^{-203}:\\ \;\;\;\;z \cdot \frac{x + y}{t\_1}\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{+125}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{t + y}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ t (+ x y))) (t_2 (- (+ a z) b)))
   (if (<= a -3e-60)
     t_2
     (if (<= a 2e-203)
       (* z (/ (+ x y) t_1))
       (if (<= a 9.5e+125) t_2 (* a (/ (+ t y) t_1)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t + (x + y);
	double t_2 = (a + z) - b;
	double tmp;
	if (a <= -3e-60) {
		tmp = t_2;
	} else if (a <= 2e-203) {
		tmp = z * ((x + y) / t_1);
	} else if (a <= 9.5e+125) {
		tmp = t_2;
	} else {
		tmp = a * ((t + y) / t_1);
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = t + (x + y)
    t_2 = (a + z) - b
    if (a <= (-3d-60)) then
        tmp = t_2
    else if (a <= 2d-203) then
        tmp = z * ((x + y) / t_1)
    else if (a <= 9.5d+125) then
        tmp = t_2
    else
        tmp = a * ((t + y) / t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t + (x + y);
	double t_2 = (a + z) - b;
	double tmp;
	if (a <= -3e-60) {
		tmp = t_2;
	} else if (a <= 2e-203) {
		tmp = z * ((x + y) / t_1);
	} else if (a <= 9.5e+125) {
		tmp = t_2;
	} else {
		tmp = a * ((t + y) / t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = t + (x + y)
	t_2 = (a + z) - b
	tmp = 0
	if a <= -3e-60:
		tmp = t_2
	elif a <= 2e-203:
		tmp = z * ((x + y) / t_1)
	elif a <= 9.5e+125:
		tmp = t_2
	else:
		tmp = a * ((t + y) / t_1)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t + Float64(x + y))
	t_2 = Float64(Float64(a + z) - b)
	tmp = 0.0
	if (a <= -3e-60)
		tmp = t_2;
	elseif (a <= 2e-203)
		tmp = Float64(z * Float64(Float64(x + y) / t_1));
	elseif (a <= 9.5e+125)
		tmp = t_2;
	else
		tmp = Float64(a * Float64(Float64(t + y) / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = t + (x + y);
	t_2 = (a + z) - b;
	tmp = 0.0;
	if (a <= -3e-60)
		tmp = t_2;
	elseif (a <= 2e-203)
		tmp = z * ((x + y) / t_1);
	elseif (a <= 9.5e+125)
		tmp = t_2;
	else
		tmp = a * ((t + y) / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[a, -3e-60], t$95$2, If[LessEqual[a, 2e-203], N[(z * N[(N[(x + y), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 9.5e+125], t$95$2, N[(a * N[(N[(t + y), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.00000000000000019e-60 or 2.0000000000000001e-203 < a < 9.50000000000000041e125

   1. Initial program 66.3%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

         \[\leadsto \left(a + z\right) - \color{blue}{b} \]

      2. lower-+.f64 60.0

         \[\leadsto \left(a + z\right) - b \]

   5. Applied rewrites 60.0%

      \[\leadsto \color{blue}{\left(a + z\right) - b} \]

   if -3.00000000000000019e-60 < a < 2.0000000000000001e-203

   1. Initial program 77.0%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   4. Applied rewrites 93.4%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-+.f64 69.7

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

   7. Applied rewrites 69.7%

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

   if 9.50000000000000041e125 < a

   1. Initial program 48.3%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   4. Applied rewrites 53.1%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-+.f64 80.1

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

   7. Applied rewrites 80.1%

      \[\leadsto \color{blue}{a \cdot \frac{t + y}{t + \left(x + y\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3 \cdot 10^{-60}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;a \leq 2 \cdot 10^{-203}:\\ \;\;\;\;z \cdot \frac{x + y}{t + \left(x + y\right)}\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{+125}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{t + y}{t + \left(x + y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 69.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y + x\right) + t\\ t_2 := b \cdot \frac{y}{t\_1}\\ \mathbf{if}\;b \leq -1.75 \cdot 10^{+53}:\\ \;\;\;\;a - t\_2\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-31}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{t\_1}, z, a\right)\\ \mathbf{else}:\\ \;\;\;\;z - t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ y x) t)) (t_2 (* b (/ y t_1))))
   (if (<= b -1.75e+53)
     (- a t_2)
     (if (<= b 7e-31) (fma (/ (+ y x) t_1) z a) (- z t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y + x) + t;
	double t_2 = b * (y / t_1);
	double tmp;
	if (b <= -1.75e+53) {
		tmp = a - t_2;
	} else if (b <= 7e-31) {
		tmp = fma(((y + x) / t_1), z, a);
	} else {
		tmp = z - t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y + x) + t)
	t_2 = Float64(b * Float64(y / t_1))
	tmp = 0.0
	if (b <= -1.75e+53)
		tmp = Float64(a - t_2);
	elseif (b <= 7e-31)
		tmp = fma(Float64(Float64(y + x) / t_1), z, a);
	else
		tmp = Float64(z - t_2);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$2 = N[(b * N[(y / t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.75e+53], N[(a - t$95$2), $MachinePrecision], If[LessEqual[b, 7e-31], N[(N[(N[(y + x), $MachinePrecision] / t$95$1), $MachinePrecision] * z + a), $MachinePrecision], N[(z - t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.75000000000000009e53

   1. Initial program 65.3%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   4. Applied rewrites 75.5%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 67.3%

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

   if -1.75000000000000009e53 < b < 6.99999999999999971e-31

   1. Initial program 65.0%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   4. Applied rewrites 78.5%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 86.6%

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

   if 6.99999999999999971e-31 < b

   1. Initial program 70.3%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   4. Applied rewrites 72.7%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 69.2%

      \[\leadsto \color{blue}{z} - b \cdot \frac{y}{\left(y + x\right) + t} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.75 \cdot 10^{+53}:\\ \;\;\;\;a - b \cdot \frac{y}{\left(y + x\right) + t}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-31}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, a\right)\\ \mathbf{else}:\\ \;\;\;\;z - b \cdot \frac{y}{\left(y + x\right) + t}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 60.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \frac{y}{\left(y + x\right) + t}\\ \mathbf{if}\;b \leq -2.3 \cdot 10^{+48}:\\ \;\;\;\;a - t\_1\\ \mathbf{elif}\;b \leq 1.08 \cdot 10^{-38}:\\ \;\;\;\;a + z\\ \mathbf{else}:\\ \;\;\;\;z - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (/ y (+ (+ y x) t)))))
   (if (<= b -2.3e+48) (- a t_1) (if (<= b 1.08e-38) (+ a z) (- z t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (y / ((y + x) + t));
	double tmp;
	if (b <= -2.3e+48) {
		tmp = a - t_1;
	} else if (b <= 1.08e-38) {
		tmp = a + z;
	} else {
		tmp = z - t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = b * (y / ((y + x) + t))
    if (b <= (-2.3d+48)) then
        tmp = a - t_1
    else if (b <= 1.08d-38) then
        tmp = a + z
    else
        tmp = z - t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (y / ((y + x) + t));
	double tmp;
	if (b <= -2.3e+48) {
		tmp = a - t_1;
	} else if (b <= 1.08e-38) {
		tmp = a + z;
	} else {
		tmp = z - t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = b * (y / ((y + x) + t))
	tmp = 0
	if b <= -2.3e+48:
		tmp = a - t_1
	elif b <= 1.08e-38:
		tmp = a + z
	else:
		tmp = z - t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(y / Float64(Float64(y + x) + t)))
	tmp = 0.0
	if (b <= -2.3e+48)
		tmp = Float64(a - t_1);
	elseif (b <= 1.08e-38)
		tmp = Float64(a + z);
	else
		tmp = Float64(z - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = b * (y / ((y + x) + t));
	tmp = 0.0;
	if (b <= -2.3e+48)
		tmp = a - t_1;
	elseif (b <= 1.08e-38)
		tmp = a + z;
	else
		tmp = z - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(y / N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2.3e+48], N[(a - t$95$1), $MachinePrecision], If[LessEqual[b, 1.08e-38], N[(a + z), $MachinePrecision], N[(z - t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.3e48

   1. Initial program 66.7%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   4. Applied rewrites 76.5%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 68.6%

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

   if -2.3e48 < b < 1.08e-38

   1. Initial program 64.3%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

         \[\leadsto \left(a + z\right) - \color{blue}{b} \]

      2. lower-+.f64 69.3

         \[\leadsto \left(a + z\right) - b \]

   5. Applied rewrites 69.3%

      \[\leadsto \color{blue}{\left(a + z\right) - b} \]

   6. Taylor expanded in b around 0

      \[\leadsto a + \color{blue}{z} \]
   7. Step-by-step derivation

      1. lift-+.f64 71.1

         \[\leadsto a + z \]

   8. Applied rewrites 71.1%

      \[\leadsto a + \color{blue}{z} \]

   if 1.08e-38 < b

   1. Initial program 70.7%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   4. Applied rewrites 73.1%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 68.2%

      \[\leadsto \color{blue}{z} - b \cdot \frac{y}{\left(y + x\right) + t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 65.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \frac{x + y}{t + \left(x + y\right)}\\ \mathbf{if}\;z \leq -7 \cdot 10^{+115}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.65 \cdot 10^{+26}:\\ \;\;\;\;a - b \cdot \frac{y}{\left(y + x\right) + t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* z (/ (+ x y) (+ t (+ x y))))))
   (if (<= z -7e+115)
     t_1
     (if (<= z 2.65e+26) (- a (* b (/ y (+ (+ y x) t)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = z * ((x + y) / (t + (x + y)));
	double tmp;
	if (z <= -7e+115) {
		tmp = t_1;
	} else if (z <= 2.65e+26) {
		tmp = a - (b * (y / ((y + x) + t)));
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = z * ((x + y) / (t + (x + y)))
    if (z <= (-7d+115)) then
        tmp = t_1
    else if (z <= 2.65d+26) then
        tmp = a - (b * (y / ((y + x) + t)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = z * ((x + y) / (t + (x + y)));
	double tmp;
	if (z <= -7e+115) {
		tmp = t_1;
	} else if (z <= 2.65e+26) {
		tmp = a - (b * (y / ((y + x) + t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = z * ((x + y) / (t + (x + y)))
	tmp = 0
	if z <= -7e+115:
		tmp = t_1
	elif z <= 2.65e+26:
		tmp = a - (b * (y / ((y + x) + t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(z * Float64(Float64(x + y) / Float64(t + Float64(x + y))))
	tmp = 0.0
	if (z <= -7e+115)
		tmp = t_1;
	elseif (z <= 2.65e+26)
		tmp = Float64(a - Float64(b * Float64(y / Float64(Float64(y + x) + t))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = z * ((x + y) / (t + (x + y)));
	tmp = 0.0;
	if (z <= -7e+115)
		tmp = t_1;
	elseif (z <= 2.65e+26)
		tmp = a - (b * (y / ((y + x) + t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(N[(x + y), $MachinePrecision] / N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7e+115], t$95$1, If[LessEqual[z, 2.65e+26], N[(a - N[(b * N[(y / N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.00000000000000011e115 or 2.64999999999999984e26 < z

   1. Initial program 55.5%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   4. Applied rewrites 83.7%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-+.f64 70.5

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

   7. Applied rewrites 70.5%

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

   if -7.00000000000000011e115 < z < 2.64999999999999984e26

   1. Initial program 72.8%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   4. Applied rewrites 76.3%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 66.1%

      \[\leadsto \color{blue}{a} - b \cdot \frac{y}{\left(y + x\right) + t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+115}:\\ \;\;\;\;z \cdot \frac{x + y}{t + \left(x + y\right)}\\ \mathbf{elif}\;z \leq 2.65 \cdot 10^{+26}:\\ \;\;\;\;a - b \cdot \frac{y}{\left(y + x\right) + t}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{x + y}{t + \left(x + y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 58.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{+260}:\\ \;\;\;\;a \cdot \frac{t}{t + x}\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{+108}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{t + y}{t + \left(x + y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -1.8e+260)
   (* a (/ t (+ t x)))
   (if (<= t 1.02e+108) (- (+ a z) b) (* a (/ (+ t y) (+ t (+ x y)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -1.8e+260) {
		tmp = a * (t / (t + x));
	} else if (t <= 1.02e+108) {
		tmp = (a + z) - b;
	} else {
		tmp = a * ((t + y) / (t + (x + y)));
	}
	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 (t <= (-1.8d+260)) then
        tmp = a * (t / (t + x))
    else if (t <= 1.02d+108) then
        tmp = (a + z) - b
    else
        tmp = a * ((t + y) / (t + (x + y)))
    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 (t <= -1.8e+260) {
		tmp = a * (t / (t + x));
	} else if (t <= 1.02e+108) {
		tmp = (a + z) - b;
	} else {
		tmp = a * ((t + y) / (t + (x + y)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -1.8e+260:
		tmp = a * (t / (t + x))
	elif t <= 1.02e+108:
		tmp = (a + z) - b
	else:
		tmp = a * ((t + y) / (t + (x + y)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -1.8e+260)
		tmp = Float64(a * Float64(t / Float64(t + x)));
	elseif (t <= 1.02e+108)
		tmp = Float64(Float64(a + z) - b);
	else
		tmp = Float64(a * Float64(Float64(t + y) / Float64(t + Float64(x + y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= -1.8e+260)
		tmp = a * (t / (t + x));
	elseif (t <= 1.02e+108)
		tmp = (a + z) - b;
	else
		tmp = a * ((t + y) / (t + (x + y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -1.8e+260], N[(a * N[(t / N[(t + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.02e+108], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision], N[(a * N[(N[(t + y), $MachinePrecision] / N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.7999999999999999e260

   1. Initial program 65.3%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   4. Applied rewrites 65.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-+.f64 73.8

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

   7. Applied rewrites 73.8%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

         \[\leadsto a \cdot \frac{t}{t + \color{blue}{x}} \]

      2. lower-+.f64 73.8

         \[\leadsto a \cdot \frac{t}{t + x} \]

   10. Applied rewrites 73.8%

       \[\leadsto a \cdot \frac{t}{\color{blue}{t + x}} \]

   if -1.7999999999999999e260 < t < 1.02e108

   1. Initial program 69.1%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

         \[\leadsto \left(a + z\right) - \color{blue}{b} \]

      2. lower-+.f64 62.7

         \[\leadsto \left(a + z\right) - b \]

   5. Applied rewrites 62.7%

      \[\leadsto \color{blue}{\left(a + z\right) - b} \]

   if 1.02e108 < t

   1. Initial program 57.3%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   4. Applied rewrites 61.0%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-+.f64 55.7

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

   7. Applied rewrites 55.7%

      \[\leadsto \color{blue}{a \cdot \frac{t + y}{t + \left(x + y\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{+260}:\\ \;\;\;\;a \cdot \frac{t}{t + x}\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{+108}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{t + y}{t + \left(x + y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 58.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \frac{t}{t + x}\\ \mathbf{if}\;t \leq -1.8 \cdot 10^{+260}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+108}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* a (/ t (+ t x)))))
   (if (<= t -1.8e+260) t_1 (if (<= t 2.8e+108) (- (+ a z) b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a * (t / (t + x));
	double tmp;
	if (t <= -1.8e+260) {
		tmp = t_1;
	} else if (t <= 2.8e+108) {
		tmp = (a + z) - b;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = a * (t / (t + x))
    if (t <= (-1.8d+260)) then
        tmp = t_1
    else if (t <= 2.8d+108) then
        tmp = (a + z) - b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a * (t / (t + x));
	double tmp;
	if (t <= -1.8e+260) {
		tmp = t_1;
	} else if (t <= 2.8e+108) {
		tmp = (a + z) - b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a * (t / (t + x))
	tmp = 0
	if t <= -1.8e+260:
		tmp = t_1
	elif t <= 2.8e+108:
		tmp = (a + z) - b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a * Float64(t / Float64(t + x)))
	tmp = 0.0
	if (t <= -1.8e+260)
		tmp = t_1;
	elseif (t <= 2.8e+108)
		tmp = Float64(Float64(a + z) - b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a * (t / (t + x));
	tmp = 0.0;
	if (t <= -1.8e+260)
		tmp = t_1;
	elseif (t <= 2.8e+108)
		tmp = (a + z) - b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a * N[(t / N[(t + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.8e+260], t$95$1, If[LessEqual[t, 2.8e+108], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.7999999999999999e260 or 2.7999999999999998e108 < t

   1. Initial program 58.7%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   4. Applied rewrites 61.7%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-+.f64 58.8

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

   7. Applied rewrites 58.8%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

         \[\leadsto a \cdot \frac{t}{t + \color{blue}{x}} \]

      2. lower-+.f64 56.1

         \[\leadsto a \cdot \frac{t}{t + x} \]

   10. Applied rewrites 56.1%

       \[\leadsto a \cdot \frac{t}{\color{blue}{t + x}} \]

   if -1.7999999999999999e260 < t < 2.7999999999999998e108

   1. Initial program 69.1%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

         \[\leadsto \left(a + z\right) - \color{blue}{b} \]

      2. lower-+.f64 62.7

         \[\leadsto \left(a + z\right) - b \]

   5. Applied rewrites 62.7%

      \[\leadsto \color{blue}{\left(a + z\right) - b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{+260}:\\ \;\;\;\;a \cdot \frac{t}{t + x}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+108}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{t}{t + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 57.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+162}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-38}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;a + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x -1.75e+162) z (if (<= x 9.2e-38) (- (+ a z) b) (+ a z))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -1.75e+162) {
		tmp = z;
	} else if (x <= 9.2e-38) {
		tmp = (a + z) - b;
	} else {
		tmp = a + 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 (x <= (-1.75d+162)) then
        tmp = z
    else if (x <= 9.2d-38) then
        tmp = (a + z) - b
    else
        tmp = a + 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 (x <= -1.75e+162) {
		tmp = z;
	} else if (x <= 9.2e-38) {
		tmp = (a + z) - b;
	} else {
		tmp = a + z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= -1.75e+162:
		tmp = z
	elif x <= 9.2e-38:
		tmp = (a + z) - b
	else:
		tmp = a + z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= -1.75e+162)
		tmp = z;
	elseif (x <= 9.2e-38)
		tmp = Float64(Float64(a + z) - b);
	else
		tmp = Float64(a + z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= -1.75e+162)
		tmp = z;
	elseif (x <= 9.2e-38)
		tmp = (a + z) - b;
	else
		tmp = a + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, -1.75e+162], z, If[LessEqual[x, 9.2e-38], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision], N[(a + z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.75000000000000009e162

   1. Initial program 40.0%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 66.5%

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

   if -1.75000000000000009e162 < x < 9.20000000000000007e-38

   1. Initial program 72.9%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

         \[\leadsto \left(a + z\right) - \color{blue}{b} \]

      2. lower-+.f64 60.2

         \[\leadsto \left(a + z\right) - b \]

   5. Applied rewrites 60.2%

      \[\leadsto \color{blue}{\left(a + z\right) - b} \]

   if 9.20000000000000007e-38 < x

   1. Initial program 63.9%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

         \[\leadsto \left(a + z\right) - \color{blue}{b} \]

      2. lower-+.f64 52.1

         \[\leadsto \left(a + z\right) - b \]

   5. Applied rewrites 52.1%

      \[\leadsto \color{blue}{\left(a + z\right) - b} \]

   6. Taylor expanded in b around 0

      \[\leadsto a + \color{blue}{z} \]
   7. Step-by-step derivation

      1. lift-+.f64 59.1

         \[\leadsto a + z \]

   8. Applied rewrites 59.1%

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

Alternative 19: 44.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+20}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+81}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x -8e+20) z (if (<= x 1.75e+81) a z)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -8e+20) {
		tmp = z;
	} else if (x <= 1.75e+81) {
		tmp = a;
	} else {
		tmp = 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 (x <= (-8d+20)) then
        tmp = z
    else if (x <= 1.75d+81) then
        tmp = a
    else
        tmp = 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 (x <= -8e+20) {
		tmp = z;
	} else if (x <= 1.75e+81) {
		tmp = a;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= -8e+20:
		tmp = z
	elif x <= 1.75e+81:
		tmp = a
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= -8e+20)
		tmp = z;
	elseif (x <= 1.75e+81)
		tmp = a;
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= -8e+20)
		tmp = z;
	elseif (x <= 1.75e+81)
		tmp = a;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, -8e+20], z, If[LessEqual[x, 1.75e+81], a, z]]

Derivation

1. Split input into 2 regimes

2. if x < -8e20 or 1.75e81 < x

   1. Initial program 56.8%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 53.1%

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

   if -8e20 < x < 1.75e81

   1. Initial program 72.6%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 45.8%

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

Alternative 20: 50.4% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{+48}:\\ \;\;\;\;a - b\\ \mathbf{else}:\\ \;\;\;\;a + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (if (<= b -2.5e+48) (- a b) (+ a z)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -2.5e+48) {
		tmp = a - b;
	} else {
		tmp = a + 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 (b <= (-2.5d+48)) then
        tmp = a - b
    else
        tmp = a + 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 (b <= -2.5e+48) {
		tmp = a - b;
	} else {
		tmp = a + z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -2.5e+48:
		tmp = a - b
	else:
		tmp = a + z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -2.5e+48)
		tmp = Float64(a - b);
	else
		tmp = Float64(a + z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -2.5e+48)
		tmp = a - b;
	else
		tmp = a + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -2.5e+48], N[(a - b), $MachinePrecision], N[(a + z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.49999999999999987e48

   1. Initial program 66.7%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

         \[\leadsto \left(a + z\right) - \color{blue}{b} \]

      2. lower-+.f64 49.1

         \[\leadsto \left(a + z\right) - b \]

   5. Applied rewrites 49.1%

      \[\leadsto \color{blue}{\left(a + z\right) - b} \]

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 45.9%

      \[\leadsto a - b \]

   if -2.49999999999999987e48 < b

   1. Initial program 66.4%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

         \[\leadsto \left(a + z\right) - \color{blue}{b} \]

      2. lower-+.f64 58.0

         \[\leadsto \left(a + z\right) - b \]

   5. Applied rewrites 58.0%

      \[\leadsto \color{blue}{\left(a + z\right) - b} \]

   6. Taylor expanded in b around 0

      \[\leadsto a + \color{blue}{z} \]
   7. Step-by-step derivation

      1. lift-+.f64 57.3

         \[\leadsto a + z \]

   8. Applied rewrites 57.3%

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

Alternative 21: 52.2% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{+162}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;a + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (if (<= x -2.15e+162) z (+ a z)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -2.15e+162) {
		tmp = z;
	} else {
		tmp = a + 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 (x <= (-2.15d+162)) then
        tmp = z
    else
        tmp = a + 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 (x <= -2.15e+162) {
		tmp = z;
	} else {
		tmp = a + z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= -2.15e+162:
		tmp = z
	else:
		tmp = a + z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= -2.15e+162)
		tmp = z;
	else
		tmp = Float64(a + z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= -2.15e+162)
		tmp = z;
	else
		tmp = a + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, -2.15e+162], z, N[(a + z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1500000000000001e162

   1. Initial program 40.0%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 66.5%

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

   if -2.1500000000000001e162 < x

   1. Initial program 70.2%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

         \[\leadsto \left(a + z\right) - \color{blue}{b} \]

      2. lower-+.f64 57.8

         \[\leadsto \left(a + z\right) - b \]

   5. Applied rewrites 57.8%

      \[\leadsto \color{blue}{\left(a + z\right) - b} \]

   6. Taylor expanded in b around 0

      \[\leadsto a + \color{blue}{z} \]
   7. Step-by-step derivation

      1. lift-+.f64 51.9

         \[\leadsto a + z \]

   8. Applied rewrites 51.9%

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

Alternative 22: 32.8% accurate, 45.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := a

Derivation

1. Initial program 66.5%

   \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
2. Add Preprocessing

3. Taylor expanded in t around inf

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

5. Applied rewrites 33.7%

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

Developer Target 1: 81.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b\\ t_3 := \frac{t\_2}{t\_1}\\ t_4 := \left(z + a\right) - b\\ \mathbf{if}\;t\_3 < -3.5813117084150564 \cdot 10^{+153}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 < 1.2285964308315609 \cdot 10^{+82}:\\ \;\;\;\;\frac{1}{\frac{t\_1}{t\_2}}\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ x t) y))
        (t_2 (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)))
        (t_3 (/ t_2 t_1))
        (t_4 (- (+ z a) b)))
   (if (< t_3 -3.5813117084150564e+153)
     t_4
     (if (< t_3 1.2285964308315609e+82) (/ 1.0 (/ t_1 t_2)) t_4))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + t) + y;
	double t_2 = (((x + y) * z) + ((t + y) * a)) - (y * b);
	double t_3 = t_2 / t_1;
	double t_4 = (z + a) - b;
	double tmp;
	if (t_3 < -3.5813117084150564e+153) {
		tmp = t_4;
	} else if (t_3 < 1.2285964308315609e+82) {
		tmp = 1.0 / (t_1 / t_2);
	} else {
		tmp = t_4;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = (x + t) + y
    t_2 = (((x + y) * z) + ((t + y) * a)) - (y * b)
    t_3 = t_2 / t_1
    t_4 = (z + a) - b
    if (t_3 < (-3.5813117084150564d+153)) then
        tmp = t_4
    else if (t_3 < 1.2285964308315609d+82) then
        tmp = 1.0d0 / (t_1 / t_2)
    else
        tmp = t_4
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + t) + y;
	double t_2 = (((x + y) * z) + ((t + y) * a)) - (y * b);
	double t_3 = t_2 / t_1;
	double t_4 = (z + a) - b;
	double tmp;
	if (t_3 < -3.5813117084150564e+153) {
		tmp = t_4;
	} else if (t_3 < 1.2285964308315609e+82) {
		tmp = 1.0 / (t_1 / t_2);
	} else {
		tmp = t_4;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x + t) + y
	t_2 = (((x + y) * z) + ((t + y) * a)) - (y * b)
	t_3 = t_2 / t_1
	t_4 = (z + a) - b
	tmp = 0
	if t_3 < -3.5813117084150564e+153:
		tmp = t_4
	elif t_3 < 1.2285964308315609e+82:
		tmp = 1.0 / (t_1 / t_2)
	else:
		tmp = t_4
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + t) + y)
	t_2 = Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b))
	t_3 = Float64(t_2 / t_1)
	t_4 = Float64(Float64(z + a) - b)
	tmp = 0.0
	if (t_3 < -3.5813117084150564e+153)
		tmp = t_4;
	elseif (t_3 < 1.2285964308315609e+82)
		tmp = Float64(1.0 / Float64(t_1 / t_2));
	else
		tmp = t_4;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (x + t) + y;
	t_2 = (((x + y) * z) + ((t + y) * a)) - (y * b);
	t_3 = t_2 / t_1;
	t_4 = (z + a) - b;
	tmp = 0.0;
	if (t_3 < -3.5813117084150564e+153)
		tmp = t_4;
	elseif (t_3 < 1.2285964308315609e+82)
		tmp = 1.0 / (t_1 / t_2);
	else
		tmp = t_4;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision]}, If[Less[t$95$3, -3.5813117084150564e+153], t$95$4, If[Less[t$95$3, 1.2285964308315609e+82], N[(1.0 / N[(t$95$1 / t$95$2), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]

Reproduce

herbie shell --seed 2025050 
(FPCore (x y z t a b)
  :name "AI.Clustering.Hierarchical.Internal:ward from clustering-0.2.1"
  :precision binary64
  :pre (TRUE)

  :alt
  (! :herbie-platform default (if (< (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) -3581311708415056400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (+ z a) b) (if (< (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) 12285964308315609000000000000000000000000000000000000000000000000000000000000000000) (/ 1 (/ (+ (+ x t) y) (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)))) (- (+ z a) b))))

  (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))