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

Percentage Accurate: 59.9% → 93.6%

Time: 5.9s

Alternatives: 18

Speedup: 2.1×

Specification

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: 59.9% 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: 93.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t + \left(x + y\right)\\ t_2 := \frac{y + x}{\left(y + x\right) + t}\\ t_3 := \frac{t + y}{t\_1}\\ t_4 := \mathsf{fma}\left(t\_2, z, a \cdot \mathsf{fma}\left(-1, \frac{b}{a} \cdot \frac{y}{t\_1}, t\_3\right)\right)\\ \mathbf{if}\;a \leq -2.3 \cdot 10^{-32}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, z, \mathsf{fma}\left(-1, \frac{b \cdot y}{t\_1}, a \cdot t\_3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ t (+ x y)))
        (t_2 (/ (+ y x) (+ (+ y x) t)))
        (t_3 (/ (+ t y) t_1))
        (t_4 (fma t_2 z (* a (fma -1.0 (* (/ b a) (/ y t_1)) t_3)))))
   (if (<= a -2.3e-32)
     t_4
     (if (<= a 2.2e-30)
       (fma t_2 z (fma -1.0 (/ (* b y) t_1) (* a t_3)))
       t_4))))

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 t_3 = (t + y) / t_1;
	double t_4 = fma(t_2, z, (a * fma(-1.0, ((b / a) * (y / t_1)), t_3)));
	double tmp;
	if (a <= -2.3e-32) {
		tmp = t_4;
	} else if (a <= 2.2e-30) {
		tmp = fma(t_2, z, fma(-1.0, ((b * y) / t_1), (a * t_3)));
	} else {
		tmp = t_4;
	}
	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))
	t_3 = Float64(Float64(t + y) / t_1)
	t_4 = fma(t_2, z, Float64(a * fma(-1.0, Float64(Float64(b / a) * Float64(y / t_1)), t_3)))
	tmp = 0.0
	if (a <= -2.3e-32)
		tmp = t_4;
	elseif (a <= 2.2e-30)
		tmp = fma(t_2, z, fma(-1.0, Float64(Float64(b * y) / t_1), Float64(a * t_3)));
	else
		tmp = t_4;
	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]}, Block[{t$95$3 = N[(N[(t + y), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * z + N[(a * N[(-1.0 * N[(N[(b / a), $MachinePrecision] * N[(y / t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2.3e-32], t$95$4, If[LessEqual[a, 2.2e-30], N[(t$95$2 * z + N[(-1.0 * N[(N[(b * y), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(a * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.3000000000000001e-32 or 2.19999999999999983e-30 < a

   1. Initial program 50.7%

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

   3. Applied rewrites 59.4%

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

   4. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, \color{blue}{a \cdot \left(-1 \cdot \frac{b \cdot y}{a \cdot \left(t + \left(x + y\right)\right)} + \left(\frac{t}{t + \left(x + y\right)} + \frac{y}{t + \left(x + y\right)}\right)\right)}\right) \]
   5. 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. lower-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. div-add-rev N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 86.7

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

   6. Applied rewrites 86.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 99.2

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

   8. Applied rewrites 99.2%

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

   if -2.3000000000000001e-32 < a < 2.19999999999999983e-30

   1. Initial program 70.9%

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

   3. Applied rewrites 86.7%

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

   4. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, \color{blue}{-1 \cdot \frac{b \cdot y}{t + \left(x + y\right)} + a \cdot \left(\frac{t}{t + \left(x + y\right)} + \frac{y}{t + \left(x + y\right)}\right)}\right) \]
   5. 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. lift-*.f64 N/A

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

      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. lower-+.f64 N/A

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

      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. lower-+.f64 86.9

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

   6. Applied rewrites 86.9%

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

Alternative 2: 92.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + z\right) - b\\ t_2 := t + \left(x + y\right)\\ \mathbf{if}\;y \leq -6.8 \cdot 10^{+127}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+179}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, \mathsf{fma}\left(-1, \frac{b \cdot y}{t\_2}, a \cdot \frac{t + y}{t\_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (+ a z) b)) (t_2 (+ t (+ x y))))
   (if (<= y -6.8e+127)
     t_1
     (if (<= y 2.05e+179)
       (fma
        (/ (+ y x) (+ (+ y x) t))
        z
        (fma -1.0 (/ (* b y) t_2) (* a (/ (+ t y) t_2))))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a + z) - b;
	double t_2 = t + (x + y);
	double tmp;
	if (y <= -6.8e+127) {
		tmp = t_1;
	} else if (y <= 2.05e+179) {
		tmp = fma(((y + x) / ((y + x) + t)), z, fma(-1.0, ((b * y) / t_2), (a * ((t + y) / t_2))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a + z) - b)
	t_2 = Float64(t + Float64(x + y))
	tmp = 0.0
	if (y <= -6.8e+127)
		tmp = t_1;
	elseif (y <= 2.05e+179)
		tmp = fma(Float64(Float64(y + x) / Float64(Float64(y + x) + t)), z, fma(-1.0, Float64(Float64(b * y) / t_2), Float64(a * Float64(Float64(t + y) / t_2))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, Block[{t$95$2 = N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.8e+127], t$95$1, If[LessEqual[y, 2.05e+179], N[(N[(N[(y + x), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] * z + N[(-1.0 * N[(N[(b * y), $MachinePrecision] / t$95$2), $MachinePrecision] + N[(a * N[(N[(t + y), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.79999999999999955e127 or 2.05000000000000005e179 < y

   1. Initial program 28.6%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 83.8

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

   4. Applied rewrites 83.8%

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

   if -6.79999999999999955e127 < y < 2.05000000000000005e179

   1. Initial program 70.7%

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

   3. Applied rewrites 82.0%

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

   4. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, \color{blue}{-1 \cdot \frac{b \cdot y}{t + \left(x + y\right)} + a \cdot \left(\frac{t}{t + \left(x + y\right)} + \frac{y}{t + \left(x + y\right)}\right)}\right) \]
   5. 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. lift-*.f64 N/A

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

      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. lower-+.f64 N/A

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

      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. lower-+.f64 95.1

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

   6. Applied rewrites 95.1%

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

Alternative 3: 90.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ t_2 := \left(y + x\right) + t\\ t_3 := \mathsf{fma}\left(\frac{y + x}{t\_2}, z, a \cdot \frac{t + y}{t + \left(x + y\right)}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+276}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t + y}{t\_2}, a, \frac{\left(y + x\right) \cdot z - b \cdot y}{t\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

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 (fma (/ (+ y x) t_2) z (* a (/ (+ t y) (+ t (+ x y)))))))
   (if (<= t_1 (- INFINITY))
     t_3
     (if (<= t_1 2e+276)
       (fma (/ (+ t y) t_2) a (/ (- (* (+ y x) z) (* b y)) t_2))
       t_3))))

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 = fma(((y + x) / t_2), z, (a * ((t + y) / (t + (x + y)))));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_1 <= 2e+276) {
		tmp = fma(((t + y) / t_2), a, ((((y + x) * z) - (b * y)) / t_2));
	} else {
		tmp = t_3;
	}
	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 = fma(Float64(Float64(y + x) / t_2), z, Float64(a * Float64(Float64(t + y) / Float64(t + Float64(x + y)))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_1 <= 2e+276)
		tmp = fma(Float64(Float64(t + y) / t_2), a, Float64(Float64(Float64(Float64(y + x) * z) - Float64(b * y)) / t_2));
	else
		tmp = t_3;
	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[(N[(y + x), $MachinePrecision] / t$95$2), $MachinePrecision] * 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$3, If[LessEqual[t$95$1, 2e+276], N[(N[(N[(t + y), $MachinePrecision] / t$95$2), $MachinePrecision] * a + N[(N[(N[(N[(y + x), $MachinePrecision] * z), $MachinePrecision] - N[(b * y), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $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 2.0000000000000001e276 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 6.5%

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

   3. Applied rewrites 34.7%

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

   4. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, \color{blue}{-1 \cdot \frac{b \cdot y}{t + \left(x + y\right)} + a \cdot \left(\frac{t}{t + \left(x + y\right)} + \frac{y}{t + \left(x + y\right)}\right)}\right) \]
   5. 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. lift-*.f64 N/A

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

      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. lower-+.f64 N/A

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

      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. lower-+.f64 71.1

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

   6. Applied rewrites 71.1%

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

   7. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, a \cdot \color{blue}{\left(\frac{t}{t + \left(x + y\right)} + \frac{y}{t + \left(x + y\right)}\right)}\right) \]
   8. 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

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

      3. lift-+.f64 N/A

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

      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 79.2

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

   9. Applied rewrites 79.2%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 2.0000000000000001e276

   1. Initial program 99.5%

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

   3. Applied rewrites 99.4%

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

Alternative 4: 90.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ t_3 := \mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, a \cdot \frac{t + y}{t + \left(x + y\right)}\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+276}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, \mathsf{fma}\left(a + \left(z + -1 \cdot b\right), y, z \cdot x\right)\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

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 2e+276)
       (/ (fma a t (fma (+ a (+ z (* -1.0 b))) y (* z x))) 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 <= 2e+276) {
		tmp = fma(a, t, fma((a + (z + (-1.0 * b))), y, (z * x))) / 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 <= 2e+276)
		tmp = Float64(fma(a, t, fma(Float64(a + Float64(z + Float64(-1.0 * b))), y, Float64(z * x))) / 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, 2e+276], N[(N[(a * t + N[(N[(a + N[(z + N[(-1.0 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + N[(z * x), $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 2.0000000000000001e276 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 6.5%

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

   3. Applied rewrites 34.7%

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

   4. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, \color{blue}{-1 \cdot \frac{b \cdot y}{t + \left(x + y\right)} + a \cdot \left(\frac{t}{t + \left(x + y\right)} + \frac{y}{t + \left(x + y\right)}\right)}\right) \]
   5. 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. lift-*.f64 N/A

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

      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. lower-+.f64 N/A

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

      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. lower-+.f64 71.1

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

   6. Applied rewrites 71.1%

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

   7. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, a \cdot \color{blue}{\left(\frac{t}{t + \left(x + y\right)} + \frac{y}{t + \left(x + y\right)}\right)}\right) \]
   8. 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

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

      3. lift-+.f64 N/A

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

      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 79.2

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

   9. Applied rewrites 79.2%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 2.0000000000000001e276

   1. Initial program 99.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in b around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 99.5

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

   7. Applied rewrites 99.5%

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

Alternative 5: 88.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, a\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+276}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, \mathsf{fma}\left(a + \left(z + -1 \cdot b\right), y, z \cdot x\right)\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \end{array} \]

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 2e+276)
       (/ (fma a t (fma (+ a (+ z (* -1.0 b))) y (* z x))) 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 <= 2e+276) {
		tmp = fma(a, t, fma((a + (z + (-1.0 * b))), y, (z * x))) / 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 <= 2e+276)
		tmp = Float64(fma(a, t, fma(Float64(a + Float64(z + Float64(-1.0 * b))), y, Float64(z * x))) / 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, 2e+276], N[(N[(a * t + N[(N[(a + N[(z + N[(-1.0 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + N[(z * x), $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%

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

   3. Applied rewrites 38.8%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 72.4%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 2.0000000000000001e276

   1. Initial program 99.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in b around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 99.5

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

   7. Applied rewrites 99.5%

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

   if 2.0000000000000001e276 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 6.8%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 72.6

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

   4. Applied rewrites 72.6%

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

Alternative 6: 88.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + z\right) - b\\ t_2 := \left(x + t\right) + y\\ t_3 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_2}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, a\right)\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+276}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, \mathsf{fma}\left(t\_1, y, z \cdot x\right)\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (+ a z) b))
        (t_2 (+ (+ x t) y))
        (t_3 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) t_2)))
   (if (<= t_3 (- INFINITY))
     (fma (/ (+ y x) (+ (+ y x) t)) z a)
     (if (<= t_3 2e+276) (/ (fma a t (fma t_1 y (* z x))) t_2) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a + z) - b;
	double t_2 = (x + t) + y;
	double t_3 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / t_2;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = fma(((y + x) / ((y + x) + t)), z, a);
	} else if (t_3 <= 2e+276) {
		tmp = fma(a, t, fma(t_1, y, (z * x))) / t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a + z) - b)
	t_2 = Float64(Float64(x + t) + y)
	t_3 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / t_2)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = fma(Float64(Float64(y + x) / Float64(Float64(y + x) + t)), z, a);
	elseif (t_3 <= 2e+276)
		tmp = Float64(fma(a, t, fma(t_1, y, Float64(z * x))) / t_2);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$3 = 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$2), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(N[(y + x), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] * z + a), $MachinePrecision], If[LessEqual[t$95$3, 2e+276], N[(N[(a * t + N[(t$95$1 * y + N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], t$95$1]]]]]

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%

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

   3. Applied rewrites 38.8%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 72.4%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 2.0000000000000001e276

   1. Initial program 99.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

   if 2.0000000000000001e276 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 6.8%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 72.6

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

   4. Applied rewrites 72.6%

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

Alternative 7: 77.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y + x\right) + t\\ t_2 := \left(a + z\right) - b\\ \mathbf{if}\;y \leq -1.35 \cdot 10^{+85}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-113}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{t\_1}, z, a \cdot \frac{t}{t + x}\right)\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \frac{\left(t + y\right) \cdot a - b \cdot y}{t\_1}\right)\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{+175}:\\ \;\;\;\;a - b \cdot \frac{y}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ y x) t)) (t_2 (- (+ a z) b)))
   (if (<= y -1.35e+85)
     t_2
     (if (<= y 2e-113)
       (fma (/ (+ y x) t_1) z (* a (/ t (+ t x))))
       (if (<= y 8.6e+31)
         (fma 1.0 z (/ (- (* (+ t y) a) (* b y)) t_1))
         (if (<= y 6.6e+175) (- a (* b (/ y t_1))) 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 = (a + z) - b;
	double tmp;
	if (y <= -1.35e+85) {
		tmp = t_2;
	} else if (y <= 2e-113) {
		tmp = fma(((y + x) / t_1), z, (a * (t / (t + x))));
	} else if (y <= 8.6e+31) {
		tmp = fma(1.0, z, ((((t + y) * a) - (b * y)) / t_1));
	} else if (y <= 6.6e+175) {
		tmp = a - (b * (y / t_1));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y + x) + t)
	t_2 = Float64(Float64(a + z) - b)
	tmp = 0.0
	if (y <= -1.35e+85)
		tmp = t_2;
	elseif (y <= 2e-113)
		tmp = fma(Float64(Float64(y + x) / t_1), z, Float64(a * Float64(t / Float64(t + x))));
	elseif (y <= 8.6e+31)
		tmp = fma(1.0, z, Float64(Float64(Float64(Float64(t + y) * a) - Float64(b * y)) / t_1));
	elseif (y <= 6.6e+175)
		tmp = Float64(a - Float64(b * Float64(y / t_1)));
	else
		tmp = 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[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[y, -1.35e+85], t$95$2, If[LessEqual[y, 2e-113], N[(N[(N[(y + x), $MachinePrecision] / t$95$1), $MachinePrecision] * z + N[(a * N[(t / N[(t + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.6e+31], N[(1.0 * z + N[(N[(N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision] - N[(b * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.6e+175], N[(a - N[(b * N[(y / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.34999999999999992e85 or 6.6000000000000005e175 < y

   1. Initial program 31.3%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 81.4

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

   4. Applied rewrites 81.4%

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

   if -1.34999999999999992e85 < y < 1.99999999999999996e-113

   1. Initial program 75.7%

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

   3. Applied rewrites 85.9%

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

   4. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, \color{blue}{a \cdot \left(-1 \cdot \frac{b \cdot y}{a \cdot \left(t + \left(x + y\right)\right)} + \left(\frac{t}{t + \left(x + y\right)} + \frac{y}{t + \left(x + y\right)}\right)\right)}\right) \]
   5. 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. lower-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. div-add-rev N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 93.9

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

   6. Applied rewrites 93.9%

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

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, a \cdot \frac{t}{\color{blue}{t + x}}\right) \]
   8. 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. lower-+.f64 81.5

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

   9. Applied rewrites 81.5%

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

   if 1.99999999999999996e-113 < y < 8.59999999999999978e31

   1. Initial program 76.5%

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

   3. Applied rewrites 86.8%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 70.8%

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

   if 8.59999999999999978e31 < y < 6.6000000000000005e175

   1. Initial program 50.4%

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

   3. Applied rewrites 61.0%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 55.1%

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

Alternative 8: 73.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y + x\right) + t\\ t_2 := \left(a + z\right) - b\\ \mathbf{if}\;y \leq -5.5 \cdot 10^{+90}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 6.1 \cdot 10^{-114}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{t\_1}, z, a\right)\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \frac{\left(t + y\right) \cdot a - b \cdot y}{t\_1}\right)\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{+175}:\\ \;\;\;\;a - b \cdot \frac{y}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ y x) t)) (t_2 (- (+ a z) b)))
   (if (<= y -5.5e+90)
     t_2
     (if (<= y 6.1e-114)
       (fma (/ (+ y x) t_1) z a)
       (if (<= y 8.6e+31)
         (fma 1.0 z (/ (- (* (+ t y) a) (* b y)) t_1))
         (if (<= y 6.6e+175) (- a (* b (/ y t_1))) 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 = (a + z) - b;
	double tmp;
	if (y <= -5.5e+90) {
		tmp = t_2;
	} else if (y <= 6.1e-114) {
		tmp = fma(((y + x) / t_1), z, a);
	} else if (y <= 8.6e+31) {
		tmp = fma(1.0, z, ((((t + y) * a) - (b * y)) / t_1));
	} else if (y <= 6.6e+175) {
		tmp = a - (b * (y / t_1));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y + x) + t)
	t_2 = Float64(Float64(a + z) - b)
	tmp = 0.0
	if (y <= -5.5e+90)
		tmp = t_2;
	elseif (y <= 6.1e-114)
		tmp = fma(Float64(Float64(y + x) / t_1), z, a);
	elseif (y <= 8.6e+31)
		tmp = fma(1.0, z, Float64(Float64(Float64(Float64(t + y) * a) - Float64(b * y)) / t_1));
	elseif (y <= 6.6e+175)
		tmp = Float64(a - Float64(b * Float64(y / t_1)));
	else
		tmp = 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[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[y, -5.5e+90], t$95$2, If[LessEqual[y, 6.1e-114], N[(N[(N[(y + x), $MachinePrecision] / t$95$1), $MachinePrecision] * z + a), $MachinePrecision], If[LessEqual[y, 8.6e+31], N[(1.0 * z + N[(N[(N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision] - N[(b * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.6e+175], N[(a - N[(b * N[(y / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -5.49999999999999999e90 or 6.6000000000000005e175 < y

   1. Initial program 30.8%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 81.7

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

   4. Applied rewrites 81.7%

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

   if -5.49999999999999999e90 < y < 6.09999999999999977e-114

   1. Initial program 75.6%

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

   3. Applied rewrites 85.8%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 66.9%

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

   if 6.09999999999999977e-114 < y < 8.59999999999999978e31

   1. Initial program 76.5%

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

   3. Applied rewrites 86.8%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 70.8%

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

   if 8.59999999999999978e31 < y < 6.6000000000000005e175

   1. Initial program 50.4%

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

   3. Applied rewrites 61.0%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 55.1%

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

Alternative 9: 70.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y + x\right) + t\\ t_2 := \mathsf{fma}\left(\frac{y + x}{t\_1}, z, a\right)\\ \mathbf{if}\;z \leq -1.1 \cdot 10^{+55}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 4400:\\ \;\;\;\;a - b \cdot \frac{y}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ y x) t)) (t_2 (fma (/ (+ y x) t_1) z a)))
   (if (<= z -1.1e+55) t_2 (if (<= z 4400.0) (- a (* b (/ y t_1))) 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 = fma(((y + x) / t_1), z, a);
	double tmp;
	if (z <= -1.1e+55) {
		tmp = t_2;
	} else if (z <= 4400.0) {
		tmp = a - (b * (y / t_1));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y + x) + t)
	t_2 = fma(Float64(Float64(y + x) / t_1), z, a)
	tmp = 0.0
	if (z <= -1.1e+55)
		tmp = t_2;
	elseif (z <= 4400.0)
		tmp = Float64(a - Float64(b * Float64(y / t_1)));
	else
		tmp = 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[(N[(N[(y + x), $MachinePrecision] / t$95$1), $MachinePrecision] * z + a), $MachinePrecision]}, If[LessEqual[z, -1.1e+55], t$95$2, If[LessEqual[z, 4400.0], N[(a - N[(b * N[(y / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.10000000000000005e55 or 4400 < z

   1. Initial program 46.5%

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

   3. Applied rewrites 71.9%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 81.1%

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

   if -1.10000000000000005e55 < z < 4400

   1. Initial program 71.1%

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

   3. Applied rewrites 78.1%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 66.2%

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

Alternative 10: 65.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \frac{x + y}{t + \left(x + y\right)}\\ \mathbf{if}\;z \leq -7.8 \cdot 10^{+157}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+56}:\\ \;\;\;\;a - b \cdot \frac{y}{\left(y + x\right) + t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* z (/ (+ x y) (+ t (+ x y))))))
   (if (<= z -7.8e+157)
     t_1
     (if (<= z 6.2e+56) (- 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 <= -7.8e+157) {
		tmp = t_1;
	} else if (z <= 6.2e+56) {
		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 <= (-7.8d+157)) then
        tmp = t_1
    else if (z <= 6.2d+56) 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 <= -7.8e+157) {
		tmp = t_1;
	} else if (z <= 6.2e+56) {
		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 <= -7.8e+157:
		tmp = t_1
	elif z <= 6.2e+56:
		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 <= -7.8e+157)
		tmp = t_1;
	elseif (z <= 6.2e+56)
		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 <= -7.8e+157)
		tmp = t_1;
	elseif (z <= 6.2e+56)
		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, -7.8e+157], t$95$1, If[LessEqual[z, 6.2e+56], 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 < -7.79999999999999941e157 or 6.20000000000000009e56 < z

   1. Initial program 41.9%

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

   3. Applied rewrites 43.5%

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

   4. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(\frac{x}{t + \left(x + y\right)} + \frac{y}{t + \left(x + y\right)}\right)} \]
   5. 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. lower-+.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. lower-+.f64 71.1

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

   6. Applied rewrites 71.1%

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

   if -7.79999999999999941e157 < z < 6.20000000000000009e56

   1. Initial program 68.8%

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

   3. Applied rewrites 75.2%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 62.4%

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

Alternative 11: 64.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + z\right) - b\\ \mathbf{if}\;y \leq -1.5 \cdot 10^{-36}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{-46}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, z \cdot x\right)}{t + x}\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{+175}:\\ \;\;\;\;a - b \cdot \frac{y}{\left(y + x\right) + t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (+ a z) b)))
   (if (<= y -1.5e-36)
     t_1
     (if (<= y 2.05e-46)
       (/ (fma a t (* z x)) (+ t x))
       (if (<= y 6.6e+175) (- 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 = (a + z) - b;
	double tmp;
	if (y <= -1.5e-36) {
		tmp = t_1;
	} else if (y <= 2.05e-46) {
		tmp = fma(a, t, (z * x)) / (t + x);
	} else if (y <= 6.6e+175) {
		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(Float64(a + z) - b)
	tmp = 0.0
	if (y <= -1.5e-36)
		tmp = t_1;
	elseif (y <= 2.05e-46)
		tmp = Float64(fma(a, t, Float64(z * x)) / Float64(t + x));
	elseif (y <= 6.6e+175)
		tmp = Float64(a - Float64(b * Float64(y / Float64(Float64(y + x) + t))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[y, -1.5e-36], t$95$1, If[LessEqual[y, 2.05e-46], N[(N[(a * t + N[(z * x), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.6e+175], N[(a - N[(b * N[(y / N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.5000000000000001e-36 or 6.6000000000000005e175 < y

   1. Initial program 41.0%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 74.1

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

   4. Applied rewrites 74.1%

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

   if -1.5000000000000001e-36 < y < 2.05e-46

   1. Initial program 76.7%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 58.3

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

   4. Applied rewrites 58.3%

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

   if 2.05e-46 < y < 6.6000000000000005e175

   1. Initial program 59.9%

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

   3. Applied rewrites 67.0%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 52.5%

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

Alternative 12: 63.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + z\right) - b\\ \mathbf{if}\;y \leq -1.5 \cdot 10^{-36}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 10^{-98}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, z \cdot x\right)}{t + x}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (+ a z) b)))
   (if (<= y -1.5e-36)
     t_1
     (if (<= y 1e-98) (/ (fma a t (* z x)) (+ t x)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a + z) - b;
	double tmp;
	if (y <= -1.5e-36) {
		tmp = t_1;
	} else if (y <= 1e-98) {
		tmp = fma(a, t, (z * x)) / (t + x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a + z) - b)
	tmp = 0.0
	if (y <= -1.5e-36)
		tmp = t_1;
	elseif (y <= 1e-98)
		tmp = Float64(fma(a, t, Float64(z * x)) / Float64(t + x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[y, -1.5e-36], t$95$1, If[LessEqual[y, 1e-98], N[(N[(a * t + N[(z * x), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.5000000000000001e-36 or 9.99999999999999939e-99 < y

   1. Initial program 48.8%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 67.7

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

   4. Applied rewrites 67.7%

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

   if -1.5000000000000001e-36 < y < 9.99999999999999939e-99

   1. Initial program 76.9%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 59.7

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

   4. Applied rewrites 59.7%

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

Alternative 13: 59.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + z\right) - b\\ \mathbf{if}\;y \leq -5 \cdot 10^{+90}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{-44}:\\ \;\;\;\;a + z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (+ a z) b)))
   (if (<= y -5e+90) t_1 (if (<= y 4.1e-44) (+ a z) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a + z) - b;
	double tmp;
	if (y <= -5e+90) {
		tmp = t_1;
	} else if (y <= 4.1e-44) {
		tmp = a + z;
	} 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 + z) - b
    if (y <= (-5d+90)) then
        tmp = t_1
    else if (y <= 4.1d-44) then
        tmp = a + z
    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 + z) - b;
	double tmp;
	if (y <= -5e+90) {
		tmp = t_1;
	} else if (y <= 4.1e-44) {
		tmp = a + z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a + z) - b
	tmp = 0
	if y <= -5e+90:
		tmp = t_1
	elif y <= 4.1e-44:
		tmp = a + z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a + z) - b)
	tmp = 0.0
	if (y <= -5e+90)
		tmp = t_1;
	elseif (y <= 4.1e-44)
		tmp = Float64(a + z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a + z) - b;
	tmp = 0.0;
	if (y <= -5e+90)
		tmp = t_1;
	elseif (y <= 4.1e-44)
		tmp = a + z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[y, -5e+90], t$95$1, If[LessEqual[y, 4.1e-44], N[(a + z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.0000000000000004e90 or 4.09999999999999992e-44 < y

   1. Initial program 41.7%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 73.1

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

   4. Applied rewrites 73.1%

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

   if -5.0000000000000004e90 < y < 4.09999999999999992e-44

   1. Initial program 75.7%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 40.3

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

   4. Applied rewrites 40.3%

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

   5. Taylor expanded in b around 0

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

      1. lift-+.f64 48.4

         \[\leadsto a + z \]

   7. Applied rewrites 48.4%

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

Alternative 14: 52.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{+176}:\\ \;\;\;\;a + z\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (if (<= t 2.3e+176) (+ a z) a))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 2.3e+176) {
		tmp = a + z;
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
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 <= 2.3d+176) then
        tmp = a + z
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 2.3e+176) {
		tmp = a + z;
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= 2.3e+176:
		tmp = a + z
	else:
		tmp = a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= 2.3e+176)
		tmp = Float64(a + z);
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= 2.3e+176)
		tmp = a + z;
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, 2.3e+176], N[(a + z), $MachinePrecision], a]

Derivation

1. Split input into 2 regimes

2. if t < 2.29999999999999996e176

   1. Initial program 61.8%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 57.7

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

   4. Applied rewrites 57.7%

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

   5. Taylor expanded in b around 0

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

      1. lift-+.f64 52.7

         \[\leadsto a + z \]

   7. Applied rewrites 52.7%

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

   if 2.29999999999999996e176 < t

   1. Initial program 44.2%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 54.9%

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

Alternative 15: 48.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.3 \cdot 10^{-75}:\\ \;\;\;\;a - b\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{+55}:\\ \;\;\;\;z - b\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -1.3e-75) (- a b) (if (<= a 4.2e+55) (- z b) a)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -1.3e-75) {
		tmp = a - b;
	} else if (a <= 4.2e+55) {
		tmp = z - b;
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
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 (a <= (-1.3d-75)) then
        tmp = a - b
    else if (a <= 4.2d+55) then
        tmp = z - b
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -1.3e-75) {
		tmp = a - b;
	} else if (a <= 4.2e+55) {
		tmp = z - b;
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -1.3e-75:
		tmp = a - b
	elif a <= 4.2e+55:
		tmp = z - b
	else:
		tmp = a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -1.3e-75)
		tmp = Float64(a - b);
	elseif (a <= 4.2e+55)
		tmp = Float64(z - b);
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -1.3e-75)
		tmp = a - b;
	elseif (a <= 4.2e+55)
		tmp = z - b;
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -1.3e-75], N[(a - b), $MachinePrecision], If[LessEqual[a, 4.2e+55], N[(z - b), $MachinePrecision], a]]

Derivation

1. Split input into 3 regimes

2. if a < -1.3e-75

   1. Initial program 53.0%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 57.8

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

   4. Applied rewrites 57.8%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 46.0%

      \[\leadsto a - b \]

   if -1.3e-75 < a < 4.2000000000000001e55

   1. Initial program 70.5%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 53.0

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

   4. Applied rewrites 53.0%

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

   5. Taylor expanded in z around inf

      \[\leadsto z - b \]
   6. Step-by-step derivation

   7. Applied rewrites 48.1%

      \[\leadsto z - b \]

   if 4.2000000000000001e55 < a

   1. Initial program 44.9%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 50.6%

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

Alternative 16: 44.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.45 \cdot 10^{-100}:\\ \;\;\;\;a - b\\ \mathbf{elif}\;a \leq 2.6 \cdot 10^{+56}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -1.45e-100) (- a b) (if (<= a 2.6e+56) z a)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -1.45e-100) {
		tmp = a - b;
	} else if (a <= 2.6e+56) {
		tmp = z;
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
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 (a <= (-1.45d-100)) then
        tmp = a - b
    else if (a <= 2.6d+56) then
        tmp = z
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -1.45e-100) {
		tmp = a - b;
	} else if (a <= 2.6e+56) {
		tmp = z;
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -1.45e-100:
		tmp = a - b
	elif a <= 2.6e+56:
		tmp = z
	else:
		tmp = a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -1.45e-100)
		tmp = Float64(a - b);
	elseif (a <= 2.6e+56)
		tmp = z;
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -1.45e-100)
		tmp = a - b;
	elseif (a <= 2.6e+56)
		tmp = z;
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -1.45e-100], N[(a - b), $MachinePrecision], If[LessEqual[a, 2.6e+56], z, a]]

Derivation

1. Split input into 3 regimes

2. if a < -1.44999999999999988e-100

   1. Initial program 53.9%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 57.1

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

   4. Applied rewrites 57.1%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 44.7%

      \[\leadsto a - b \]

   if -1.44999999999999988e-100 < a < 2.60000000000000011e56

   1. Initial program 70.6%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 42.3%

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

   if 2.60000000000000011e56 < a

   1. Initial program 44.9%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 50.7%

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

Alternative 17: 44.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.2 \cdot 10^{-75}:\\ \;\;\;\;a\\ \mathbf{elif}\;a \leq 2.6 \cdot 10^{+56}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -1.2e-75) a (if (<= a 2.6e+56) z a)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -1.2e-75) {
		tmp = a;
	} else if (a <= 2.6e+56) {
		tmp = z;
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
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 (a <= (-1.2d-75)) then
        tmp = a
    else if (a <= 2.6d+56) then
        tmp = z
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -1.2e-75) {
		tmp = a;
	} else if (a <= 2.6e+56) {
		tmp = z;
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -1.2e-75:
		tmp = a
	elif a <= 2.6e+56:
		tmp = z
	else:
		tmp = a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -1.2e-75)
		tmp = a;
	elseif (a <= 2.6e+56)
		tmp = z;
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -1.2e-75)
		tmp = a;
	elseif (a <= 2.6e+56)
		tmp = z;
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -1.2e-75], a, If[LessEqual[a, 2.6e+56], z, a]]

Derivation

1. Split input into 2 regimes

2. if a < -1.2000000000000001e-75 or 2.60000000000000011e56 < a

   1. Initial program 49.8%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 47.0%

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

   if -1.2000000000000001e-75 < a < 2.60000000000000011e56

   1. Initial program 70.5%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 42.2%

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

Alternative 18: 32.0% accurate, 29.5× 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 59.9%

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

2. Taylor expanded in t around inf

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

4. Applied rewrites 32.0%

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

Reproduce

herbie shell --seed 2025114 
(FPCore (x y z t a b)
  :name "AI.Clustering.Hierarchical.Internal:ward from clustering-0.2.1"
  :precision binary64
  (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))