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

Percentage Accurate: 60.5% → 93.9%

Time: 7.3s

Alternatives: 14

Speedup: 1.5×

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: 60.5% 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: 93.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t + \left(x + y\right)\\ t_2 := \left(y + x\right) + t\\ t_3 := \frac{y + x}{t\_2}\\ t_4 := \mathsf{fma}\left(t\_3, z, a \cdot \mathsf{fma}\left(-1, \frac{b}{a} \cdot \frac{y}{t\_1}, \frac{t + y}{t\_1}\right)\right)\\ \mathbf{if}\;a \leq -1.8 \cdot 10^{-71}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;a \leq 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(t\_3, z, \frac{\left(t + y\right) \cdot a - b \cdot y}{t\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ t (+ x y)))
        (t_2 (+ (+ y x) t))
        (t_3 (/ (+ y x) t_2))
        (t_4
         (fma t_3 z (* a (fma -1.0 (* (/ b a) (/ y t_1)) (/ (+ t y) t_1))))))
   (if (<= a -1.8e-71)
     t_4
     (if (<= a 1e-37) (fma t_3 z (/ (- (* (+ t y) a) (* b y)) t_2)) t_4))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t + (x + y);
	double t_2 = (y + x) + t;
	double t_3 = (y + x) / t_2;
	double t_4 = fma(t_3, z, (a * fma(-1.0, ((b / a) * (y / t_1)), ((t + y) / t_1))));
	double tmp;
	if (a <= -1.8e-71) {
		tmp = t_4;
	} else if (a <= 1e-37) {
		tmp = fma(t_3, z, ((((t + y) * a) - (b * y)) / t_2));
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t + Float64(x + y))
	t_2 = Float64(Float64(y + x) + t)
	t_3 = Float64(Float64(y + x) / t_2)
	t_4 = fma(t_3, z, Float64(a * fma(-1.0, Float64(Float64(b / a) * Float64(y / t_1)), Float64(Float64(t + y) / t_1))))
	tmp = 0.0
	if (a <= -1.8e-71)
		tmp = t_4;
	elseif (a <= 1e-37)
		tmp = fma(t_3, z, Float64(Float64(Float64(Float64(t + y) * a) - Float64(b * y)) / t_2));
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y + x), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * z + N[(a * N[(-1.0 * N[(N[(b / a), $MachinePrecision] * N[(y / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.8e-71], t$95$4, If[LessEqual[a, 1e-37], N[(t$95$3 * z + N[(N[(N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision] - N[(b * y), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.8e-71 or 1.00000000000000007e-37 < a

   1. Initial program 60.5%

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

   3. Applied rewrites 72.1%

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

   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 83.2

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

   6. Applied rewrites 83.2%

      \[\leadsto \mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, \color{blue}{a \cdot \mathsf{fma}\left(-1, \frac{b \cdot y}{a \cdot \left(t + \left(x + y\right)\right)}, \frac{t + y}{t + \left(x + y\right)}\right)}\right) \]
   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 87.6

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

   8. Applied rewrites 87.6%

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

   if -1.8e-71 < a < 1.00000000000000007e-37

   1. Initial program 60.5%

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

   3. Applied rewrites 72.1%

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

Alternative 2: 90.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + y\right) \cdot a\\ t_2 := \left(x + t\right) + y\\ t_3 := \frac{\left(\left(x + y\right) \cdot z + t\_1\right) - y \cdot b}{t\_2}\\ t_4 := \frac{y + x}{\left(y + x\right) + t}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(t\_4, z, a - b\right)\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+214}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y + x, z, t\_1 - b \cdot y\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_4, z, a \cdot \mathsf{fma}\left(-1, \frac{b}{a} \cdot \frac{y}{t + \left(x + y\right)}, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (+ t y) a))
        (t_2 (+ (+ x t) y))
        (t_3 (/ (- (+ (* (+ x y) z) t_1) (* y b)) t_2))
        (t_4 (/ (+ y x) (+ (+ y x) t))))
   (if (<= t_3 (- INFINITY))
     (fma t_4 z (- a b))
     (if (<= t_3 2e+214)
       (/ (fma (+ y x) z (- t_1 (* b y))) t_2)
       (fma t_4 z (* a (fma -1.0 (* (/ b a) (/ y (+ t (+ x y)))) 1.0)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + y) * a;
	double t_2 = (x + t) + y;
	double t_3 = ((((x + y) * z) + t_1) - (y * b)) / t_2;
	double t_4 = (y + x) / ((y + x) + t);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = fma(t_4, z, (a - b));
	} else if (t_3 <= 2e+214) {
		tmp = fma((y + x), z, (t_1 - (b * y))) / t_2;
	} else {
		tmp = fma(t_4, z, (a * fma(-1.0, ((b / a) * (y / (t + (x + y)))), 1.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t + y) * a)
	t_2 = Float64(Float64(x + t) + y)
	t_3 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + t_1) - Float64(y * b)) / t_2)
	t_4 = Float64(Float64(y + x) / Float64(Float64(y + x) + t))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = fma(t_4, z, Float64(a - b));
	elseif (t_3 <= 2e+214)
		tmp = Float64(fma(Float64(y + x), z, Float64(t_1 - Float64(b * y))) / t_2);
	else
		tmp = fma(t_4, z, Float64(a * fma(-1.0, Float64(Float64(b / a) * Float64(y / Float64(t + Float64(x + y)))), 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t + y), $MachinePrecision] * a), $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] + t$95$1), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(y + x), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(t$95$4 * z + N[(a - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+214], N[(N[(N[(y + x), $MachinePrecision] * z + N[(t$95$1 - N[(b * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(t$95$4 * z + N[(a * N[(-1.0 * N[(N[(b / a), $MachinePrecision] * N[(y / N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 60.5%

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in y around inf

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

      1. lower--.f64 64.4

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

   6. Applied rewrites 64.4%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1.9999999999999999e214

   1. Initial program 60.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate--l+ N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-*.f64 60.6

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

   3. Applied rewrites 60.6%

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

   if 1.9999999999999999e214 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 60.5%

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

   3. Applied rewrites 72.1%

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

   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 83.2

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

   6. Applied rewrites 83.2%

      \[\leadsto \mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, \color{blue}{a \cdot \mathsf{fma}\left(-1, \frac{b \cdot y}{a \cdot \left(t + \left(x + y\right)\right)}, \frac{t + y}{t + \left(x + y\right)}\right)}\right) \]
   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 87.6

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

   8. Applied rewrites 87.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 73.9%

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

Alternative 3: 90.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + y\right) \cdot a\\ t_2 := \left(x + t\right) + y\\ t_3 := \frac{\left(\left(x + y\right) \cdot z + t\_1\right) - y \cdot b}{t\_2}\\ t_4 := \mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, a - b\right)\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+214}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y + x, z, t\_1 - b \cdot y\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (+ t y) a))
        (t_2 (+ (+ x t) y))
        (t_3 (/ (- (+ (* (+ x y) z) t_1) (* y b)) t_2))
        (t_4 (fma (/ (+ y x) (+ (+ y x) t)) z (- a b))))
   (if (<= t_3 (- INFINITY))
     t_4
     (if (<= t_3 2e+214) (/ (fma (+ y x) z (- t_1 (* b y))) t_2) t_4))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + y) * a;
	double t_2 = (x + t) + y;
	double t_3 = ((((x + y) * z) + t_1) - (y * b)) / t_2;
	double t_4 = fma(((y + x) / ((y + x) + t)), z, (a - b));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_3 <= 2e+214) {
		tmp = fma((y + x), z, (t_1 - (b * y))) / t_2;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t + y) * a)
	t_2 = Float64(Float64(x + t) + y)
	t_3 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + t_1) - Float64(y * b)) / t_2)
	t_4 = fma(Float64(Float64(y + x) / Float64(Float64(y + x) + t)), z, Float64(a - b))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_3 <= 2e+214)
		tmp = Float64(fma(Float64(y + x), z, Float64(t_1 - Float64(b * y))) / t_2);
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t + y), $MachinePrecision] * a), $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] + t$95$1), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(y + x), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] * z + N[(a - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$4, If[LessEqual[t$95$3, 2e+214], N[(N[(N[(y + x), $MachinePrecision] * z + N[(t$95$1 - N[(b * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], t$95$4]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -inf.0 or 1.9999999999999999e214 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 60.5%

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in y around inf

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

      1. lower--.f64 64.4

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

   6. Applied rewrites 64.4%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1.9999999999999999e214

   1. Initial program 60.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate--l+ N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-*.f64 60.6

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

   3. Applied rewrites 60.6%

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

Alternative 4: 75.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y + x\right) \cdot z\\ t_2 := \left(x + t\right) + y\\ t_3 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_2}\\ t_4 := \left(y + x\right) + t\\ t_5 := \frac{\mathsf{fma}\left(t + y, a, t\_1\right)}{t\_4}\\ t_6 := \mathsf{fma}\left(\frac{y + x}{t\_4}, z, a - b\right)\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+262}:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{+182}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-89}:\\ \;\;\;\;\frac{t\_1 - b \cdot y}{t\_2}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+214}:\\ \;\;\;\;t\_5\\ \mathbf{else}:\\ \;\;\;\;t\_6\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (+ y x) z))
        (t_2 (+ (+ x t) y))
        (t_3 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) t_2))
        (t_4 (+ (+ y x) t))
        (t_5 (/ (fma (+ t y) a t_1) t_4))
        (t_6 (fma (/ (+ y x) t_4) z (- a b))))
   (if (<= t_3 -2e+262)
     t_6
     (if (<= t_3 -2e+182)
       t_5
       (if (<= t_3 -2e-89)
         (/ (- t_1 (* b y)) t_2)
         (if (<= t_3 2e+214) t_5 t_6))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y + x) * z;
	double t_2 = (x + t) + y;
	double t_3 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / t_2;
	double t_4 = (y + x) + t;
	double t_5 = fma((t + y), a, t_1) / t_4;
	double t_6 = fma(((y + x) / t_4), z, (a - b));
	double tmp;
	if (t_3 <= -2e+262) {
		tmp = t_6;
	} else if (t_3 <= -2e+182) {
		tmp = t_5;
	} else if (t_3 <= -2e-89) {
		tmp = (t_1 - (b * y)) / t_2;
	} else if (t_3 <= 2e+214) {
		tmp = t_5;
	} else {
		tmp = t_6;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y + x) * z)
	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)
	t_4 = Float64(Float64(y + x) + t)
	t_5 = Float64(fma(Float64(t + y), a, t_1) / t_4)
	t_6 = fma(Float64(Float64(y + x) / t_4), z, Float64(a - b))
	tmp = 0.0
	if (t_3 <= -2e+262)
		tmp = t_6;
	elseif (t_3 <= -2e+182)
		tmp = t_5;
	elseif (t_3 <= -2e-89)
		tmp = Float64(Float64(t_1 - Float64(b * y)) / t_2);
	elseif (t_3 <= 2e+214)
		tmp = t_5;
	else
		tmp = t_6;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y + x), $MachinePrecision] * z), $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]}, Block[{t$95$4 = N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(t + y), $MachinePrecision] * a + t$95$1), $MachinePrecision] / t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(y + x), $MachinePrecision] / t$95$4), $MachinePrecision] * z + N[(a - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e+262], t$95$6, If[LessEqual[t$95$3, -2e+182], t$95$5, If[LessEqual[t$95$3, -2e-89], N[(N[(t$95$1 - N[(b * y), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$3, 2e+214], t$95$5, t$95$6]]]]]]]]]]

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)) < -2e262 or 1.9999999999999999e214 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 60.5%

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in y around inf

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

      1. lower--.f64 64.4

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

   6. Applied rewrites 64.4%

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

   if -2e262 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -2.0000000000000001e182 or -2.00000000000000008e-89 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1.9999999999999999e214

   1. Initial program 60.5%

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

   2. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 48.0

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

   4. Applied rewrites 48.0%

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

   if -2.0000000000000001e182 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -2.00000000000000008e-89

   1. Initial program 60.5%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 37.7

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

   4. Applied rewrites 37.7%

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

Alternative 5: 69.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + y\right) \cdot a\\ t_2 := \left(x + t\right) + y\\ t_3 := \frac{t\_1 - y \cdot b}{t\_2}\\ t_4 := \frac{\left(\left(x + y\right) \cdot z + t\_1\right) - y \cdot b}{t\_2}\\ t_5 := \mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, a - b\right)\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq -2 \cdot 10^{+182}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq 10^{+37}:\\ \;\;\;\;\frac{\left(y + x\right) \cdot z - b \cdot y}{t\_2}\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+214}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (+ t y) a))
        (t_2 (+ (+ x t) y))
        (t_3 (/ (- t_1 (* y b)) t_2))
        (t_4 (/ (- (+ (* (+ x y) z) t_1) (* y b)) t_2))
        (t_5 (fma (/ (+ y x) (+ (+ y x) t)) z (- a b))))
   (if (<= t_4 (- INFINITY))
     t_5
     (if (<= t_4 -2e+182)
       t_3
       (if (<= t_4 1e+37)
         (/ (- (* (+ y x) z) (* b y)) t_2)
         (if (<= t_4 2e+214) t_3 t_5))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + y) * a;
	double t_2 = (x + t) + y;
	double t_3 = (t_1 - (y * b)) / t_2;
	double t_4 = ((((x + y) * z) + t_1) - (y * b)) / t_2;
	double t_5 = fma(((y + x) / ((y + x) + t)), z, (a - b));
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = t_5;
	} else if (t_4 <= -2e+182) {
		tmp = t_3;
	} else if (t_4 <= 1e+37) {
		tmp = (((y + x) * z) - (b * y)) / t_2;
	} else if (t_4 <= 2e+214) {
		tmp = t_3;
	} else {
		tmp = t_5;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t + y) * a)
	t_2 = Float64(Float64(x + t) + y)
	t_3 = Float64(Float64(t_1 - Float64(y * b)) / t_2)
	t_4 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + t_1) - Float64(y * b)) / t_2)
	t_5 = fma(Float64(Float64(y + x) / Float64(Float64(y + x) + t)), z, Float64(a - b))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = t_5;
	elseif (t_4 <= -2e+182)
		tmp = t_3;
	elseif (t_4 <= 1e+37)
		tmp = Float64(Float64(Float64(Float64(y + x) * z) - Float64(b * y)) / t_2);
	elseif (t_4 <= 2e+214)
		tmp = t_3;
	else
		tmp = t_5;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + t$95$1), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(y + x), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] * z + N[(a - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], t$95$5, If[LessEqual[t$95$4, -2e+182], t$95$3, If[LessEqual[t$95$4, 1e+37], N[(N[(N[(N[(y + x), $MachinePrecision] * z), $MachinePrecision] - N[(b * y), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, 2e+214], t$95$3, t$95$5]]]]]]]]]

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 or 1.9999999999999999e214 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 60.5%

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in y around inf

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

      1. lower--.f64 64.4

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

   6. Applied rewrites 64.4%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -2.0000000000000001e182 or 9.99999999999999954e36 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1.9999999999999999e214

   1. Initial program 60.5%

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

   2. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 37.4

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

   4. Applied rewrites 37.4%

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

   if -2.0000000000000001e182 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 9.99999999999999954e36

   1. Initial program 60.5%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 37.7

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

   4. Applied rewrites 37.7%

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

Alternative 6: 68.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ t_3 := \mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, a - b\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+262}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+182}:\\ \;\;\;\;a \cdot \frac{t + y}{t + \left(x + y\right)}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+95}:\\ \;\;\;\;\frac{\left(y + x\right) \cdot z - b \cdot y}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

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 b))))
   (if (<= t_2 -2e+262)
     t_3
     (if (<= t_2 -2e+182)
       (* a (/ (+ t y) (+ t (+ x y))))
       (if (<= t_2 4e+95) (/ (- (* (+ y x) z) (* b y)) t_1) t_3)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + t) + y;
	double t_2 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / t_1;
	double t_3 = fma(((y + x) / ((y + x) + t)), z, (a - b));
	double tmp;
	if (t_2 <= -2e+262) {
		tmp = t_3;
	} else if (t_2 <= -2e+182) {
		tmp = a * ((t + y) / (t + (x + y)));
	} else if (t_2 <= 4e+95) {
		tmp = (((y + x) * z) - (b * y)) / t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + t) + y)
	t_2 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / t_1)
	t_3 = fma(Float64(Float64(y + x) / Float64(Float64(y + x) + t)), z, Float64(a - b))
	tmp = 0.0
	if (t_2 <= -2e+262)
		tmp = t_3;
	elseif (t_2 <= -2e+182)
		tmp = Float64(a * Float64(Float64(t + y) / Float64(t + Float64(x + y))));
	elseif (t_2 <= 4e+95)
		tmp = Float64(Float64(Float64(Float64(y + x) * z) - Float64(b * y)) / t_1);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(y + x), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] * z + N[(a - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+262], t$95$3, If[LessEqual[t$95$2, -2e+182], N[(a * N[(N[(t + y), $MachinePrecision] / N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+95], N[(N[(N[(N[(y + x), $MachinePrecision] * z), $MachinePrecision] - N[(b * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$3]]]]]]

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)) < -2e262 or 4.00000000000000008e95 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 60.5%

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in y around inf

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

      1. lower--.f64 64.4

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

   6. Applied rewrites 64.4%

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

   if -2e262 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -2.0000000000000001e182

   1. Initial program 60.5%

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 40.2

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

   6. Applied rewrites 40.2%

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

   if -2.0000000000000001e182 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4.00000000000000008e95

   1. Initial program 60.5%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 37.7

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

   4. Applied rewrites 37.7%

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

Alternative 7: 67.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ t_3 := \mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, a - b\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-104}:\\ \;\;\;\;\frac{\left(\left(a + z\right) - b\right) \cdot y}{t\_1}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+95}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, z \cdot x\right)}{t + x}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

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 b))))
   (if (<= t_2 (- INFINITY))
     t_3
     (if (<= t_2 -1e-104)
       (/ (* (- (+ a z) b) y) t_1)
       (if (<= t_2 4e+95) (/ (fma a t (* z x)) (+ t x)) 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 - b));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= -1e-104) {
		tmp = (((a + z) - b) * y) / t_1;
	} else if (t_2 <= 4e+95) {
		tmp = fma(a, t, (z * x)) / (t + x);
	} 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 - b))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= -1e-104)
		tmp = Float64(Float64(Float64(Float64(a + z) - b) * y) / t_1);
	elseif (t_2 <= 4e+95)
		tmp = Float64(fma(a, t, Float64(z * x)) / Float64(t + x));
	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 - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, -1e-104], N[(N[(N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision] * y), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 4e+95], N[(N[(a * t + N[(z * x), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

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 or 4.00000000000000008e95 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 60.5%

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in y around inf

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

      1. lower--.f64 64.4

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

   6. Applied rewrites 64.4%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -9.99999999999999927e-105

   1. Initial program 60.5%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 30.4

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

   4. Applied rewrites 30.4%

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

   if -9.99999999999999927e-105 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4.00000000000000008e95

   1. Initial program 60.5%

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

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{a \cdot t + x \cdot z}{t + x}} \]
   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 40.8

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

   4. Applied rewrites 40.8%

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

Alternative 8: 66.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + z\right) - b\\ t_2 := t + \left(x + y\right)\\ t_3 := a \cdot \frac{t + y}{t\_2}\\ \mathbf{if}\;a \leq -2.65 \cdot 10^{+141}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;a \leq -3.4 \cdot 10^{-105}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{-94}:\\ \;\;\;\;z \cdot \frac{x + y}{t\_2}\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{+92}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (+ a z) b)) (t_2 (+ t (+ x y))) (t_3 (* a (/ (+ t y) t_2))))
   (if (<= a -2.65e+141)
     t_3
     (if (<= a -3.4e-105)
       t_1
       (if (<= a 1.4e-94)
         (* z (/ (+ x y) t_2))
         (if (<= a 1.25e+92) t_1 t_3))))))

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 t_3 = a * ((t + y) / t_2);
	double tmp;
	if (a <= -2.65e+141) {
		tmp = t_3;
	} else if (a <= -3.4e-105) {
		tmp = t_1;
	} else if (a <= 1.4e-94) {
		tmp = z * ((x + y) / t_2);
	} else if (a <= 1.25e+92) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (a + z) - b
    t_2 = t + (x + y)
    t_3 = a * ((t + y) / t_2)
    if (a <= (-2.65d+141)) then
        tmp = t_3
    else if (a <= (-3.4d-105)) then
        tmp = t_1
    else if (a <= 1.4d-94) then
        tmp = z * ((x + y) / t_2)
    else if (a <= 1.25d+92) then
        tmp = t_1
    else
        tmp = t_3
    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 t_2 = t + (x + y);
	double t_3 = a * ((t + y) / t_2);
	double tmp;
	if (a <= -2.65e+141) {
		tmp = t_3;
	} else if (a <= -3.4e-105) {
		tmp = t_1;
	} else if (a <= 1.4e-94) {
		tmp = z * ((x + y) / t_2);
	} else if (a <= 1.25e+92) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a + z) - b
	t_2 = t + (x + y)
	t_3 = a * ((t + y) / t_2)
	tmp = 0
	if a <= -2.65e+141:
		tmp = t_3
	elif a <= -3.4e-105:
		tmp = t_1
	elif a <= 1.4e-94:
		tmp = z * ((x + y) / t_2)
	elif a <= 1.25e+92:
		tmp = t_1
	else:
		tmp = t_3
	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))
	t_3 = Float64(a * Float64(Float64(t + y) / t_2))
	tmp = 0.0
	if (a <= -2.65e+141)
		tmp = t_3;
	elseif (a <= -3.4e-105)
		tmp = t_1;
	elseif (a <= 1.4e-94)
		tmp = Float64(z * Float64(Float64(x + y) / t_2));
	elseif (a <= 1.25e+92)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a + z) - b;
	t_2 = t + (x + y);
	t_3 = a * ((t + y) / t_2);
	tmp = 0.0;
	if (a <= -2.65e+141)
		tmp = t_3;
	elseif (a <= -3.4e-105)
		tmp = t_1;
	elseif (a <= 1.4e-94)
		tmp = z * ((x + y) / t_2);
	elseif (a <= 1.25e+92)
		tmp = t_1;
	else
		tmp = t_3;
	end
	tmp_2 = 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]}, Block[{t$95$3 = N[(a * N[(N[(t + y), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2.65e+141], t$95$3, If[LessEqual[a, -3.4e-105], t$95$1, If[LessEqual[a, 1.4e-94], N[(z * N[(N[(x + y), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.25e+92], t$95$1, t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.65e141 or 1.25000000000000005e92 < a

   1. Initial program 60.5%

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 40.2

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

   6. Applied rewrites 40.2%

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

   if -2.65e141 < a < -3.39999999999999992e-105 or 1.3999999999999999e-94 < a < 1.25000000000000005e92

   1. Initial program 60.5%

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

   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 55.9

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

   4. Applied rewrites 55.9%

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

   if -3.39999999999999992e-105 < a < 1.3999999999999999e-94

   1. Initial program 60.5%

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in 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 40.4

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

   6. Applied rewrites 40.4%

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

Alternative 9: 63.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ t_3 := \left(a + z\right) - b\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-104}:\\ \;\;\;\;\frac{t\_3 \cdot y}{t\_1}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+95}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, z \cdot x\right)}{t + x}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

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 (- (+ a z) b)))
   (if (<= t_2 (- INFINITY))
     t_3
     (if (<= t_2 -1e-104)
       (/ (* t_3 y) t_1)
       (if (<= t_2 4e+95) (/ (fma a t (* z x)) (+ t x)) 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 = (a + z) - b;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= -1e-104) {
		tmp = (t_3 * y) / t_1;
	} else if (t_2 <= 4e+95) {
		tmp = fma(a, t, (z * x)) / (t + x);
	} 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 = Float64(Float64(a + z) - b)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= -1e-104)
		tmp = Float64(Float64(t_3 * y) / t_1);
	elseif (t_2 <= 4e+95)
		tmp = Float64(fma(a, t, Float64(z * x)) / Float64(t + x));
	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[(a + z), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, -1e-104], N[(N[(t$95$3 * y), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 4e+95], N[(N[(a * t + N[(z * x), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

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 or 4.00000000000000008e95 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 60.5%

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

   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 55.9

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

   4. Applied rewrites 55.9%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -9.99999999999999927e-105

   1. Initial program 60.5%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 30.4

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

   4. Applied rewrites 30.4%

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

   if -9.99999999999999927e-105 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4.00000000000000008e95

   1. Initial program 60.5%

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

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{a \cdot t + x \cdot z}{t + x}} \]
   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 40.8

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

   4. Applied rewrites 40.8%

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

Alternative 10: 61.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ t_2 := \left(a + z\right) - b\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+131}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+95}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, z \cdot x\right)}{t + x}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))
        (t_2 (- (+ a z) b)))
   (if (<= t_1 -2e+131)
     t_2
     (if (<= t_1 4e+95) (/ (fma a t (* z x)) (+ t x)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
	double t_2 = (a + z) - b;
	double tmp;
	if (t_1 <= -2e+131) {
		tmp = t_2;
	} else if (t_1 <= 4e+95) {
		tmp = fma(a, t, (z * x)) / (t + x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / Float64(Float64(x + t) + y))
	t_2 = Float64(Float64(a + z) - b)
	tmp = 0.0
	if (t_1 <= -2e+131)
		tmp = t_2;
	elseif (t_1 <= 4e+95)
		tmp = Float64(fma(a, t, Float64(z * x)) / Float64(t + x));
	else
		tmp = t_2;
	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[(a + z), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+131], t$95$2, If[LessEqual[t$95$1, 4e+95], N[(N[(a * t + N[(z * x), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -1.9999999999999998e131 or 4.00000000000000008e95 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 60.5%

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

   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 55.9

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

   4. Applied rewrites 55.9%

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

   if -1.9999999999999998e131 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4.00000000000000008e95

   1. Initial program 60.5%

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

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{a \cdot t + x \cdot z}{t + x}} \]
   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 40.8

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

   4. Applied rewrites 40.8%

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

Alternative 11: 60.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z + t \cdot \frac{a - z}{x}\\ \mathbf{if}\;x \leq -1 \cdot 10^{+88}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{+109}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ z (* t (/ (- a z) x)))))
   (if (<= x -1e+88) t_1 (if (<= x 9.2e+109) (- (+ a z) b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = z + (t * ((a - z) / x));
	double tmp;
	if (x <= -1e+88) {
		tmp = t_1;
	} else if (x <= 9.2e+109) {
		tmp = (a + z) - b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = z + (t * ((a - z) / x))
    if (x <= (-1d+88)) then
        tmp = t_1
    else if (x <= 9.2d+109) then
        tmp = (a + z) - b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = z + (t * ((a - z) / x));
	double tmp;
	if (x <= -1e+88) {
		tmp = t_1;
	} else if (x <= 9.2e+109) {
		tmp = (a + z) - b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = z + (t * ((a - z) / x))
	tmp = 0
	if x <= -1e+88:
		tmp = t_1
	elif x <= 9.2e+109:
		tmp = (a + z) - b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(z + Float64(t * Float64(Float64(a - z) / x)))
	tmp = 0.0
	if (x <= -1e+88)
		tmp = t_1;
	elseif (x <= 9.2e+109)
		tmp = Float64(Float64(a + z) - b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = z + (t * ((a - z) / x));
	tmp = 0.0;
	if (x <= -1e+88)
		tmp = t_1;
	elseif (x <= 9.2e+109)
		tmp = (a + z) - b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z + N[(t * N[(N[(a - z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1e+88], t$95$1, If[LessEqual[x, 9.2e+109], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.99999999999999959e87 or 9.20000000000000042e109 < x

   1. Initial program 60.5%

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

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{a \cdot t + x \cdot z}{t + x}} \]
   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 40.8

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

   4. Applied rewrites 40.8%

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

   5. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. sub-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 29.4

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

   7. Applied rewrites 29.4%

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

   if -9.99999999999999959e87 < x < 9.20000000000000042e109

   1. Initial program 60.5%

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

   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 55.9

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

   4. Applied rewrites 55.9%

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

Alternative 12: 58.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+88}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+116}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x -1e+88) z (if (<= x 4.2e+116) (- (+ a z) b) z)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -1e+88) {
		tmp = z;
	} else if (x <= 4.2e+116) {
		tmp = (a + z) - b;
	} else {
		tmp = z;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (x <= (-1d+88)) then
        tmp = z
    else if (x <= 4.2d+116) then
        tmp = (a + z) - b
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -1e+88) {
		tmp = z;
	} else if (x <= 4.2e+116) {
		tmp = (a + z) - b;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= -1e+88:
		tmp = z
	elif x <= 4.2e+116:
		tmp = (a + z) - b
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= -1e+88)
		tmp = z;
	elseif (x <= 4.2e+116)
		tmp = Float64(Float64(a + z) - b);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= -1e+88)
		tmp = z;
	elseif (x <= 4.2e+116)
		tmp = (a + z) - b;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, -1e+88], z, If[LessEqual[x, 4.2e+116], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision], z]]

Derivation

1. Split input into 2 regimes

2. if x < -9.99999999999999959e87 or 4.2000000000000002e116 < x

   1. Initial program 60.5%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 32.8%

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

   if -9.99999999999999959e87 < x < 4.2000000000000002e116

   1. Initial program 60.5%

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

   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 55.9

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

   4. Applied rewrites 55.9%

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

Alternative 13: 44.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.4 \cdot 10^{+122}:\\ \;\;\;\;a\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{+91}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -3.4e+122) a (if (<= a 7.5e+91) z a)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -3.4e+122) {
		tmp = a;
	} else if (a <= 7.5e+91) {
		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 <= (-3.4d+122)) then
        tmp = a
    else if (a <= 7.5d+91) 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 <= -3.4e+122) {
		tmp = a;
	} else if (a <= 7.5e+91) {
		tmp = z;
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -3.4e+122:
		tmp = a
	elif a <= 7.5e+91:
		tmp = z
	else:
		tmp = a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -3.4e+122)
		tmp = a;
	elseif (a <= 7.5e+91)
		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 <= -3.4e+122)
		tmp = a;
	elseif (a <= 7.5e+91)
		tmp = z;
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -3.4e+122], a, If[LessEqual[a, 7.5e+91], z, a]]

Derivation

1. Split input into 2 regimes

2. if a < -3.4e122 or 7.50000000000000033e91 < a

   1. Initial program 60.5%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 32.9%

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

   if -3.4e122 < a < 7.50000000000000033e91

   1. Initial program 60.5%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 32.8%

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

Alternative 14: 32.9% 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 60.5%

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

2. Taylor expanded in t around inf

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

4. Applied rewrites 32.9%

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

Reproduce

herbie shell --seed 2025142 
(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)))