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

Percentage Accurate: 60.7% → 98.6%

Time: 7.3s

Alternatives: 20

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: 60.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y + x\right) + t\\ \mathsf{fma}\left(\frac{y + x}{t\_1}, z, \frac{t + y}{t\_1} \cdot a - \frac{y}{t\_1} \cdot b\right) \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ y x) t)))
   (fma (/ (+ y x) t_1) z (- (* (/ (+ t y) t_1) a) (* (/ y t_1) b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y + x) + t;
	return fma(((y + x) / t_1), z, ((((t + y) / t_1) * a) - ((y / t_1) * b)));
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y + x) + t)
	return fma(Float64(Float64(y + x) / t_1), z, Float64(Float64(Float64(Float64(t + y) / t_1) * a) - Float64(Float64(y / t_1) * b)))
end

Wolfram

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

Derivation

1. Initial program 60.7%

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

2. Taylor expanded in a around 0

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

   1. lower--.f64 N/A

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

4. Applied rewrites 72.4%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. associate--l+ N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

6. Applied rewrites 98.6%

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

Alternative 2: 95.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ t_2 := \left(y + x\right) + t\\ t_3 := \mathsf{fma}\left(\frac{y + x}{t\_2}, z, a - \frac{y}{t\_2} \cdot b\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+228}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t + y}{t\_2}, a, \frac{\left(y + x\right) \cdot z - b \cdot y}{t\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))
        (t_2 (+ (+ y x) t))
        (t_3 (fma (/ (+ y x) t_2) z (- a (* (/ y t_2) b)))))
   (if (<= t_1 (- INFINITY))
     t_3
     (if (<= t_1 2e+228)
       (fma (/ (+ t y) t_2) a (/ (- (* (+ y x) z) (* b y)) t_2))
       t_3))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
	double t_2 = (y + x) + t;
	double t_3 = fma(((y + x) / t_2), z, (a - ((y / t_2) * b)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_1 <= 2e+228) {
		tmp = fma(((t + y) / t_2), a, ((((y + x) * z) - (b * y)) / t_2));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / Float64(Float64(x + t) + y))
	t_2 = Float64(Float64(y + x) + t)
	t_3 = fma(Float64(Float64(y + x) / t_2), z, Float64(a - Float64(Float64(y / t_2) * b)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_1 <= 2e+228)
		tmp = fma(Float64(Float64(t + y) / t_2), a, Float64(Float64(Float64(Float64(y + x) * z) - Float64(b * y)) / t_2));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(y + x), $MachinePrecision] / t$95$2), $MachinePrecision] * z + N[(a - N[(N[(y / t$95$2), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$3, If[LessEqual[t$95$1, 2e+228], N[(N[(N[(t + y), $MachinePrecision] / t$95$2), $MachinePrecision] * a + N[(N[(N[(N[(y + x), $MachinePrecision] * z), $MachinePrecision] - N[(b * y), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -inf.0 or 1.9999999999999998e228 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 60.7%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 98.6%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 84.3%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1.9999999999999998e228

   1. Initial program 60.7%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   6. Applied rewrites 72.4%

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

Alternative 3: 95.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ t_2 := \left(y + x\right) + t\\ t_3 := \mathsf{fma}\left(\frac{y + x}{t\_2}, z, a - \frac{y}{t\_2} \cdot b\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+304}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+228}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))
        (t_2 (+ (+ y x) t))
        (t_3 (fma (/ (+ y x) t_2) z (- a (* (/ y t_2) b)))))
   (if (<= t_1 -2e+304) t_3 (if (<= t_1 2e+228) t_1 t_3))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
	double t_2 = (y + x) + t;
	double t_3 = fma(((y + x) / t_2), z, (a - ((y / t_2) * b)));
	double tmp;
	if (t_1 <= -2e+304) {
		tmp = t_3;
	} else if (t_1 <= 2e+228) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / Float64(Float64(x + t) + y))
	t_2 = Float64(Float64(y + x) + t)
	t_3 = fma(Float64(Float64(y + x) / t_2), z, Float64(a - Float64(Float64(y / t_2) * b)))
	tmp = 0.0
	if (t_1 <= -2e+304)
		tmp = t_3;
	elseif (t_1 <= 2e+228)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(y + x), $MachinePrecision] / t$95$2), $MachinePrecision] * z + N[(a - N[(N[(y / t$95$2), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+304], t$95$3, If[LessEqual[t$95$1, 2e+228], t$95$1, t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -1.9999999999999999e304 or 1.9999999999999998e228 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 60.7%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 98.6%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 84.3%

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

   if -1.9999999999999999e304 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1.9999999999999998e228

   1. Initial program 60.7%

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

Alternative 4: 95.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ t_2 := \left(y + x\right) + t\\ t_3 := \mathsf{fma}\left(\frac{y + x}{t\_2}, z, a - y \cdot \frac{b}{t\_2}\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+304}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+282}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))
        (t_2 (+ (+ y x) t))
        (t_3 (fma (/ (+ y x) t_2) z (- a (* y (/ b t_2))))))
   (if (<= t_1 -2e+304) t_3 (if (<= t_1 4e+282) t_1 t_3))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
	double t_2 = (y + x) + t;
	double t_3 = fma(((y + x) / t_2), z, (a - (y * (b / t_2))));
	double tmp;
	if (t_1 <= -2e+304) {
		tmp = t_3;
	} else if (t_1 <= 4e+282) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / Float64(Float64(x + t) + y))
	t_2 = Float64(Float64(y + x) + t)
	t_3 = fma(Float64(Float64(y + x) / t_2), z, Float64(a - Float64(y * Float64(b / t_2))))
	tmp = 0.0
	if (t_1 <= -2e+304)
		tmp = t_3;
	elseif (t_1 <= 4e+282)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(y + x), $MachinePrecision] / t$95$2), $MachinePrecision] * z + N[(a - N[(y * N[(b / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+304], t$95$3, If[LessEqual[t$95$1, 4e+282], t$95$1, t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -1.9999999999999999e304 or 4.00000000000000013e282 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 60.7%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 98.6%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 84.3%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*l/ N/A

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

       4. associate-/l* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 82.9

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

   11. Applied rewrites 82.9%

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

   if -1.9999999999999999e304 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4.00000000000000013e282

   1. Initial program 60.7%

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

Alternative 5: 90.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ t_2 := \frac{y}{y + t}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+304}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, z, a - t\_2 \cdot b\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+241}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, z, a - \frac{y}{\left(y + x\right) + t} \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))
        (t_2 (/ y (+ y t))))
   (if (<= t_1 -2e+304)
     (fma t_2 z (- a (* t_2 b)))
     (if (<= t_1 1e+241) t_1 (fma 1.0 z (- a (* (/ y (+ (+ y x) t)) b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
	double t_2 = y / (y + t);
	double tmp;
	if (t_1 <= -2e+304) {
		tmp = fma(t_2, z, (a - (t_2 * b)));
	} else if (t_1 <= 1e+241) {
		tmp = t_1;
	} else {
		tmp = fma(1.0, z, (a - ((y / ((y + x) + t)) * b)));
	}
	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(y / Float64(y + t))
	tmp = 0.0
	if (t_1 <= -2e+304)
		tmp = fma(t_2, z, Float64(a - Float64(t_2 * b)));
	elseif (t_1 <= 1e+241)
		tmp = t_1;
	else
		tmp = fma(1.0, z, Float64(a - Float64(Float64(y / Float64(Float64(y + x) + t)) * b)));
	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[(y / N[(y + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+304], N[(t$95$2 * z + N[(a - N[(t$95$2 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+241], t$95$1, N[(1.0 * z + N[(a - N[(N[(y / N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] * b), $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)) < -1.9999999999999999e304

   1. Initial program 60.7%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 98.6%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 84.3%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 65.9%

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

   13. Taylor expanded in x around 0

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

   15. Applied rewrites 72.6%

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

   16. Taylor expanded in x around 0

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

   18. Applied rewrites 67.6%

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

   if -1.9999999999999999e304 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1.0000000000000001e241

   1. Initial program 60.7%

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

   if 1.0000000000000001e241 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 60.7%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 98.6%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 84.3%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 70.4%

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

Alternative 6: 82.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{y + t}\\ \mathbf{if}\;y \leq -2.1 \cdot 10^{-166}:\\ \;\;\;\;\mathsf{fma}\left(1, z, a - \frac{y}{\left(y + x\right) + t} \cdot b\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-77}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x}{t + x}, a \cdot \frac{t}{t + x}\right)\\ \mathbf{elif}\;y \leq 7.5:\\ \;\;\;\;\left(z + \frac{a \cdot y}{x + y}\right) - \frac{b \cdot y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, z, a - t\_1 \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ y (+ y t))))
   (if (<= y -2.1e-166)
     (fma 1.0 z (- a (* (/ y (+ (+ y x) t)) b)))
     (if (<= y 5e-77)
       (fma z (/ x (+ t x)) (* a (/ t (+ t x))))
       (if (<= y 7.5)
         (- (+ z (/ (* a y) (+ x y))) (/ (* b y) (+ x y)))
         (fma t_1 z (- a (* t_1 b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y / (y + t);
	double tmp;
	if (y <= -2.1e-166) {
		tmp = fma(1.0, z, (a - ((y / ((y + x) + t)) * b)));
	} else if (y <= 5e-77) {
		tmp = fma(z, (x / (t + x)), (a * (t / (t + x))));
	} else if (y <= 7.5) {
		tmp = (z + ((a * y) / (x + y))) - ((b * y) / (x + y));
	} else {
		tmp = fma(t_1, z, (a - (t_1 * b)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y / Float64(y + t))
	tmp = 0.0
	if (y <= -2.1e-166)
		tmp = fma(1.0, z, Float64(a - Float64(Float64(y / Float64(Float64(y + x) + t)) * b)));
	elseif (y <= 5e-77)
		tmp = fma(z, Float64(x / Float64(t + x)), Float64(a * Float64(t / Float64(t + x))));
	elseif (y <= 7.5)
		tmp = Float64(Float64(z + Float64(Float64(a * y) / Float64(x + y))) - Float64(Float64(b * y) / Float64(x + y)));
	else
		tmp = fma(t_1, z, Float64(a - Float64(t_1 * b)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y / N[(y + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.1e-166], N[(1.0 * z + N[(a - N[(N[(y / N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e-77], N[(z * N[(x / N[(t + x), $MachinePrecision]), $MachinePrecision] + N[(a * N[(t / N[(t + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.5], N[(N[(z + N[(N[(a * y), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(b * y), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * z + N[(a - N[(t$95$1 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.0999999999999999e-166

   1. Initial program 60.7%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 98.6%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 84.3%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 70.4%

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

   if -2.0999999999999999e-166 < y < 4.99999999999999963e-77

   1. Initial program 60.7%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 40.2

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

   4. Applied rewrites 40.2%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 59.5

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

   6. Applied rewrites 59.5%

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

   if 4.99999999999999963e-77 < y < 7.5

   1. Initial program 60.7%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 72.4%

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

   5. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 50.7

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

   7. Applied rewrites 50.7%

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

   if 7.5 < y

   1. Initial program 60.7%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 98.6%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 84.3%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 65.9%

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

   13. Taylor expanded in x around 0

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

   15. Applied rewrites 72.6%

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

   16. Taylor expanded in x around 0

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

   18. Applied rewrites 67.6%

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

Alternative 7: 79.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y + x\right) + t\\ \mathbf{if}\;y \leq -2.1 \cdot 10^{-166}:\\ \;\;\;\;\mathsf{fma}\left(1, z, a - \frac{y}{t\_1} \cdot b\right)\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-61}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x}{t + x}, a \cdot \frac{t}{t + x}\right)\\ \mathbf{elif}\;y \leq 700000000:\\ \;\;\;\;\frac{y \cdot \left(\left(a + z\right) - b\right)}{\left(x + t\right) + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{t\_1}, z, a - b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ y x) t)))
   (if (<= y -2.1e-166)
     (fma 1.0 z (- a (* (/ y t_1) b)))
     (if (<= y 2.3e-61)
       (fma z (/ x (+ t x)) (* a (/ t (+ t x))))
       (if (<= y 700000000.0)
         (/ (* y (- (+ a z) b)) (+ (+ x t) y))
         (fma (/ (+ y x) t_1) z (- a b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y + x) + t;
	double tmp;
	if (y <= -2.1e-166) {
		tmp = fma(1.0, z, (a - ((y / t_1) * b)));
	} else if (y <= 2.3e-61) {
		tmp = fma(z, (x / (t + x)), (a * (t / (t + x))));
	} else if (y <= 700000000.0) {
		tmp = (y * ((a + z) - b)) / ((x + t) + y);
	} else {
		tmp = fma(((y + x) / t_1), z, (a - b));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y + x) + t)
	tmp = 0.0
	if (y <= -2.1e-166)
		tmp = fma(1.0, z, Float64(a - Float64(Float64(y / t_1) * b)));
	elseif (y <= 2.3e-61)
		tmp = fma(z, Float64(x / Float64(t + x)), Float64(a * Float64(t / Float64(t + x))));
	elseif (y <= 700000000.0)
		tmp = Float64(Float64(y * Float64(Float64(a + z) - b)) / Float64(Float64(x + t) + y));
	else
		tmp = fma(Float64(Float64(y + x) / t_1), z, Float64(a - b));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[y, -2.1e-166], N[(1.0 * z + N[(a - N[(N[(y / t$95$1), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.3e-61], N[(z * N[(x / N[(t + x), $MachinePrecision]), $MachinePrecision] + N[(a * N[(t / N[(t + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 700000000.0], N[(N[(y * N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y + x), $MachinePrecision] / t$95$1), $MachinePrecision] * z + N[(a - b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.0999999999999999e-166

   1. Initial program 60.7%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 98.6%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 84.3%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 70.4%

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

   if -2.0999999999999999e-166 < y < 2.29999999999999992e-61

   1. Initial program 60.7%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 40.2

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

   4. Applied rewrites 40.2%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 59.5

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

   6. Applied rewrites 59.5%

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

   if 2.29999999999999992e-61 < y < 7e8

   1. Initial program 60.7%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 30.9

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

   4. Applied rewrites 30.9%

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

   if 7e8 < y

   1. Initial program 60.7%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 98.6%

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

   7. Taylor expanded in y around inf

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

      1. lower--.f64 63.6

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

   9. Applied rewrites 63.6%

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

Alternative 8: 79.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \left(x + y\right) \cdot z\\ t_3 := \frac{\left(t\_2 + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ t_4 := \left(y + x\right) + t\\ \mathbf{if}\;t\_3 \leq -4 \cdot 10^{+126}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{t\_4}, z, a - b\right)\\ \mathbf{elif}\;t\_3 \leq 10^{+103}:\\ \;\;\;\;\frac{\left(t\_2 + a \cdot t\right) - y \cdot b}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, z, a - \frac{y}{t\_4} \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ x t) y))
        (t_2 (* (+ x y) z))
        (t_3 (/ (- (+ t_2 (* (+ t y) a)) (* y b)) t_1))
        (t_4 (+ (+ y x) t)))
   (if (<= t_3 -4e+126)
     (fma (/ (+ y x) t_4) z (- a b))
     (if (<= t_3 1e+103)
       (/ (- (+ t_2 (* a t)) (* y b)) t_1)
       (fma 1.0 z (- a (* (/ y t_4) b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + t) + y;
	double t_2 = (x + y) * z;
	double t_3 = ((t_2 + ((t + y) * a)) - (y * b)) / t_1;
	double t_4 = (y + x) + t;
	double tmp;
	if (t_3 <= -4e+126) {
		tmp = fma(((y + x) / t_4), z, (a - b));
	} else if (t_3 <= 1e+103) {
		tmp = ((t_2 + (a * t)) - (y * b)) / t_1;
	} else {
		tmp = fma(1.0, z, (a - ((y / t_4) * b)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + t) + y)
	t_2 = Float64(Float64(x + y) * z)
	t_3 = Float64(Float64(Float64(t_2 + Float64(Float64(t + y) * a)) - Float64(y * b)) / t_1)
	t_4 = Float64(Float64(y + x) + t)
	tmp = 0.0
	if (t_3 <= -4e+126)
		tmp = fma(Float64(Float64(y + x) / t_4), z, Float64(a - b));
	elseif (t_3 <= 1e+103)
		tmp = Float64(Float64(Float64(t_2 + Float64(a * t)) - Float64(y * b)) / t_1);
	else
		tmp = fma(1.0, z, Float64(a - Float64(Float64(y / t_4) * b)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$2 + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[t$95$3, -4e+126], N[(N[(N[(y + x), $MachinePrecision] / t$95$4), $MachinePrecision] * z + N[(a - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+103], N[(N[(N[(t$95$2 + N[(a * t), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(1.0 * z + N[(a - N[(N[(y / t$95$4), $MachinePrecision] * b), $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)) < -3.9999999999999997e126

   1. Initial program 60.7%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 98.6%

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

   7. Taylor expanded in y around inf

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

      1. lower--.f64 63.6

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

   9. Applied rewrites 63.6%

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

   if -3.9999999999999997e126 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1e103

   1. Initial program 60.7%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 52.9

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

   4. Applied rewrites 52.9%

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

   if 1e103 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 60.7%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 98.6%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 84.3%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 70.4%

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

Alternative 9: 76.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{y + t}\\ \mathbf{if}\;y \leq -2.1 \cdot 10^{-166}:\\ \;\;\;\;\mathsf{fma}\left(1, z, a - \frac{y}{\left(y + x\right) + t} \cdot b\right)\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x}{t + x}, a \cdot \frac{t}{t + x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, z, a - t\_1 \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ y (+ y t))))
   (if (<= y -2.1e-166)
     (fma 1.0 z (- a (* (/ y (+ (+ y x) t)) b)))
     (if (<= y 2.8e-52)
       (fma z (/ x (+ t x)) (* a (/ t (+ t x))))
       (fma t_1 z (- a (* t_1 b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y / (y + t);
	double tmp;
	if (y <= -2.1e-166) {
		tmp = fma(1.0, z, (a - ((y / ((y + x) + t)) * b)));
	} else if (y <= 2.8e-52) {
		tmp = fma(z, (x / (t + x)), (a * (t / (t + x))));
	} else {
		tmp = fma(t_1, z, (a - (t_1 * b)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y / Float64(y + t))
	tmp = 0.0
	if (y <= -2.1e-166)
		tmp = fma(1.0, z, Float64(a - Float64(Float64(y / Float64(Float64(y + x) + t)) * b)));
	elseif (y <= 2.8e-52)
		tmp = fma(z, Float64(x / Float64(t + x)), Float64(a * Float64(t / Float64(t + x))));
	else
		tmp = fma(t_1, z, Float64(a - Float64(t_1 * b)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y / N[(y + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.1e-166], N[(1.0 * z + N[(a - N[(N[(y / N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.8e-52], N[(z * N[(x / N[(t + x), $MachinePrecision]), $MachinePrecision] + N[(a * N[(t / N[(t + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * z + N[(a - N[(t$95$1 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.0999999999999999e-166

   1. Initial program 60.7%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 98.6%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 84.3%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 70.4%

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

   if -2.0999999999999999e-166 < y < 2.79999999999999995e-52

   1. Initial program 60.7%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 40.2

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

   4. Applied rewrites 40.2%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 59.5

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

   6. Applied rewrites 59.5%

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

   if 2.79999999999999995e-52 < y

   1. Initial program 60.7%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 98.6%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 84.3%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 65.9%

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

   13. Taylor expanded in x around 0

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

   15. Applied rewrites 72.6%

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

   16. Taylor expanded in x around 0

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

   18. Applied rewrites 67.6%

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

Alternative 10: 71.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ t_3 := \left(y + x\right) + t\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+126}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{t\_3}, z, a - b\right)\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-162}:\\ \;\;\;\;\frac{z \cdot \left(x + y\right) - b \cdot y}{t\_1}\\ \mathbf{elif}\;t\_2 \leq 10^{+22}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, x \cdot z\right)}{t + x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, z, a - \frac{y}{t\_3} \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ x t) y))
        (t_2 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) t_1))
        (t_3 (+ (+ y x) t)))
   (if (<= t_2 -4e+126)
     (fma (/ (+ y x) t_3) z (- a b))
     (if (<= t_2 -1e-162)
       (/ (- (* z (+ x y)) (* b y)) t_1)
       (if (<= t_2 1e+22)
         (/ (fma a t (* x z)) (+ t x))
         (fma 1.0 z (- a (* (/ y t_3) b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + t) + y;
	double t_2 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / t_1;
	double t_3 = (y + x) + t;
	double tmp;
	if (t_2 <= -4e+126) {
		tmp = fma(((y + x) / t_3), z, (a - b));
	} else if (t_2 <= -1e-162) {
		tmp = ((z * (x + y)) - (b * y)) / t_1;
	} else if (t_2 <= 1e+22) {
		tmp = fma(a, t, (x * z)) / (t + x);
	} else {
		tmp = fma(1.0, z, (a - ((y / t_3) * b)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + t) + y)
	t_2 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / t_1)
	t_3 = Float64(Float64(y + x) + t)
	tmp = 0.0
	if (t_2 <= -4e+126)
		tmp = fma(Float64(Float64(y + x) / t_3), z, Float64(a - b));
	elseif (t_2 <= -1e-162)
		tmp = Float64(Float64(Float64(z * Float64(x + y)) - Float64(b * y)) / t_1);
	elseif (t_2 <= 1e+22)
		tmp = Float64(fma(a, t, Float64(x * z)) / Float64(t + x));
	else
		tmp = fma(1.0, z, Float64(a - Float64(Float64(y / t_3) * b)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+126], N[(N[(N[(y + x), $MachinePrecision] / t$95$3), $MachinePrecision] * z + N[(a - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1e-162], N[(N[(N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision] - N[(b * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 1e+22], N[(N[(a * t + N[(x * z), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], N[(1.0 * z + N[(a - N[(N[(y / t$95$3), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -3.9999999999999997e126

   1. Initial program 60.7%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 98.6%

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

   7. Taylor expanded in y around inf

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

      1. lower--.f64 63.6

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

   9. Applied rewrites 63.6%

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

   if -3.9999999999999997e126 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -9.99999999999999954e-163

   1. Initial program 60.7%

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

   2. Taylor expanded in 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. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 37.8

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

   4. Applied rewrites 37.8%

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

   if -9.99999999999999954e-163 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1e22

   1. Initial program 60.7%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 40.2

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

   4. Applied rewrites 40.2%

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

   if 1e22 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 60.7%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 98.6%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 84.3%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 70.4%

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

Alternative 11: 69.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y + x\right) + t\\ \mathbf{if}\;y \leq -2.1 \cdot 10^{-166}:\\ \;\;\;\;\mathsf{fma}\left(1, z, a - \frac{y}{t\_1} \cdot b\right)\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-105}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, x \cdot z\right)}{t + x}\\ \mathbf{elif}\;y \leq 700000000:\\ \;\;\;\;\frac{y \cdot \left(\left(a + z\right) - b\right)}{\left(x + t\right) + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{t\_1}, z, a - b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ y x) t)))
   (if (<= y -2.1e-166)
     (fma 1.0 z (- a (* (/ y t_1) b)))
     (if (<= y 5.5e-105)
       (/ (fma a t (* x z)) (+ t x))
       (if (<= y 700000000.0)
         (/ (* y (- (+ a z) b)) (+ (+ x t) y))
         (fma (/ (+ y x) t_1) z (- a b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y + x) + t;
	double tmp;
	if (y <= -2.1e-166) {
		tmp = fma(1.0, z, (a - ((y / t_1) * b)));
	} else if (y <= 5.5e-105) {
		tmp = fma(a, t, (x * z)) / (t + x);
	} else if (y <= 700000000.0) {
		tmp = (y * ((a + z) - b)) / ((x + t) + y);
	} else {
		tmp = fma(((y + x) / t_1), z, (a - b));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y + x) + t)
	tmp = 0.0
	if (y <= -2.1e-166)
		tmp = fma(1.0, z, Float64(a - Float64(Float64(y / t_1) * b)));
	elseif (y <= 5.5e-105)
		tmp = Float64(fma(a, t, Float64(x * z)) / Float64(t + x));
	elseif (y <= 700000000.0)
		tmp = Float64(Float64(y * Float64(Float64(a + z) - b)) / Float64(Float64(x + t) + y));
	else
		tmp = fma(Float64(Float64(y + x) / t_1), z, Float64(a - b));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[y, -2.1e-166], N[(1.0 * z + N[(a - N[(N[(y / t$95$1), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.5e-105], N[(N[(a * t + N[(x * z), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 700000000.0], N[(N[(y * N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y + x), $MachinePrecision] / t$95$1), $MachinePrecision] * z + N[(a - b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.0999999999999999e-166

   1. Initial program 60.7%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 98.6%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 84.3%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 70.4%

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

   if -2.0999999999999999e-166 < y < 5.50000000000000029e-105

   1. Initial program 60.7%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 40.2

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

   4. Applied rewrites 40.2%

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

   if 5.50000000000000029e-105 < y < 7e8

   1. Initial program 60.7%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 30.9

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

   4. Applied rewrites 30.9%

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

   if 7e8 < y

   1. Initial program 60.7%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 98.6%

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

   7. Taylor expanded in y around inf

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

      1. lower--.f64 63.6

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

   9. Applied rewrites 63.6%

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

Alternative 12: 68.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, a - b\right)\\ \mathbf{if}\;y \leq -1.45 \cdot 10^{+105}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{-166}:\\ \;\;\;\;\left(a + z\right) - \frac{b \cdot y}{t + \left(x + y\right)}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-105}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, x \cdot z\right)}{t + x}\\ \mathbf{elif}\;y \leq 700000000:\\ \;\;\;\;\frac{y \cdot \left(\left(a + z\right) - b\right)}{\left(x + t\right) + y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (/ (+ y x) (+ (+ y x) t)) z (- a b))))
   (if (<= y -1.45e+105)
     t_1
     (if (<= y -2.1e-166)
       (- (+ a z) (/ (* b y) (+ t (+ x y))))
       (if (<= y 5.5e-105)
         (/ (fma a t (* x z)) (+ t x))
         (if (<= y 700000000.0) (/ (* y (- (+ a z) b)) (+ (+ x t) y)) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(((y + x) / ((y + x) + t)), z, (a - b));
	double tmp;
	if (y <= -1.45e+105) {
		tmp = t_1;
	} else if (y <= -2.1e-166) {
		tmp = (a + z) - ((b * y) / (t + (x + y)));
	} else if (y <= 5.5e-105) {
		tmp = fma(a, t, (x * z)) / (t + x);
	} else if (y <= 700000000.0) {
		tmp = (y * ((a + z) - b)) / ((x + t) + y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(Float64(y + x) / Float64(Float64(y + x) + t)), z, Float64(a - b))
	tmp = 0.0
	if (y <= -1.45e+105)
		tmp = t_1;
	elseif (y <= -2.1e-166)
		tmp = Float64(Float64(a + z) - Float64(Float64(b * y) / Float64(t + Float64(x + y))));
	elseif (y <= 5.5e-105)
		tmp = Float64(fma(a, t, Float64(x * z)) / Float64(t + x));
	elseif (y <= 700000000.0)
		tmp = Float64(Float64(y * Float64(Float64(a + z) - b)) / Float64(Float64(x + t) + y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y + x), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] * z + N[(a - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.45e+105], t$95$1, If[LessEqual[y, -2.1e-166], N[(N[(a + z), $MachinePrecision] - N[(N[(b * y), $MachinePrecision] / N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.5e-105], N[(N[(a * t + N[(x * z), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 700000000.0], N[(N[(y * N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.45000000000000005e105 or 7e8 < y

   1. Initial program 60.7%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 98.6%

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

   7. Taylor expanded in y around inf

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

      1. lower--.f64 63.6

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

   9. Applied rewrites 63.6%

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

   if -1.45000000000000005e105 < y < -2.0999999999999999e-166

   1. Initial program 60.7%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 72.4%

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

   5. Taylor expanded in y around inf

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

      1. lower-+.f64 59.9

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

   7. Applied rewrites 59.9%

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

   if -2.0999999999999999e-166 < y < 5.50000000000000029e-105

   1. Initial program 60.7%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 40.2

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

   4. Applied rewrites 40.2%

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

   if 5.50000000000000029e-105 < y < 7e8

   1. Initial program 60.7%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 30.9

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

   4. Applied rewrites 30.9%

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

Alternative 13: 68.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + z\right) - b\\ \mathbf{if}\;y \leq -3.5 \cdot 10^{+142}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{-166}:\\ \;\;\;\;\left(a + z\right) - \frac{b \cdot y}{t + \left(x + y\right)}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-105}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, x \cdot z\right)}{t + x}\\ \mathbf{elif}\;y \leq 700000000:\\ \;\;\;\;\frac{y \cdot t\_1}{\left(x + t\right) + y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (+ a z) b)))
   (if (<= y -3.5e+142)
     t_1
     (if (<= y -2.1e-166)
       (- (+ a z) (/ (* b y) (+ t (+ x y))))
       (if (<= y 5.5e-105)
         (/ (fma a t (* x z)) (+ t x))
         (if (<= y 700000000.0) (/ (* y t_1) (+ (+ x t) y)) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a + z) - b;
	double tmp;
	if (y <= -3.5e+142) {
		tmp = t_1;
	} else if (y <= -2.1e-166) {
		tmp = (a + z) - ((b * y) / (t + (x + y)));
	} else if (y <= 5.5e-105) {
		tmp = fma(a, t, (x * z)) / (t + x);
	} else if (y <= 700000000.0) {
		tmp = (y * t_1) / ((x + t) + y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a + z) - b)
	tmp = 0.0
	if (y <= -3.5e+142)
		tmp = t_1;
	elseif (y <= -2.1e-166)
		tmp = Float64(Float64(a + z) - Float64(Float64(b * y) / Float64(t + Float64(x + y))));
	elseif (y <= 5.5e-105)
		tmp = Float64(fma(a, t, Float64(x * z)) / Float64(t + x));
	elseif (y <= 700000000.0)
		tmp = Float64(Float64(y * t_1) / Float64(Float64(x + t) + y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[y, -3.5e+142], t$95$1, If[LessEqual[y, -2.1e-166], N[(N[(a + z), $MachinePrecision] - N[(N[(b * y), $MachinePrecision] / N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.5e-105], N[(N[(a * t + N[(x * z), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 700000000.0], N[(N[(y * t$95$1), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.49999999999999997e142 or 7e8 < y

   1. Initial program 60.7%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 55.6

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

   4. Applied rewrites 55.6%

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

   if -3.49999999999999997e142 < y < -2.0999999999999999e-166

   1. Initial program 60.7%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 72.4%

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

   5. Taylor expanded in y around inf

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

      1. lower-+.f64 59.9

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

   7. Applied rewrites 59.9%

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

   if -2.0999999999999999e-166 < y < 5.50000000000000029e-105

   1. Initial program 60.7%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 40.2

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

   4. Applied rewrites 40.2%

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

   if 5.50000000000000029e-105 < y < 7e8

   1. Initial program 60.7%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 30.9

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

   4. Applied rewrites 30.9%

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

Alternative 14: 66.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + z\right) - \frac{b \cdot y}{t + \left(x + y\right)}\\ t_2 := \left(a + z\right) - b\\ \mathbf{if}\;y \leq -3.5 \cdot 10^{+142}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{-166}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.55 \cdot 10^{-102}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, x \cdot z\right)}{t + x}\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+14}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (+ a z) (/ (* b y) (+ t (+ x y))))) (t_2 (- (+ a z) b)))
   (if (<= y -3.5e+142)
     t_2
     (if (<= y -2.1e-166)
       t_1
       (if (<= y 2.55e-102)
         (/ (fma a t (* x z)) (+ t x))
         (if (<= y 1.12e+14) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a + z) - ((b * y) / (t + (x + y)));
	double t_2 = (a + z) - b;
	double tmp;
	if (y <= -3.5e+142) {
		tmp = t_2;
	} else if (y <= -2.1e-166) {
		tmp = t_1;
	} else if (y <= 2.55e-102) {
		tmp = fma(a, t, (x * z)) / (t + x);
	} else if (y <= 1.12e+14) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a + z) - Float64(Float64(b * y) / Float64(t + Float64(x + y))))
	t_2 = Float64(Float64(a + z) - b)
	tmp = 0.0
	if (y <= -3.5e+142)
		tmp = t_2;
	elseif (y <= -2.1e-166)
		tmp = t_1;
	elseif (y <= 2.55e-102)
		tmp = Float64(fma(a, t, Float64(x * z)) / Float64(t + x));
	elseif (y <= 1.12e+14)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a + z), $MachinePrecision] - N[(N[(b * y), $MachinePrecision] / N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[y, -3.5e+142], t$95$2, If[LessEqual[y, -2.1e-166], t$95$1, If[LessEqual[y, 2.55e-102], N[(N[(a * t + N[(x * z), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.12e+14], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.49999999999999997e142 or 1.12e14 < y

   1. Initial program 60.7%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 55.6

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

   4. Applied rewrites 55.6%

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

   if -3.49999999999999997e142 < y < -2.0999999999999999e-166 or 2.55e-102 < y < 1.12e14

   1. Initial program 60.7%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 72.4%

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

   5. Taylor expanded in y around inf

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

      1. lower-+.f64 59.9

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

   7. Applied rewrites 59.9%

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

   if -2.0999999999999999e-166 < y < 2.55e-102

   1. Initial program 60.7%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 40.2

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

   4. Applied rewrites 40.2%

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

Alternative 15: 63.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + z\right) - b\\ \mathbf{if}\;y \leq -5 \cdot 10^{-162}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.55 \cdot 10^{-102}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, x \cdot z\right)}{t + x}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (+ a z) b)))
   (if (<= y -5e-162)
     t_1
     (if (<= y 2.55e-102) (/ (fma a t (* x z)) (+ t x)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a + z) - b;
	double tmp;
	if (y <= -5e-162) {
		tmp = t_1;
	} else if (y <= 2.55e-102) {
		tmp = fma(a, t, (x * z)) / (t + x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a + z) - b)
	tmp = 0.0
	if (y <= -5e-162)
		tmp = t_1;
	elseif (y <= 2.55e-102)
		tmp = Float64(fma(a, t, Float64(x * z)) / Float64(t + x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[y, -5e-162], t$95$1, If[LessEqual[y, 2.55e-102], N[(N[(a * t + N[(x * z), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.00000000000000014e-162 or 2.55e-102 < y

   1. Initial program 60.7%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 55.6

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

   4. Applied rewrites 55.6%

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

   if -5.00000000000000014e-162 < y < 2.55e-102

   1. Initial program 60.7%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 40.2

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

   4. Applied rewrites 40.2%

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

Alternative 16: 60.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z - a}{t}, x, a\right)\\ \mathbf{if}\;t \leq -2.15 \cdot 10^{+110}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{+160}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (/ (- z a) t) x a)))
   (if (<= t -2.15e+110) t_1 (if (<= t 2.3e+160) (- (+ a z) b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(((z - a) / t), x, a);
	double tmp;
	if (t <= -2.15e+110) {
		tmp = t_1;
	} else if (t <= 2.3e+160) {
		tmp = (a + z) - b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(Float64(z - a) / t), x, a)
	tmp = 0.0
	if (t <= -2.15e+110)
		tmp = t_1;
	elseif (t <= 2.3e+160)
		tmp = Float64(Float64(a + z) - b);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] * x + a), $MachinePrecision]}, If[LessEqual[t, -2.15e+110], t$95$1, If[LessEqual[t, 2.3e+160], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.15000000000000003e110 or 2.29999999999999987e160 < t

   1. Initial program 60.7%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 40.2

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

   4. Applied rewrites 40.2%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 29.1

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

   7. Applied rewrites 29.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 29.1

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

      6. lift--.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. sub-div N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 29.2

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

   9. Applied rewrites 29.2%

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

   if -2.15000000000000003e110 < t < 2.29999999999999987e160

   1. Initial program 60.7%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 55.6

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

   4. Applied rewrites 55.6%

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

Alternative 17: 57.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.4 \cdot 10^{+165}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (if (<= x 3.4e+165) (- (+ a z) b) z))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= 3.4e+165) {
		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 <= 3.4d+165) 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 <= 3.4e+165) {
		tmp = (a + z) - b;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= 3.4e+165:
		tmp = (a + z) - b
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= 3.4e+165)
		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 <= 3.4e+165)
		tmp = (a + z) - b;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, 3.4e+165], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if x < 3.40000000000000011e165

   1. Initial program 60.7%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 55.6

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

   4. Applied rewrites 55.6%

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

   if 3.40000000000000011e165 < x

   1. Initial program 60.7%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 32.6%

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

Alternative 18: 47.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -31500000:\\ \;\;\;\;a\\ \mathbf{elif}\;a \leq 8.5 \cdot 10^{+36}:\\ \;\;\;\;z - b\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -31500000.0) a (if (<= a 8.5e+36) (- z b) a)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -31500000.0) {
		tmp = a;
	} else if (a <= 8.5e+36) {
		tmp = z - b;
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-31500000.0d0)) then
        tmp = a
    else if (a <= 8.5d+36) then
        tmp = z - b
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -31500000.0) {
		tmp = a;
	} else if (a <= 8.5e+36) {
		tmp = z - b;
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -31500000.0:
		tmp = a
	elif a <= 8.5e+36:
		tmp = z - b
	else:
		tmp = a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -31500000.0)
		tmp = a;
	elseif (a <= 8.5e+36)
		tmp = Float64(z - b);
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -31500000.0)
		tmp = a;
	elseif (a <= 8.5e+36)
		tmp = z - b;
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -31500000.0], a, If[LessEqual[a, 8.5e+36], N[(z - b), $MachinePrecision], a]]

Derivation

1. Split input into 2 regimes

2. if a < -3.15e7 or 8.50000000000000014e36 < a

   1. Initial program 60.7%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 32.8%

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

   if -3.15e7 < a < 8.50000000000000014e36

   1. Initial program 60.7%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 55.6

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

   4. Applied rewrites 55.6%

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

   5. Taylor expanded in a around 0

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

      1. lower--.f64 37.2

         \[\leadsto z - b \]

   7. Applied rewrites 37.2%

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

Alternative 19: 44.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -15600000:\\ \;\;\;\;a\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{+18}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -15600000.0) a (if (<= a 4.5e+18) z a)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -15600000.0) {
		tmp = a;
	} else if (a <= 4.5e+18) {
		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 <= (-15600000.0d0)) then
        tmp = a
    else if (a <= 4.5d+18) 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 <= -15600000.0) {
		tmp = a;
	} else if (a <= 4.5e+18) {
		tmp = z;
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -15600000.0:
		tmp = a
	elif a <= 4.5e+18:
		tmp = z
	else:
		tmp = a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -15600000.0)
		tmp = a;
	elseif (a <= 4.5e+18)
		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 <= -15600000.0)
		tmp = a;
	elseif (a <= 4.5e+18)
		tmp = z;
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -15600000.0], a, If[LessEqual[a, 4.5e+18], z, a]]

Derivation

1. Split input into 2 regimes

2. if a < -1.56e7 or 4.5e18 < a

   1. Initial program 60.7%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 32.8%

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

   if -1.56e7 < a < 4.5e18

   1. Initial program 60.7%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 32.6%

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

Alternative 20: 32.8% 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 60.7%

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

2. Taylor expanded in t around inf

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

4. Applied rewrites 32.8%

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

Reproduce

herbie shell --seed 2025151 
(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)))