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

Percentage Accurate: 61.1% → 99.0%

Time: 5.5s

Alternatives: 14

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.1% 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: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ t_3 := \left(t + x\right) + y\\ t_4 := \mathsf{fma}\left(t + y, \frac{a}{t\_3}, \left(x + y\right) \cdot \frac{z}{t\_1} - b \cdot \frac{y}{t\_1}\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq 10^{+307}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t + y, z \cdot \left(y + x\right)\right)}{t\_3} - \frac{y}{t\_3} \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ x t) y))
        (t_2 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) t_1))
        (t_3 (+ (+ t x) y))
        (t_4
         (fma (+ t y) (/ a t_3) (- (* (+ x y) (/ z t_1)) (* b (/ y t_1))))))
   (if (<= t_2 (- INFINITY))
     t_4
     (if (<= t_2 1e+307)
       (- (/ (fma a (+ t y) (* z (+ y x))) t_3) (* (/ y t_3) b))
       t_4))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + t) + y;
	double t_2 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / t_1;
	double t_3 = (t + x) + y;
	double t_4 = fma((t + y), (a / t_3), (((x + y) * (z / t_1)) - (b * (y / t_1))));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_2 <= 1e+307) {
		tmp = (fma(a, (t + y), (z * (y + x))) / t_3) - ((y / t_3) * b);
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + t) + y)
	t_2 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / t_1)
	t_3 = Float64(Float64(t + x) + y)
	t_4 = fma(Float64(t + y), Float64(a / t_3), Float64(Float64(Float64(x + y) * Float64(z / t_1)) - Float64(b * Float64(y / t_1))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_2 <= 1e+307)
		tmp = Float64(Float64(fma(a, Float64(t + y), Float64(z * Float64(y + x))) / t_3) - Float64(Float64(y / t_3) * b));
	else
		tmp = t_4;
	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[(t + x), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t + y), $MachinePrecision] * N[(a / t$95$3), $MachinePrecision] + N[(N[(N[(x + y), $MachinePrecision] * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision] - N[(b * N[(y / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$4, If[LessEqual[t$95$2, 1e+307], N[(N[(N[(a * N[(t + y), $MachinePrecision] + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] - N[(N[(y / t$95$3), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -inf.0 or 9.99999999999999986e306 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 61.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. div-add N/A

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

      7. associate--l+ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. remove-double-neg N/A

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

      11. mult-flip-rev N/A

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

   3. Applied rewrites 68.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

   5. Applied rewrites 90.4%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 9.99999999999999986e306

   1. Initial program 61.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

   3. Applied rewrites 65.8%

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

Alternative 2: 90.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + x\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ t_3 := \frac{y}{y + t}\\ t_4 := \left(a + z \cdot t\_3\right) - b \cdot t\_3\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+266}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+296}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t + y, z \cdot \left(y + x\right)\right)}{t\_1} - \frac{y}{t\_1} \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ t x) y))
        (t_2 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))
        (t_3 (/ y (+ y t)))
        (t_4 (- (+ a (* z t_3)) (* b t_3))))
   (if (<= t_2 -4e+266)
     t_4
     (if (<= t_2 5e+296)
       (- (/ (fma a (+ t y) (* z (+ y x))) t_1) (* (/ y t_1) b))
       t_4))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + x) + y;
	double t_2 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
	double t_3 = y / (y + t);
	double t_4 = (a + (z * t_3)) - (b * t_3);
	double tmp;
	if (t_2 <= -4e+266) {
		tmp = t_4;
	} else if (t_2 <= 5e+296) {
		tmp = (fma(a, (t + y), (z * (y + x))) / t_1) - ((y / t_1) * b);
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t + x) + y)
	t_2 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / Float64(Float64(x + t) + y))
	t_3 = Float64(y / Float64(y + t))
	t_4 = Float64(Float64(a + Float64(z * t_3)) - Float64(b * t_3))
	tmp = 0.0
	if (t_2 <= -4e+266)
		tmp = t_4;
	elseif (t_2 <= 5e+296)
		tmp = Float64(Float64(fma(a, Float64(t + y), Float64(z * Float64(y + x))) / t_1) - Float64(Float64(y / t_1) * b));
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t + x), $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] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(y / N[(y + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(a + N[(z * t$95$3), $MachinePrecision]), $MachinePrecision] - N[(b * t$95$3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+266], t$95$4, If[LessEqual[t$95$2, 5e+296], N[(N[(N[(a * N[(t + y), $MachinePrecision] + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] - N[(N[(y / t$95$1), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -4.0000000000000001e266 or 5.0000000000000001e296 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 61.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 40.7

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

   4. Applied rewrites 40.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. add-to-fraction-rev N/A

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

      7. lower-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 68.2

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 68.2

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

   6. Applied rewrites 68.2%

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

   if -4.0000000000000001e266 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 5.0000000000000001e296

   1. Initial program 61.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

   3. Applied rewrites 65.8%

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

Alternative 3: 89.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ t_3 := \frac{y}{y + t}\\ t_4 := \left(a + z \cdot t\_3\right) - b \cdot t\_3\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+266}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+296}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y + x, z, a \cdot \left(t + y\right) - b \cdot y\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ x t) y))
        (t_2 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) t_1))
        (t_3 (/ y (+ y t)))
        (t_4 (- (+ a (* z t_3)) (* b t_3))))
   (if (<= t_2 -4e+266)
     t_4
     (if (<= t_2 5e+296)
       (/ (fma (+ y x) z (- (* a (+ t y)) (* b y))) t_1)
       t_4))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + t) + y;
	double t_2 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / t_1;
	double t_3 = y / (y + t);
	double t_4 = (a + (z * t_3)) - (b * t_3);
	double tmp;
	if (t_2 <= -4e+266) {
		tmp = t_4;
	} else if (t_2 <= 5e+296) {
		tmp = fma((y + x), z, ((a * (t + y)) - (b * y))) / t_1;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + t) + y)
	t_2 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / t_1)
	t_3 = Float64(y / Float64(y + t))
	t_4 = Float64(Float64(a + Float64(z * t_3)) - Float64(b * t_3))
	tmp = 0.0
	if (t_2 <= -4e+266)
		tmp = t_4;
	elseif (t_2 <= 5e+296)
		tmp = Float64(fma(Float64(y + x), z, Float64(Float64(a * Float64(t + y)) - Float64(b * y))) / t_1);
	else
		tmp = t_4;
	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[(y / N[(y + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(a + N[(z * t$95$3), $MachinePrecision]), $MachinePrecision] - N[(b * t$95$3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+266], t$95$4, If[LessEqual[t$95$2, 5e+296], N[(N[(N[(y + x), $MachinePrecision] * z + N[(N[(a * N[(t + y), $MachinePrecision]), $MachinePrecision] - N[(b * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$4]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -4.0000000000000001e266 or 5.0000000000000001e296 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 61.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 40.7

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

   4. Applied rewrites 40.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. add-to-fraction-rev N/A

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

      7. lower-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 68.2

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 68.2

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

   6. Applied rewrites 68.2%

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

   if -4.0000000000000001e266 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 5.0000000000000001e296

   1. Initial program 61.1%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. add-flip N/A

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

      5. sub-flip-reverse N/A

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

      6. lift-*.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. sub-negate-rev N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. remove-double-neg N/A

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

      14. lower--.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. remove-double-neg 61.3

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 61.3

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

   3. Applied rewrites 61.3%

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

Alternative 4: 78.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{y + t}\\ \mathbf{if}\;x \leq -3.8 \cdot 10^{+166}:\\ \;\;\;\;\frac{x + y}{\left(x + t\right) + y} \cdot z\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-22}:\\ \;\;\;\;\left(a + z \cdot t\_1\right) - b \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t + y, \frac{a}{\left(t + x\right) + y}, z - b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ y (+ y t))))
   (if (<= x -3.8e+166)
     (* (/ (+ x y) (+ (+ x t) y)) z)
     (if (<= x 2.5e-22)
       (- (+ a (* z t_1)) (* b t_1))
       (fma (+ t y) (/ a (+ (+ t x) y)) (- z 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 (x <= -3.8e+166) {
		tmp = ((x + y) / ((x + t) + y)) * z;
	} else if (x <= 2.5e-22) {
		tmp = (a + (z * t_1)) - (b * t_1);
	} else {
		tmp = fma((t + y), (a / ((t + x) + y)), (z - b));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y / Float64(y + t))
	tmp = 0.0
	if (x <= -3.8e+166)
		tmp = Float64(Float64(Float64(x + y) / Float64(Float64(x + t) + y)) * z);
	elseif (x <= 2.5e-22)
		tmp = Float64(Float64(a + Float64(z * t_1)) - Float64(b * t_1));
	else
		tmp = fma(Float64(t + y), Float64(a / Float64(Float64(t + x) + y)), Float64(z - 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[x, -3.8e+166], N[(N[(N[(x + y), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[x, 2.5e-22], N[(N[(a + N[(z * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(b * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(t + y), $MachinePrecision] * N[(a / N[(N[(t + x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + N[(z - b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.80000000000000007e166

   1. Initial program 61.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 25.4

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

   4. Applied rewrites 25.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l+ N/A

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 40.1

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 40.1

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

   6. Applied rewrites 40.1%

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

   if -3.80000000000000007e166 < x < 2.49999999999999977e-22

   1. Initial program 61.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 40.7

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

   4. Applied rewrites 40.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. add-to-fraction-rev N/A

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

      7. lower-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 68.2

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 68.2

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

   6. Applied rewrites 68.2%

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

   if 2.49999999999999977e-22 < x

   1. Initial program 61.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. div-add N/A

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

      7. associate--l+ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. remove-double-neg N/A

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

      11. mult-flip-rev N/A

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

   3. Applied rewrites 68.2%

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

   4. Taylor expanded in y around inf

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

      1. lower--.f64 62.8

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

   6. Applied rewrites 62.8%

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

Alternative 5: 75.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ t_2 := \mathsf{fma}\left(a, \frac{t + y}{\left(t + x\right) + y}, z - b\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+266}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+210}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t + y, z \cdot \left(x + y\right)\right)}{t + \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))
        (t_2 (fma a (/ (+ t y) (+ (+ t x) y)) (- z b))))
   (if (<= t_1 -4e+266)
     t_2
     (if (<= t_1 1e+210)
       (/ (fma a (+ t y) (* z (+ x y))) (+ t (+ x y)))
       t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
	double t_2 = fma(a, ((t + y) / ((t + x) + y)), (z - b));
	double tmp;
	if (t_1 <= -4e+266) {
		tmp = t_2;
	} else if (t_1 <= 1e+210) {
		tmp = fma(a, (t + y), (z * (x + y))) / (t + (x + y));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / Float64(Float64(x + t) + y))
	t_2 = fma(a, Float64(Float64(t + y) / Float64(Float64(t + x) + y)), Float64(z - b))
	tmp = 0.0
	if (t_1 <= -4e+266)
		tmp = t_2;
	elseif (t_1 <= 1e+210)
		tmp = Float64(fma(a, Float64(t + y), Float64(z * Float64(x + y))) / Float64(t + Float64(x + y)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[(N[(t + y), $MachinePrecision] / N[(N[(t + x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + N[(z - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+266], t$95$2, If[LessEqual[t$95$1, 1e+210], N[(N[(a * N[(t + y), $MachinePrecision] + N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -4.0000000000000001e266 or 9.99999999999999927e209 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 61.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. div-add N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 72.6%

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

   4. Taylor expanded in y around inf

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

      1. lower--.f64 64.9

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

   6. Applied rewrites 64.9%

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

   if -4.0000000000000001e266 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 9.99999999999999927e209

   1. Initial program 61.1%

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

   2. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 47.9

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

   4. Applied rewrites 47.9%

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

Alternative 6: 70.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a - \frac{y \cdot \left(b - z\right)}{y + t}\\ t_2 := a - \left(b - z\right)\\ \mathbf{if}\;y \leq -1.1 \cdot 10^{+101}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{-165}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-176}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, x \cdot z\right)}{t + x}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+103}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- a (/ (* y (- b z)) (+ y t)))) (t_2 (- a (- b z))))
   (if (<= y -1.1e+101)
     t_2
     (if (<= y -2.1e-165)
       t_1
       (if (<= y 9.5e-176)
         (/ (fma a t (* x z)) (+ t x))
         (if (<= y 1.3e+103) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a - ((y * (b - z)) / (y + t));
	double t_2 = a - (b - z);
	double tmp;
	if (y <= -1.1e+101) {
		tmp = t_2;
	} else if (y <= -2.1e-165) {
		tmp = t_1;
	} else if (y <= 9.5e-176) {
		tmp = fma(a, t, (x * z)) / (t + x);
	} else if (y <= 1.3e+103) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a - Float64(Float64(y * Float64(b - z)) / Float64(y + t)))
	t_2 = Float64(a - Float64(b - z))
	tmp = 0.0
	if (y <= -1.1e+101)
		tmp = t_2;
	elseif (y <= -2.1e-165)
		tmp = t_1;
	elseif (y <= 9.5e-176)
		tmp = Float64(fma(a, t, Float64(x * z)) / Float64(t + x));
	elseif (y <= 1.3e+103)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a - N[(N[(y * N[(b - z), $MachinePrecision]), $MachinePrecision] / N[(y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a - N[(b - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.1e+101], t$95$2, If[LessEqual[y, -2.1e-165], t$95$1, If[LessEqual[y, 9.5e-176], N[(N[(a * t + N[(x * z), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3e+103], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.1e101 or 1.3000000000000001e103 < y

   1. Initial program 61.1%

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

   2. 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 56.7

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

   4. Applied rewrites 56.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. add-flip N/A

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

      5. sub-negate-rev N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 56.7

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

   6. Applied rewrites 56.7%

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

   if -1.1e101 < y < -2.09999999999999995e-165 or 9.5e-176 < y < 1.3000000000000001e103

   1. Initial program 61.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 40.7

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

   4. Applied rewrites 40.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. add-flip N/A

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

      5. associate--l- N/A

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

      6. sub-to-fraction-rev N/A

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

      7. add-flip-rev N/A

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

      8. sub-flip N/A

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

      9. distribute-neg-in N/A

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

      10. sub-flip N/A

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

      11. sub-negate-rev N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

   6. Applied rewrites 51.2%

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

   if -2.09999999999999995e-165 < y < 9.5e-176

   1. Initial program 61.1%

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

   2. 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 40.6

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

   4. Applied rewrites 40.6%

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

Alternative 7: 68.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ t_2 := \mathsf{fma}\left(a, \frac{t + y}{\left(t + x\right) + y}, z - b\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+50}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+182}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, x \cdot z\right)}{t + x}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))
        (t_2 (fma a (/ (+ t y) (+ (+ t x) y)) (- z b))))
   (if (<= t_1 -2e+50)
     t_2
     (if (<= t_1 5e+182) (/ (fma a t (* x z)) (+ t x)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
	double t_2 = fma(a, ((t + y) / ((t + x) + y)), (z - b));
	double tmp;
	if (t_1 <= -2e+50) {
		tmp = t_2;
	} else if (t_1 <= 5e+182) {
		tmp = fma(a, t, (x * z)) / (t + x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / Float64(Float64(x + t) + y))
	t_2 = fma(a, Float64(Float64(t + y) / Float64(Float64(t + x) + y)), Float64(z - b))
	tmp = 0.0
	if (t_1 <= -2e+50)
		tmp = t_2;
	elseif (t_1 <= 5e+182)
		tmp = Float64(fma(a, t, Float64(x * z)) / Float64(t + x));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[(N[(t + y), $MachinePrecision] / N[(N[(t + x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + N[(z - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+50], t$95$2, If[LessEqual[t$95$1, 5e+182], N[(N[(a * t + N[(x * z), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -2.0000000000000002e50 or 4.99999999999999973e182 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 61.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. div-add N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 72.6%

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

   4. Taylor expanded in y around inf

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

      1. lower--.f64 64.9

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

   6. Applied rewrites 64.9%

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

   if -2.0000000000000002e50 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4.99999999999999973e182

   1. Initial program 61.1%

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

   2. 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 40.6

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

   4. Applied rewrites 40.6%

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

Alternative 8: 66.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ t_2 := a - \left(b - z\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+145}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+182}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, x \cdot z\right)}{t + x}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))
        (t_2 (- a (- b z))))
   (if (<= t_1 -2e+145)
     t_2
     (if (<= t_1 5e+182) (/ (fma a t (* x z)) (+ t x)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
	double t_2 = a - (b - z);
	double tmp;
	if (t_1 <= -2e+145) {
		tmp = t_2;
	} else if (t_1 <= 5e+182) {
		tmp = fma(a, t, (x * z)) / (t + x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / Float64(Float64(x + t) + y))
	t_2 = Float64(a - Float64(b - z))
	tmp = 0.0
	if (t_1 <= -2e+145)
		tmp = t_2;
	elseif (t_1 <= 5e+182)
		tmp = Float64(fma(a, t, Float64(x * z)) / Float64(t + x));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a - N[(b - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+145], t$95$2, If[LessEqual[t$95$1, 5e+182], N[(N[(a * t + N[(x * z), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -2e145 or 4.99999999999999973e182 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 61.1%

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

   2. 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 56.7

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

   4. Applied rewrites 56.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. add-flip N/A

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

      5. sub-negate-rev N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 56.7

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

   6. Applied rewrites 56.7%

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

   if -2e145 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4.99999999999999973e182

   1. Initial program 61.1%

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

   2. 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 40.6

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

   4. Applied rewrites 40.6%

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

Alternative 9: 63.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.1 \cdot 10^{+99}:\\ \;\;\;\;a - \frac{b - z}{t} \cdot y\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{+132}:\\ \;\;\;\;a - \left(b - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - b}{t}, y, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -3.1e+99)
   (- a (* (/ (- b z) t) y))
   (if (<= t 7.4e+132) (- a (- b z)) (fma (/ (- z b) t) y a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -3.1e+99) {
		tmp = a - (((b - z) / t) * y);
	} else if (t <= 7.4e+132) {
		tmp = a - (b - z);
	} else {
		tmp = fma(((z - b) / t), y, a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -3.1e+99)
		tmp = Float64(a - Float64(Float64(Float64(b - z) / t) * y));
	elseif (t <= 7.4e+132)
		tmp = Float64(a - Float64(b - z));
	else
		tmp = fma(Float64(Float64(z - b) / t), y, a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -3.1e+99], N[(a - N[(N[(N[(b - z), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7.4e+132], N[(a - N[(b - z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z - b), $MachinePrecision] / t), $MachinePrecision] * y + a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.1000000000000001e99

   1. Initial program 61.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 40.7

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

   4. Applied rewrites 40.7%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 32.8

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

   7. Applied rewrites 32.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lower--.f64 N/A

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

      5. distribute-lft-neg-out N/A

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

      6. *-commutative N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. sub-div N/A

          \[\leadsto a - \frac{b - z}{t} \cdot y \]

      14. lower-/.f64 N/A

          \[\leadsto a - \frac{b - z}{t} \cdot y \]

      15. lower--.f64 32.8

          \[\leadsto a - \frac{b - z}{t} \cdot y \]

   9. Applied rewrites 32.8%

      \[\leadsto a - \frac{b - z}{t} \cdot \color{blue}{y} \]

   if -3.1000000000000001e99 < t < 7.40000000000000022e132

   1. Initial program 61.1%

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

   2. 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 56.7

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

   4. Applied rewrites 56.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. add-flip N/A

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

      5. sub-negate-rev N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 56.7

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

   6. Applied rewrites 56.7%

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

   if 7.40000000000000022e132 < t

   1. Initial program 61.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 40.7

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

   4. Applied rewrites 40.7%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 32.8

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

   7. Applied rewrites 32.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 32.8

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

      6. lift--.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. sub-div N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 32.8

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

   9. Applied rewrites 32.8%

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

Alternative 10: 63.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z - b}{t}, y, a\right)\\ \mathbf{if}\;t \leq -3.1 \cdot 10^{+99}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{+132}:\\ \;\;\;\;a - \left(b - z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (/ (- z b) t) y a)))
   (if (<= t -3.1e+99) t_1 (if (<= t 7.4e+132) (- a (- b z)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(((z - b) / t), y, a);
	double tmp;
	if (t <= -3.1e+99) {
		tmp = t_1;
	} else if (t <= 7.4e+132) {
		tmp = a - (b - z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(Float64(z - b) / t), y, a)
	tmp = 0.0
	if (t <= -3.1e+99)
		tmp = t_1;
	elseif (t <= 7.4e+132)
		tmp = Float64(a - Float64(b - z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(z - b), $MachinePrecision] / t), $MachinePrecision] * y + a), $MachinePrecision]}, If[LessEqual[t, -3.1e+99], t$95$1, If[LessEqual[t, 7.4e+132], N[(a - N[(b - z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.1000000000000001e99 or 7.40000000000000022e132 < t

   1. Initial program 61.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 40.7

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

   4. Applied rewrites 40.7%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 32.8

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

   7. Applied rewrites 32.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 32.8

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

      6. lift--.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. sub-div N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 32.8

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

   9. Applied rewrites 32.8%

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

   if -3.1000000000000001e99 < t < 7.40000000000000022e132

   1. Initial program 61.1%

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

   2. 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 56.7

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

   4. Applied rewrites 56.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. add-flip N/A

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

      5. sub-negate-rev N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 56.7

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

   6. Applied rewrites 56.7%

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

Alternative 11: 58.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{+181}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;a - \left(b - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x -4.3e+181) z (- a (- b z))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -4.3e+181) {
		tmp = z;
	} else {
		tmp = a - (b - 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 <= (-4.3d+181)) then
        tmp = z
    else
        tmp = a - (b - 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 <= -4.3e+181) {
		tmp = z;
	} else {
		tmp = a - (b - z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= -4.3e+181:
		tmp = z
	else:
		tmp = a - (b - z)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= -4.3e+181)
		tmp = z;
	else
		tmp = Float64(a - Float64(b - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= -4.3e+181)
		tmp = z;
	else
		tmp = a - (b - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, -4.3e+181], z, N[(a - N[(b - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.29999999999999972e181

   1. Initial program 61.1%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 32.8%

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

   if -4.29999999999999972e181 < x

   1. Initial program 61.1%

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

   2. 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 56.7

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

   4. Applied rewrites 56.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. add-flip N/A

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

      5. sub-negate-rev N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 56.7

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

   6. Applied rewrites 56.7%

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

Alternative 12: 48.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+59}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+135}:\\ \;\;\;\;a - b\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.4e+59) z (if (<= z 6e+135) (- a b) z)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.4e+59) {
		tmp = z;
	} else if (z <= 6e+135) {
		tmp = a - 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 (z <= (-1.4d+59)) then
        tmp = z
    else if (z <= 6d+135) then
        tmp = a - 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 (z <= -1.4e+59) {
		tmp = z;
	} else if (z <= 6e+135) {
		tmp = a - b;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -1.4e+59:
		tmp = z
	elif z <= 6e+135:
		tmp = a - b
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.4e+59)
		tmp = z;
	elseif (z <= 6e+135)
		tmp = Float64(a - b);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -1.4e+59)
		tmp = z;
	elseif (z <= 6e+135)
		tmp = a - b;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.4e+59], z, If[LessEqual[z, 6e+135], N[(a - b), $MachinePrecision], z]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3999999999999999e59 or 6.0000000000000001e135 < z

   1. Initial program 61.1%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 32.8%

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

   if -1.3999999999999999e59 < z < 6.0000000000000001e135

   1. Initial program 61.1%

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

   2. 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 56.7

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

   4. Applied rewrites 56.7%

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

   5. Taylor expanded in z around 0

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

      1. lower--.f64 38.1

         \[\leadsto a - b \]

   7. Applied rewrites 38.1%

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

Alternative 13: 44.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+59}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{+58}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.4e+59) z (if (<= z 9.8e+58) a z)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.4e+59) {
		tmp = z;
	} else if (z <= 9.8e+58) {
		tmp = a;
	} else {
		tmp = z;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (z <= (-1.4d+59)) then
        tmp = z
    else if (z <= 9.8d+58) then
        tmp = a
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.4e+59) {
		tmp = z;
	} else if (z <= 9.8e+58) {
		tmp = a;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -1.4e+59:
		tmp = z
	elif z <= 9.8e+58:
		tmp = a
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.4e+59)
		tmp = z;
	elseif (z <= 9.8e+58)
		tmp = a;
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -1.4e+59)
		tmp = z;
	elseif (z <= 9.8e+58)
		tmp = a;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.4e+59], z, If[LessEqual[z, 9.8e+58], a, z]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3999999999999999e59 or 9.80000000000000037e58 < z

   1. Initial program 61.1%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 32.8%

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

   if -1.3999999999999999e59 < z < 9.80000000000000037e58

   1. Initial program 61.1%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 32.5%

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

Alternative 14: 32.5% 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 61.1%

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

2. Taylor expanded in t around inf

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

4. Applied rewrites 32.5%

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

Reproduce

herbie shell --seed 2025156 
(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)))