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

Percentage Accurate: 60.2% → 97.7%

Time: 6.5s

Alternatives: 17

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 60.2% 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: 97.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t + \left(x + y\right)\\ t_2 := \frac{x + y}{t\_1}\\ t_3 := \frac{t + y}{\left(y + x\right) + t}\\ \mathbf{if}\;b \leq -6 \cdot 10^{+26} \lor \neg \left(b \leq 7.6 \cdot 10^{+85}\right):\\ \;\;\;\;\mathsf{fma}\left(t\_3, a, b \cdot \mathsf{fma}\left(-1, \frac{y}{t\_1}, \frac{z}{b} \cdot t\_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_3, a, \mathsf{fma}\left(-1, \frac{b \cdot y}{t\_1}, z \cdot t\_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ t (+ x y)))
        (t_2 (/ (+ x y) t_1))
        (t_3 (/ (+ t y) (+ (+ y x) t))))
   (if (or (<= b -6e+26) (not (<= b 7.6e+85)))
     (fma t_3 a (* b (fma -1.0 (/ y t_1) (* (/ z b) t_2))))
     (fma t_3 a (fma -1.0 (/ (* b y) t_1) (* z t_2))))))

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) / t_1;
	double t_3 = (t + y) / ((y + x) + t);
	double tmp;
	if ((b <= -6e+26) || !(b <= 7.6e+85)) {
		tmp = fma(t_3, a, (b * fma(-1.0, (y / t_1), ((z / b) * t_2))));
	} else {
		tmp = fma(t_3, a, fma(-1.0, ((b * y) / t_1), (z * t_2)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t + Float64(x + y))
	t_2 = Float64(Float64(x + y) / t_1)
	t_3 = Float64(Float64(t + y) / Float64(Float64(y + x) + t))
	tmp = 0.0
	if ((b <= -6e+26) || !(b <= 7.6e+85))
		tmp = fma(t_3, a, Float64(b * fma(-1.0, Float64(y / t_1), Float64(Float64(z / b) * t_2))));
	else
		tmp = fma(t_3, a, fma(-1.0, Float64(Float64(b * y) / t_1), Float64(z * t_2)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + y), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t + y), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[b, -6e+26], N[Not[LessEqual[b, 7.6e+85]], $MachinePrecision]], N[(t$95$3 * a + N[(b * N[(-1.0 * N[(y / t$95$1), $MachinePrecision] + N[(N[(z / b), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 * a + N[(-1.0 * N[(N[(b * y), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(z * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if b < -5.99999999999999994e26 or 7.59999999999999984e85 < b

   1. Initial program 51.4%

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

   4. Applied rewrites 58.3%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. times-frac N/A

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

      7. div-add-rev N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. div-add-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-+.f64 96.1

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

   7. Applied rewrites 96.1%

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

   if -5.99999999999999994e26 < b < 7.59999999999999984e85

   1. Initial program 64.7%

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

   4. Applied rewrites 78.3%

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

   5. Taylor expanded in z around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 99.7

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

   7. Applied rewrites 99.7%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+26} \lor \neg \left(b \leq 7.6 \cdot 10^{+85}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{t + y}{\left(y + x\right) + t}, a, b \cdot \mathsf{fma}\left(-1, \frac{y}{t + \left(x + y\right)}, \frac{z}{b} \cdot \frac{x + y}{t + \left(x + y\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t + y}{\left(y + x\right) + t}, a, \mathsf{fma}\left(-1, \frac{b \cdot y}{t + \left(x + y\right)}, z \cdot \frac{x + y}{t + \left(x + y\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ t_2 := \left(y + x\right) + t\\ t_3 := \frac{t + y}{t\_2}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(t\_3, a, z \cdot \frac{x + y}{t + \left(x + y\right)}\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+280}:\\ \;\;\;\;\mathsf{fma}\left(t\_3, a, \frac{\mathsf{fma}\left(y + x, z, \left(-b\right) \cdot y\right)}{t\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))
        (t_2 (+ (+ y x) t))
        (t_3 (/ (+ t y) t_2)))
   (if (<= t_1 (- INFINITY))
     (fma t_3 a (* z (/ (+ x y) (+ t (+ x y)))))
     (if (<= t_1 5e+280)
       (fma t_3 a (/ (fma (+ y x) z (* (- b) y)) t_2))
       (- (+ a z) b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
	double t_2 = (y + x) + t;
	double t_3 = (t + y) / t_2;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(t_3, a, (z * ((x + y) / (t + (x + y)))));
	} else if (t_1 <= 5e+280) {
		tmp = fma(t_3, a, (fma((y + x), z, (-b * y)) / t_2));
	} else {
		tmp = (a + z) - b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / Float64(Float64(x + t) + y))
	t_2 = Float64(Float64(y + x) + t)
	t_3 = Float64(Float64(t + y) / t_2)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(t_3, a, Float64(z * Float64(Float64(x + y) / Float64(t + Float64(x + y)))));
	elseif (t_1 <= 5e+280)
		tmp = fma(t_3, a, Float64(fma(Float64(y + x), z, Float64(Float64(-b) * y)) / t_2));
	else
		tmp = Float64(Float64(a + z) - b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t + y), $MachinePrecision] / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(t$95$3 * a + N[(z * N[(N[(x + y), $MachinePrecision] / N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+280], N[(t$95$3 * a + N[(N[(N[(y + x), $MachinePrecision] * z + N[((-b) * y), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 6.9%

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

   4. Applied rewrites 34.6%

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

   5. Taylor expanded in z around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 78.9

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

   7. Applied rewrites 78.9%

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

   8. Taylor expanded in z around inf

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

      1. div-add-rev N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 82.3

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

   10. Applied rewrites 82.3%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 5.0000000000000002e280

   1. Initial program 99.7%

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

   4. Applied rewrites 99.8%

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

   if 5.0000000000000002e280 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 6.2%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 79.8

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

   5. Applied rewrites 79.8%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{t + y}{\left(y + x\right) + t}, a, z \cdot \frac{x + y}{t + \left(x + y\right)}\right)\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq 5 \cdot 10^{+280}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t + y}{\left(y + x\right) + t}, a, \frac{\mathsf{fma}\left(y + x, z, \left(-b\right) \cdot y\right)}{\left(y + x\right) + t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ t_2 := \left(a + z\right) - b\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+119}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-42}:\\ \;\;\;\;\frac{t + y}{t + \left(x + y\right)} \cdot a\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+127}:\\ \;\;\;\;a + z\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))
        (t_2 (- (+ a z) b)))
   (if (<= t_1 -2e+119)
     t_2
     (if (<= t_1 -2e-42)
       (* (/ (+ t y) (+ t (+ x y))) a)
       (if (<= t_1 4e+127) (+ a z) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
	double t_2 = (a + z) - b;
	double tmp;
	if (t_1 <= -2e+119) {
		tmp = t_2;
	} else if (t_1 <= -2e-42) {
		tmp = ((t + y) / (t + (x + y))) * a;
	} else if (t_1 <= 4e+127) {
		tmp = a + z;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y)
    t_2 = (a + z) - b
    if (t_1 <= (-2d+119)) then
        tmp = t_2
    else if (t_1 <= (-2d-42)) then
        tmp = ((t + y) / (t + (x + y))) * a
    else if (t_1 <= 4d+127) then
        tmp = a + z
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
	double t_2 = (a + z) - b;
	double tmp;
	if (t_1 <= -2e+119) {
		tmp = t_2;
	} else if (t_1 <= -2e-42) {
		tmp = ((t + y) / (t + (x + y))) * a;
	} else if (t_1 <= 4e+127) {
		tmp = a + z;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y)
	t_2 = (a + z) - b
	tmp = 0
	if t_1 <= -2e+119:
		tmp = t_2
	elif t_1 <= -2e-42:
		tmp = ((t + y) / (t + (x + y))) * a
	elif t_1 <= 4e+127:
		tmp = a + z
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / Float64(Float64(x + t) + y))
	t_2 = Float64(Float64(a + z) - b)
	tmp = 0.0
	if (t_1 <= -2e+119)
		tmp = t_2;
	elseif (t_1 <= -2e-42)
		tmp = Float64(Float64(Float64(t + y) / Float64(t + Float64(x + y))) * a);
	elseif (t_1 <= 4e+127)
		tmp = Float64(a + z);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
	t_2 = (a + z) - b;
	tmp = 0.0;
	if (t_1 <= -2e+119)
		tmp = t_2;
	elseif (t_1 <= -2e-42)
		tmp = ((t + y) / (t + (x + y))) * a;
	elseif (t_1 <= 4e+127)
		tmp = a + z;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+119], t$95$2, If[LessEqual[t$95$1, -2e-42], N[(N[(N[(t + y), $MachinePrecision] / N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[t$95$1, 4e+127], N[(a + z), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -1.99999999999999989e119 or 3.99999999999999982e127 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 33.8%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 80.3

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

   5. Applied rewrites 80.3%

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

   if -1.99999999999999989e119 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -2.00000000000000008e-42

   1. Initial program 99.7%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 90.0%

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

   6. Taylor expanded in a around inf

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

      1. div-add-rev N/A

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

      2. lower-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 59.2

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

   8. Applied rewrites 59.2%

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

   if -2.00000000000000008e-42 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 3.99999999999999982e127

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 34.4

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

   5. Applied rewrites 34.4%

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

   6. Taylor expanded in b around 0

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

      1. lift-+.f64 54.7

         \[\leadsto a + z \]

   8. Applied rewrites 54.7%

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

Alternative 4: 88.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+305} \lor \neg \left(t\_2 \leq 5 \cdot 10^{+280}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y + x, z, \mathsf{fma}\left(t + y, a, \left(-b\right) \cdot y\right)\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ x t) y))
        (t_2 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) t_1)))
   (if (or (<= t_2 -2e+305) (not (<= t_2 5e+280)))
     (- (+ a z) b)
     (/ (fma (+ y x) z (fma (+ t y) a (* (- b) y))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + t) + y;
	double t_2 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / t_1;
	double tmp;
	if ((t_2 <= -2e+305) || !(t_2 <= 5e+280)) {
		tmp = (a + z) - b;
	} else {
		tmp = fma((y + x), z, fma((t + y), a, (-b * y))) / t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + t) + y)
	t_2 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / t_1)
	tmp = 0.0
	if ((t_2 <= -2e+305) || !(t_2 <= 5e+280))
		tmp = Float64(Float64(a + z) - b);
	else
		tmp = Float64(fma(Float64(y + x), z, fma(Float64(t + y), a, Float64(Float64(-b) * y))) / t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[Or[LessEqual[t$95$2, -2e+305], N[Not[LessEqual[t$95$2, 5e+280]], $MachinePrecision]], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision], N[(N[(N[(y + x), $MachinePrecision] * z + N[(N[(t + y), $MachinePrecision] * a + N[((-b) * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -1.9999999999999999e305 or 5.0000000000000002e280 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 7.3%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 79.8

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

   5. Applied rewrites 79.8%

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

   if -1.9999999999999999e305 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 5.0000000000000002e280

   1. Initial program 99.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate--l+ N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

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

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

      15. *-commutative N/A

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

      16. mul-1-neg N/A

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

      17. associate-*r* N/A

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

      18. mul-1-neg N/A

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

      19. lower-fma.f64 N/A

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

      20. lift-+.f64 N/A

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

      21. mul-1-neg N/A

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

      22. associate-*r* N/A

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

      23. lower-*.f64 N/A

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

      24. mul-1-neg N/A

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

      25. lower-neg.f64 99.7

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

   4. Applied rewrites 99.7%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq -2 \cdot 10^{+305} \lor \neg \left(\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq 5 \cdot 10^{+280}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y + x, z, \mathsf{fma}\left(t + y, a, \left(-b\right) \cdot y\right)\right)}{\left(x + t\right) + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+305}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t + y}{\left(y + x\right) + t}, a, z \cdot \frac{x + y}{t + \left(x + y\right)}\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+280}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y + x, z, \mathsf{fma}\left(t + y, a, \left(-b\right) \cdot y\right)\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ x t) y))
        (t_2 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) t_1)))
   (if (<= t_2 -2e+305)
     (fma (/ (+ t y) (+ (+ y x) t)) a (* z (/ (+ x y) (+ t (+ x y)))))
     (if (<= t_2 5e+280)
       (/ (fma (+ y x) z (fma (+ t y) a (* (- b) y))) t_1)
       (- (+ a z) b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + t) + y;
	double t_2 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / t_1;
	double tmp;
	if (t_2 <= -2e+305) {
		tmp = fma(((t + y) / ((y + x) + t)), a, (z * ((x + y) / (t + (x + y)))));
	} else if (t_2 <= 5e+280) {
		tmp = fma((y + x), z, fma((t + y), a, (-b * y))) / t_1;
	} else {
		tmp = (a + z) - b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + t) + y)
	t_2 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / t_1)
	tmp = 0.0
	if (t_2 <= -2e+305)
		tmp = fma(Float64(Float64(t + y) / Float64(Float64(y + x) + t)), a, Float64(z * Float64(Float64(x + y) / Float64(t + Float64(x + y)))));
	elseif (t_2 <= 5e+280)
		tmp = Float64(fma(Float64(y + x), z, fma(Float64(t + y), a, Float64(Float64(-b) * y))) / t_1);
	else
		tmp = Float64(Float64(a + z) - b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+305], N[(N[(N[(t + y), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] * a + N[(z * N[(N[(x + y), $MachinePrecision] / N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+280], N[(N[(N[(y + x), $MachinePrecision] * z + N[(N[(t + y), $MachinePrecision] * a + N[((-b) * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -1.9999999999999999e305

   1. Initial program 8.9%

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

   4. Applied rewrites 36.0%

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

   5. Taylor expanded in z around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 79.4

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

   7. Applied rewrites 79.4%

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

   8. Taylor expanded in z around inf

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

      1. div-add-rev N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 82.7

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

   10. Applied rewrites 82.7%

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

   if -1.9999999999999999e305 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 5.0000000000000002e280

   1. Initial program 99.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate--l+ N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

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

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

      15. *-commutative N/A

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

      16. mul-1-neg N/A

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

      17. associate-*r* N/A

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

      18. mul-1-neg N/A

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

      19. lower-fma.f64 N/A

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

      20. lift-+.f64 N/A

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

      21. mul-1-neg N/A

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

      22. associate-*r* N/A

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

      23. lower-*.f64 N/A

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

      24. mul-1-neg N/A

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

      25. lower-neg.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if 5.0000000000000002e280 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 6.2%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 79.8

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

   5. Applied rewrites 79.8%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq -2 \cdot 10^{+305}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t + y}{\left(y + x\right) + t}, a, z \cdot \frac{x + y}{t + \left(x + y\right)}\right)\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq 5 \cdot 10^{+280}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y + x, z, \mathsf{fma}\left(t + y, a, \left(-b\right) \cdot y\right)\right)}{\left(x + t\right) + y}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ t_3 := \left(a + z\right) - b\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+305} \lor \neg \left(t\_2 \leq 5 \cdot 10^{+280}\right):\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, \mathsf{fma}\left(t\_3, y, z \cdot x\right)\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ x t) y))
        (t_2 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) t_1))
        (t_3 (- (+ a z) b)))
   (if (or (<= t_2 -2e+305) (not (<= t_2 5e+280)))
     t_3
     (/ (fma a t (fma t_3 y (* z x))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + t) + y;
	double t_2 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / t_1;
	double t_3 = (a + z) - b;
	double tmp;
	if ((t_2 <= -2e+305) || !(t_2 <= 5e+280)) {
		tmp = t_3;
	} else {
		tmp = fma(a, t, fma(t_3, y, (z * x))) / t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + t) + y)
	t_2 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / t_1)
	t_3 = Float64(Float64(a + z) - b)
	tmp = 0.0
	if ((t_2 <= -2e+305) || !(t_2 <= 5e+280))
		tmp = t_3;
	else
		tmp = Float64(fma(a, t, fma(t_3, y, Float64(z * x))) / t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, If[Or[LessEqual[t$95$2, -2e+305], N[Not[LessEqual[t$95$2, 5e+280]], $MachinePrecision]], t$95$3, N[(N[(a * t + N[(t$95$3 * y + N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -1.9999999999999999e305 or 5.0000000000000002e280 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 7.3%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 79.8

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

   5. Applied rewrites 79.8%

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

   if -1.9999999999999999e305 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 5.0000000000000002e280

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq -2 \cdot 10^{+305} \lor \neg \left(\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq 5 \cdot 10^{+280}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, \mathsf{fma}\left(\left(a + z\right) - b, y, z \cdot x\right)\right)}{\left(x + t\right) + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+167} \lor \neg \left(t\_2 \leq 5 \cdot 10^{+232}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t + y, a, \left(y + x\right) \cdot z\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ x t) y))
        (t_2 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) t_1)))
   (if (or (<= t_2 -5e+167) (not (<= t_2 5e+232)))
     (- (+ a z) b)
     (/ (fma (+ t y) a (* (+ y x) z)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + t) + y;
	double t_2 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / t_1;
	double tmp;
	if ((t_2 <= -5e+167) || !(t_2 <= 5e+232)) {
		tmp = (a + z) - b;
	} else {
		tmp = fma((t + y), a, ((y + x) * z)) / t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + t) + y)
	t_2 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / t_1)
	tmp = 0.0
	if ((t_2 <= -5e+167) || !(t_2 <= 5e+232))
		tmp = Float64(Float64(a + z) - b);
	else
		tmp = Float64(fma(Float64(t + y), a, Float64(Float64(y + x) * z)) / t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[Or[LessEqual[t$95$2, -5e+167], N[Not[LessEqual[t$95$2, 5e+232]], $MachinePrecision]], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision], N[(N[(N[(t + y), $MachinePrecision] * a + N[(N[(y + x), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -4.9999999999999997e167 or 4.99999999999999987e232 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 18.9%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 81.1

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

   5. Applied rewrites 81.1%

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

   if -4.9999999999999997e167 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4.99999999999999987e232

   1. Initial program 99.7%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 83.1

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

   5. Applied rewrites 83.1%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq -5 \cdot 10^{+167} \lor \neg \left(\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq 5 \cdot 10^{+232}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t + y, a, \left(y + x\right) \cdot z\right)}{\left(x + t\right) + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+119} \lor \neg \left(t\_1 \leq 4 \cdot 10^{+127}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, z \cdot x\right)}{t + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y))))
   (if (or (<= t_1 -2e+119) (not (<= t_1 4e+127)))
     (- (+ a z) b)
     (/ (fma a t (* z x)) (+ t x)))))

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 tmp;
	if ((t_1 <= -2e+119) || !(t_1 <= 4e+127)) {
		tmp = (a + z) - b;
	} else {
		tmp = fma(a, t, (z * x)) / (t + x);
	}
	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))
	tmp = 0.0
	if ((t_1 <= -2e+119) || !(t_1 <= 4e+127))
		tmp = Float64(Float64(a + z) - b);
	else
		tmp = Float64(fma(a, t, Float64(z * x)) / Float64(t + x));
	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]}, If[Or[LessEqual[t$95$1, -2e+119], N[Not[LessEqual[t$95$1, 4e+127]], $MachinePrecision]], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision], N[(N[(a * t + N[(z * x), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -1.99999999999999989e119 or 3.99999999999999982e127 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 33.8%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 80.3

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

   5. Applied rewrites 80.3%

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

   if -1.99999999999999989e119 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 3.99999999999999982e127

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 66.6

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

   5. Applied rewrites 66.6%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq -2 \cdot 10^{+119} \lor \neg \left(\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq 4 \cdot 10^{+127}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, z \cdot x\right)}{t + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+121} \lor \neg \left(t\_1 \leq 4 \cdot 10^{+127}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;a + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y))))
   (if (or (<= t_1 -2e+121) (not (<= t_1 4e+127))) (- (+ a z) b) (+ a z))))

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 tmp;
	if ((t_1 <= -2e+121) || !(t_1 <= 4e+127)) {
		tmp = (a + z) - b;
	} else {
		tmp = a + z;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y)
    if ((t_1 <= (-2d+121)) .or. (.not. (t_1 <= 4d+127))) then
        tmp = (a + z) - b
    else
        tmp = a + z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
	double tmp;
	if ((t_1 <= -2e+121) || !(t_1 <= 4e+127)) {
		tmp = (a + z) - b;
	} else {
		tmp = a + z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y)
	tmp = 0
	if (t_1 <= -2e+121) or not (t_1 <= 4e+127):
		tmp = (a + z) - b
	else:
		tmp = a + z
	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))
	tmp = 0.0
	if ((t_1 <= -2e+121) || !(t_1 <= 4e+127))
		tmp = Float64(Float64(a + z) - b);
	else
		tmp = Float64(a + z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
	tmp = 0.0;
	if ((t_1 <= -2e+121) || ~((t_1 <= 4e+127)))
		tmp = (a + z) - b;
	else
		tmp = a + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+121], N[Not[LessEqual[t$95$1, 4e+127]], $MachinePrecision]], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision], N[(a + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -2.00000000000000007e121 or 3.99999999999999982e127 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 33.4%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 80.2

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

   5. Applied rewrites 80.2%

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

   if -2.00000000000000007e121 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 3.99999999999999982e127

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 35.4

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

   5. Applied rewrites 35.4%

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

   6. Taylor expanded in b around 0

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

      1. lift-+.f64 50.2

         \[\leadsto a + z \]

   8. Applied rewrites 50.2%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq -2 \cdot 10^{+121} \lor \neg \left(\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq 4 \cdot 10^{+127}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;a + z\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 92.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t + \left(x + y\right)\\ t_2 := \frac{x + y}{t\_1}\\ \mathbf{if}\;y \leq -2.1 \cdot 10^{+106}:\\ \;\;\;\;\mathsf{fma}\left(1, a, b \cdot \mathsf{fma}\left(-1, \frac{y}{t\_1}, \frac{z}{b} \cdot t\_2\right)\right)\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+209}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t + y}{\left(y + x\right) + t}, a, \mathsf{fma}\left(-1, \frac{b \cdot y}{t\_1}, z \cdot t\_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ t (+ x y))) (t_2 (/ (+ x y) t_1)))
   (if (<= y -2.1e+106)
     (fma 1.0 a (* b (fma -1.0 (/ y t_1) (* (/ z b) t_2))))
     (if (<= y 6.2e+209)
       (fma (/ (+ t y) (+ (+ y x) t)) a (fma -1.0 (/ (* b y) t_1) (* z t_2)))
       (- (+ a z) b)))))

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) / t_1;
	double tmp;
	if (y <= -2.1e+106) {
		tmp = fma(1.0, a, (b * fma(-1.0, (y / t_1), ((z / b) * t_2))));
	} else if (y <= 6.2e+209) {
		tmp = fma(((t + y) / ((y + x) + t)), a, fma(-1.0, ((b * y) / t_1), (z * t_2)));
	} else {
		tmp = (a + z) - b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t + Float64(x + y))
	t_2 = Float64(Float64(x + y) / t_1)
	tmp = 0.0
	if (y <= -2.1e+106)
		tmp = fma(1.0, a, Float64(b * fma(-1.0, Float64(y / t_1), Float64(Float64(z / b) * t_2))));
	elseif (y <= 6.2e+209)
		tmp = fma(Float64(Float64(t + y) / Float64(Float64(y + x) + t)), a, fma(-1.0, Float64(Float64(b * y) / t_1), Float64(z * t_2)));
	else
		tmp = Float64(Float64(a + z) - b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + y), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[y, -2.1e+106], N[(1.0 * a + N[(b * N[(-1.0 * N[(y / t$95$1), $MachinePrecision] + N[(N[(z / b), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.2e+209], N[(N[(N[(t + y), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] * a + N[(-1.0 * N[(N[(b * y), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(z * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.10000000000000005e106

   1. Initial program 15.9%

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

   4. Applied rewrites 33.7%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. times-frac N/A

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

      7. div-add-rev N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. div-add-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-+.f64 97.0

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

   7. Applied rewrites 97.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 97.0%

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

   if -2.10000000000000005e106 < y < 6.2000000000000002e209

   1. Initial program 69.2%

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

   4. Applied rewrites 79.0%

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

   5. Taylor expanded in z around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 96.1

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

   7. Applied rewrites 96.1%

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

   if 6.2000000000000002e209 < y

   1. Initial program 26.1%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 94.5

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

   5. Applied rewrites 94.5%

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

Alternative 11: 90.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t + \left(x + y\right)\\ t_2 := \frac{x + y}{t\_1}\\ \mathbf{if}\;b \leq -48000 \lor \neg \left(b \leq 4.4 \cdot 10^{-42}\right):\\ \;\;\;\;\mathsf{fma}\left(1, a, b \cdot \mathsf{fma}\left(-1, \frac{y}{t\_1}, \frac{z}{b} \cdot t\_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t + y}{\left(y + x\right) + t}, a, z \cdot t\_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ t (+ x y))) (t_2 (/ (+ x y) t_1)))
   (if (or (<= b -48000.0) (not (<= b 4.4e-42)))
     (fma 1.0 a (* b (fma -1.0 (/ y t_1) (* (/ z b) t_2))))
     (fma (/ (+ t y) (+ (+ y x) t)) a (* z t_2)))))

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) / t_1;
	double tmp;
	if ((b <= -48000.0) || !(b <= 4.4e-42)) {
		tmp = fma(1.0, a, (b * fma(-1.0, (y / t_1), ((z / b) * t_2))));
	} else {
		tmp = fma(((t + y) / ((y + x) + t)), a, (z * t_2));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t + Float64(x + y))
	t_2 = Float64(Float64(x + y) / t_1)
	tmp = 0.0
	if ((b <= -48000.0) || !(b <= 4.4e-42))
		tmp = fma(1.0, a, Float64(b * fma(-1.0, Float64(y / t_1), Float64(Float64(z / b) * t_2))));
	else
		tmp = fma(Float64(Float64(t + y) / Float64(Float64(y + x) + t)), a, Float64(z * t_2));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + y), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[Or[LessEqual[b, -48000.0], N[Not[LessEqual[b, 4.4e-42]], $MachinePrecision]], N[(1.0 * a + N[(b * N[(-1.0 * N[(y / t$95$1), $MachinePrecision] + N[(N[(z / b), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t + y), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] * a + N[(z * t$95$2), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < -48000 or 4.4000000000000001e-42 < b

   1. Initial program 54.8%

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

   4. Applied rewrites 63.5%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. times-frac N/A

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

      7. div-add-rev N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. div-add-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-+.f64 97.0

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

   7. Applied rewrites 97.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 91.2%

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

   if -48000 < b < 4.4000000000000001e-42

   1. Initial program 64.5%

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

   4. Applied rewrites 78.0%

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

   5. Taylor expanded in z around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in z around inf

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

      1. div-add-rev N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 98.0

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

   10. Applied rewrites 98.0%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -48000 \lor \neg \left(b \leq 4.4 \cdot 10^{-42}\right):\\ \;\;\;\;\mathsf{fma}\left(1, a, b \cdot \mathsf{fma}\left(-1, \frac{y}{t + \left(x + y\right)}, \frac{z}{b} \cdot \frac{x + y}{t + \left(x + y\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t + y}{\left(y + x\right) + t}, a, z \cdot \frac{x + y}{t + \left(x + y\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+106} \lor \neg \left(y \leq 5.8 \cdot 10^{+50}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t + y}{\left(y + x\right) + t}, a, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -6.8e+106) (not (<= y 5.8e+50)))
   (- (+ a z) b)
   (fma (/ (+ t y) (+ (+ y x) t)) a z)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -6.8e+106) || !(y <= 5.8e+50)) {
		tmp = (a + z) - b;
	} else {
		tmp = fma(((t + y) / ((y + x) + t)), a, z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -6.8e+106) || !(y <= 5.8e+50))
		tmp = Float64(Float64(a + z) - b);
	else
		tmp = fma(Float64(Float64(t + y) / Float64(Float64(y + x) + t)), a, z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -6.8e+106], N[Not[LessEqual[y, 5.8e+50]], $MachinePrecision]], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision], N[(N[(N[(t + y), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] * a + z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.79999999999999989e106 or 5.8e50 < y

   1. Initial program 28.8%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 84.4

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

   5. Applied rewrites 84.4%

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

   if -6.79999999999999989e106 < y < 5.8e50

   1. Initial program 75.3%

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

   4. Applied rewrites 82.7%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 73.9%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+106} \lor \neg \left(y \leq 5.8 \cdot 10^{+50}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t + y}{\left(y + x\right) + t}, a, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 44.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+127}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+21}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -6e+127) z (if (<= z 1.4e+21) a z)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -6e+127) {
		tmp = z;
	} else if (z <= 1.4e+21) {
		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 <= (-6d+127)) then
        tmp = z
    else if (z <= 1.4d+21) 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 <= -6e+127) {
		tmp = z;
	} else if (z <= 1.4e+21) {
		tmp = a;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -6e+127:
		tmp = z
	elif z <= 1.4e+21:
		tmp = a
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -6e+127)
		tmp = z;
	elseif (z <= 1.4e+21)
		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 <= -6e+127)
		tmp = z;
	elseif (z <= 1.4e+21)
		tmp = a;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -6e+127], z, If[LessEqual[z, 1.4e+21], a, z]]

Derivation

1. Split input into 2 regimes

2. if z < -6.0000000000000005e127 or 1.4e21 < z

   1. Initial program 44.7%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 63.2%

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

   if -6.0000000000000005e127 < z < 1.4e21

   1. Initial program 70.8%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 45.7%

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

Alternative 14: 52.6% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.06 \cdot 10^{+121}:\\ \;\;\;\;a + z\\ \mathbf{else}:\\ \;\;\;\;z - b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b 1.06e+121) (+ a z) (- z b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= 1.06e+121) {
		tmp = a + z;
	} else {
		tmp = z - b;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 1.06d+121) then
        tmp = a + z
    else
        tmp = z - b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= 1.06e+121) {
		tmp = a + z;
	} else {
		tmp = z - b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= 1.06e+121:
		tmp = a + z
	else:
		tmp = z - b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= 1.06e+121)
		tmp = Float64(a + z);
	else
		tmp = Float64(z - b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= 1.06e+121)
		tmp = a + z;
	else
		tmp = z - b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, 1.06e+121], N[(a + z), $MachinePrecision], N[(z - b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.05999999999999997e121

   1. Initial program 62.3%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 65.1

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

   5. Applied rewrites 65.1%

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

   6. Taylor expanded in b around 0

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

      1. lift-+.f64 67.5

         \[\leadsto a + z \]

   8. Applied rewrites 67.5%

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

   if 1.05999999999999997e121 < b

   1. Initial program 40.5%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 47.6

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

   5. Applied rewrites 47.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 44.9%

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

Alternative 15: 52.8% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.6 \cdot 10^{+235}:\\ \;\;\;\;a + z\\ \mathbf{else}:\\ \;\;\;\;a - b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (if (<= b 2.6e+235) (+ a z) (- a b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= 2.6e+235) {
		tmp = a + z;
	} else {
		tmp = a - b;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 2.6d+235) then
        tmp = a + z
    else
        tmp = a - b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= 2.6e+235) {
		tmp = a + z;
	} else {
		tmp = a - b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= 2.6e+235:
		tmp = a + z
	else:
		tmp = a - b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= 2.6e+235)
		tmp = Float64(a + z);
	else
		tmp = Float64(a - b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= 2.6e+235)
		tmp = a + z;
	else
		tmp = a - b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, 2.6e+235], N[(a + z), $MachinePrecision], N[(a - b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.5999999999999998e235

   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. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 64.3

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

   5. Applied rewrites 64.3%

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

   6. Taylor expanded in b around 0

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

      1. lift-+.f64 65.4

         \[\leadsto a + z \]

   8. Applied rewrites 65.4%

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

   if 2.5999999999999998e235 < b

   1. Initial program 30.8%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 36.2

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

   5. Applied rewrites 36.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 36.3%

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

Alternative 16: 52.3% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

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 + z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(a + z), $MachinePrecision]

Derivation

1. Initial program 59.3%

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

3. Taylor expanded in y around inf

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

   1. lower--.f64 N/A

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

   2. lower-+.f64 62.7

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

5. Applied rewrites 62.7%

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

6. Taylor expanded in b around 0

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

   1. lift-+.f64 61.7

      \[\leadsto a + z \]

8. Applied rewrites 61.7%

   \[\leadsto a + \color{blue}{z} \]
9. Add Preprocessing

Alternative 17: 31.3% accurate, 45.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.3%

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

3. Taylor expanded in t around inf

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

5. Applied rewrites 34.2%

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

Developer Target 1: 82.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025080 
(FPCore (x y z t a b)
  :name "AI.Clustering.Hierarchical.Internal:ward from clustering-0.2.1"
  :precision binary64

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

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