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

Percentage Accurate: 60.3% → 89.1%

Time: 8.0s

Alternatives: 15

Speedup: 1.4×

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: 60.3% 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: 89.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{y + x}\\ t_2 := \mathsf{fma}\left(t\_1, a, z\right)\\ t_3 := \left(x + t\right) + y\\ t_4 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_3}\\ \mathbf{if}\;t\_4 \leq -\infty \lor \neg \left(t\_4 \leq 5 \cdot 10^{+227}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(t\_1, b, a\right)}{y + x} - \frac{t\_2}{y + x}, t, t\_2\right) - t\_1 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(a + z\right) - b, y, \mathsf{fma}\left(a, t, z \cdot x\right)\right)}{t\_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ y (+ y x)))
        (t_2 (fma t_1 a z))
        (t_3 (+ (+ x t) y))
        (t_4 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) t_3)))
   (if (or (<= t_4 (- INFINITY)) (not (<= t_4 5e+227)))
     (- (fma (- (/ (fma t_1 b a) (+ y x)) (/ t_2 (+ y x))) t t_2) (* t_1 b))
     (/ (fma (- (+ a z) b) y (fma a t (* z x))) t_3))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y / (y + x);
	double t_2 = fma(t_1, a, z);
	double t_3 = (x + t) + y;
	double t_4 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / t_3;
	double tmp;
	if ((t_4 <= -((double) INFINITY)) || !(t_4 <= 5e+227)) {
		tmp = fma(((fma(t_1, b, a) / (y + x)) - (t_2 / (y + x))), t, t_2) - (t_1 * b);
	} else {
		tmp = fma(((a + z) - b), y, fma(a, t, (z * x))) / t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y / Float64(y + x))
	t_2 = fma(t_1, a, z)
	t_3 = Float64(Float64(x + t) + y)
	t_4 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / t_3)
	tmp = 0.0
	if ((t_4 <= Float64(-Inf)) || !(t_4 <= 5e+227))
		tmp = Float64(fma(Float64(Float64(fma(t_1, b, a) / Float64(y + x)) - Float64(t_2 / Float64(y + x))), t, t_2) - Float64(t_1 * b));
	else
		tmp = Float64(fma(Float64(Float64(a + z) - b), y, fma(a, t, Float64(z * x))) / t_3);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * a + z), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$4 = 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$3), $MachinePrecision]}, If[Or[LessEqual[t$95$4, (-Infinity)], N[Not[LessEqual[t$95$4, 5e+227]], $MachinePrecision]], N[(N[(N[(N[(N[(t$95$1 * b + a), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] - N[(t$95$2 / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t + t$95$2), $MachinePrecision] - N[(t$95$1 * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision] * y + N[(a * t + N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $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)) < -inf.0 or 4.9999999999999996e227 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 6.9%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(z + \left(t \cdot \left(\left(\frac{a}{x + y} + \frac{b \cdot y}{{\left(x + y\right)}^{2}}\right) - \left(\frac{z}{x + y} + \frac{a \cdot y}{{\left(x + y\right)}^{2}}\right)\right) + \frac{a \cdot y}{x + y}\right)\right) - \frac{b \cdot y}{x + y}} \]

   5. Applied rewrites 86.4%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4.9999999999999996e227

   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. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq -\infty \lor \neg \left(\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq 5 \cdot 10^{+227}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{y}{y + x}, b, a\right)}{y + x} - \frac{\mathsf{fma}\left(\frac{y}{y + x}, a, z\right)}{y + x}, t, \mathsf{fma}\left(\frac{y}{y + x}, a, z\right)\right) - \frac{y}{y + x} \cdot b\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(a + z\right) - b, y, \mathsf{fma}\left(a, t, z \cdot x\right)\right)}{\left(x + t\right) + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ \mathbf{if}\;t\_2 \leq -\infty \lor \neg \left(t\_2 \leq 2 \cdot 10^{+274}\right):\\ \;\;\;\;\left(z + a\right) - \frac{y}{y + x} \cdot b\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(a + z\right) - b, y, \mathsf{fma}\left(a, t, z \cdot x\right)\right)}{t\_1}\\ \end{array} \end{array} \]

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)))
   (if (or (<= t_2 (- INFINITY)) (not (<= t_2 2e+274)))
     (- (+ z a) (* (/ y (+ y x)) b))
     (/ (fma (- (+ a z) b) y (fma a t (* 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 tmp;
	if ((t_2 <= -((double) INFINITY)) || !(t_2 <= 2e+274)) {
		tmp = (z + a) - ((y / (y + x)) * b);
	} else {
		tmp = fma(((a + z) - b), y, fma(a, t, (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)
	tmp = 0.0
	if ((t_2 <= Float64(-Inf)) || !(t_2 <= 2e+274))
		tmp = Float64(Float64(z + a) - Float64(Float64(y / Float64(y + x)) * b));
	else
		tmp = Float64(fma(Float64(Float64(a + z) - b), y, fma(a, t, 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]}, If[Or[LessEqual[t$95$2, (-Infinity)], N[Not[LessEqual[t$95$2, 2e+274]], $MachinePrecision]], N[(N[(z + a), $MachinePrecision] - N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision] * y + N[(a * t + 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)) < -inf.0 or 1.99999999999999984e274 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 5.1%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(z + \left(t \cdot \left(\left(\frac{a}{x + y} + \frac{b \cdot y}{{\left(x + y\right)}^{2}}\right) - \left(\frac{z}{x + y} + \frac{a \cdot y}{{\left(x + y\right)}^{2}}\right)\right) + \frac{a \cdot y}{x + y}\right)\right) - \frac{b \cdot y}{x + y}} \]

   5. Applied rewrites 86.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 79.0%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1.99999999999999984e274

   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. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq -\infty \lor \neg \left(\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq 2 \cdot 10^{+274}\right):\\ \;\;\;\;\left(z + a\right) - \frac{y}{y + x} \cdot b\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(a + z\right) - b, y, \mathsf{fma}\left(a, t, z \cdot x\right)\right)}{\left(x + t\right) + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + y\right) \cdot a\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + t\_1\right) - y \cdot b}{\left(x + t\right) + y}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+70} \lor \neg \left(t\_2 \leq 5 \cdot 10^{+227}\right):\\ \;\;\;\;\left(z + a\right) - \frac{y}{y + x} \cdot b\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y + x, z, t\_1\right)}{\left(y + x\right) + t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (+ t y) a))
        (t_2 (/ (- (+ (* (+ x y) z) t_1) (* y b)) (+ (+ x t) y))))
   (if (or (<= t_2 -5e+70) (not (<= t_2 5e+227)))
     (- (+ z a) (* (/ y (+ y x)) b))
     (/ (fma (+ y x) z t_1) (+ (+ y x) t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + y) * a;
	double t_2 = ((((x + y) * z) + t_1) - (y * b)) / ((x + t) + y);
	double tmp;
	if ((t_2 <= -5e+70) || !(t_2 <= 5e+227)) {
		tmp = (z + a) - ((y / (y + x)) * b);
	} else {
		tmp = fma((y + x), z, t_1) / ((y + x) + t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t + y) * a)
	t_2 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + t_1) - Float64(y * b)) / Float64(Float64(x + t) + y))
	tmp = 0.0
	if ((t_2 <= -5e+70) || !(t_2 <= 5e+227))
		tmp = Float64(Float64(z + a) - Float64(Float64(y / Float64(y + x)) * b));
	else
		tmp = Float64(fma(Float64(y + x), z, t_1) / Float64(Float64(y + x) + t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + t$95$1), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$2, -5e+70], N[Not[LessEqual[t$95$2, 5e+227]], $MachinePrecision]], N[(N[(z + a), $MachinePrecision] - N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y + x), $MachinePrecision] * z + t$95$1), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] + t), $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)) < -5.0000000000000002e70 or 4.9999999999999996e227 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 24.3%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(z + \left(t \cdot \left(\left(\frac{a}{x + y} + \frac{b \cdot y}{{\left(x + y\right)}^{2}}\right) - \left(\frac{z}{x + y} + \frac{a \cdot y}{{\left(x + y\right)}^{2}}\right)\right) + \frac{a \cdot y}{x + y}\right)\right) - \frac{b \cdot y}{x + y}} \]

   5. Applied rewrites 80.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 79.3%

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

   if -5.0000000000000002e70 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4.9999999999999996e227

   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

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 78.8

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

   5. Applied rewrites 78.8%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq -5 \cdot 10^{+70} \lor \neg \left(\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq 5 \cdot 10^{+227}\right):\\ \;\;\;\;\left(z + a\right) - \frac{y}{y + x} \cdot b\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y + x, z, \left(t + y\right) \cdot a\right)}{\left(y + x\right) + t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+22} \lor \neg \left(t\_1 \leq 1000000\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot t}{t + x}\\ \end{array} \end{array} \]

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+22) (not (<= t_1 1000000.0)))
     (- (+ a z) b)
     (/ (* a t) (+ 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+22) || !(t_1 <= 1000000.0)) {
		tmp = (a + z) - b;
	} else {
		tmp = (a * t) / (t + x);
	}
	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+22)) .or. (.not. (t_1 <= 1000000.0d0))) then
        tmp = (a + z) - b
    else
        tmp = (a * t) / (t + x)
    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+22) || !(t_1 <= 1000000.0)) {
		tmp = (a + z) - b;
	} else {
		tmp = (a * t) / (t + x);
	}
	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+22) or not (t_1 <= 1000000.0):
		tmp = (a + z) - b
	else:
		tmp = (a * t) / (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+22) || !(t_1 <= 1000000.0))
		tmp = Float64(Float64(a + z) - b);
	else
		tmp = Float64(Float64(a * t) / Float64(t + x));
	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+22) || ~((t_1 <= 1000000.0)))
		tmp = (a + z) - b;
	else
		tmp = (a * t) / (t + x);
	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+22], N[Not[LessEqual[t$95$1, 1000000.0]], $MachinePrecision]], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision], N[(N[(a * t), $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)) < -2e22 or 1e6 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 45.2%

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

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

      2. lower-+.f64 64.3

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

   5. Applied rewrites 64.3%

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

   if -2e22 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1e6

   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 t around 0

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

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(z + \left(t \cdot \left(\left(\frac{a}{x + y} + \frac{b \cdot y}{{\left(x + y\right)}^{2}}\right) - \left(\frac{z}{x + y} + \frac{a \cdot y}{{\left(x + y\right)}^{2}}\right)\right) + \frac{a \cdot y}{x + y}\right)\right) - \frac{b \cdot y}{x + y}} \]

   5. Applied rewrites 49.6%

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

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 63.7

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

   8. Applied rewrites 63.7%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 49.4%

       \[\leadsto \frac{a \cdot t}{\color{blue}{t} + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq -2 \cdot 10^{+22} \lor \neg \left(\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq 1000000\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot t}{t + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.75 \cdot 10^{+124} \lor \neg \left(x \leq 2.4 \cdot 10^{+118}\right):\\ \;\;\;\;z - \frac{b - a}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z - b, \frac{y}{t + y}, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= x -3.75e+124) (not (<= x 2.4e+118)))
   (- z (* (/ (- b a) x) y))
   (fma (- z b) (/ y (+ t y)) a)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x <= -3.75e+124) || !(x <= 2.4e+118)) {
		tmp = z - (((b - a) / x) * y);
	} else {
		tmp = fma((z - b), (y / (t + y)), a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((x <= -3.75e+124) || !(x <= 2.4e+118))
		tmp = Float64(z - Float64(Float64(Float64(b - a) / x) * y));
	else
		tmp = fma(Float64(z - b), Float64(y / Float64(t + y)), a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[x, -3.75e+124], N[Not[LessEqual[x, 2.4e+118]], $MachinePrecision]], N[(z - N[(N[(N[(b - a), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[(z - b), $MachinePrecision] * N[(y / N[(t + y), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.75000000000000019e124 or 2.4e118 < x

   1. Initial program 41.9%

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

   3. Taylor expanded in x around -inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

   5. Applied rewrites 62.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 72.1%

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

   if -3.75000000000000019e124 < x < 2.4e118

   1. Initial program 67.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. distribute-lft-out-- N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-+.f64 50.6

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 79.1%

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

   10. Applied rewrites 81.6%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.75 \cdot 10^{+124} \lor \neg \left(x \leq 2.4 \cdot 10^{+118}\right):\\ \;\;\;\;z - \frac{b - a}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z - b, \frac{y}{t + y}, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.75 \cdot 10^{+124} \lor \neg \left(x \leq 2.4 \cdot 10^{+118}\right):\\ \;\;\;\;z - \frac{b - a}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - b}{t + y}, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= x -3.75e+124) (not (<= x 2.4e+118)))
   (- z (* (/ (- b a) x) y))
   (fma y (/ (- z b) (+ t y)) a)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x <= -3.75e+124) || !(x <= 2.4e+118)) {
		tmp = z - (((b - a) / x) * y);
	} else {
		tmp = fma(y, ((z - b) / (t + y)), a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((x <= -3.75e+124) || !(x <= 2.4e+118))
		tmp = Float64(z - Float64(Float64(Float64(b - a) / x) * y));
	else
		tmp = fma(y, Float64(Float64(z - b) / Float64(t + y)), a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[x, -3.75e+124], N[Not[LessEqual[x, 2.4e+118]], $MachinePrecision]], N[(z - N[(N[(N[(b - a), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(z - b), $MachinePrecision] / N[(t + y), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.75000000000000019e124 or 2.4e118 < x

   1. Initial program 41.9%

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

   3. Taylor expanded in x around -inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

   5. Applied rewrites 62.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 72.1%

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

   if -3.75000000000000019e124 < x < 2.4e118

   1. Initial program 67.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. distribute-lft-out-- N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-+.f64 50.6

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 79.1%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.75 \cdot 10^{+124} \lor \neg \left(x \leq 2.4 \cdot 10^{+118}\right):\\ \;\;\;\;z - \frac{b - a}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - b}{t + y}, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{+49} \lor \neg \left(t \leq 2.5 \cdot 10^{+55}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t + y}, a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -2.8e+49) (not (<= t 2.5e+55)))
   (fma y (/ z (+ t y)) a)
   (- (+ a z) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -2.8e+49) || !(t <= 2.5e+55)) {
		tmp = fma(y, (z / (t + y)), a);
	} else {
		tmp = (a + z) - b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -2.8e+49) || !(t <= 2.5e+55))
		tmp = fma(y, Float64(z / Float64(t + y)), a);
	else
		tmp = Float64(Float64(a + z) - b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -2.8e+49], N[Not[LessEqual[t, 2.5e+55]], $MachinePrecision]], N[(y * N[(z / N[(t + y), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.7999999999999998e49 or 2.50000000000000023e55 < t

   1. Initial program 60.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. distribute-lft-out-- N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-+.f64 51.9

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

   5. Applied rewrites 51.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 82.0%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 69.7%

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

   if -2.7999999999999998e49 < t < 2.50000000000000023e55

   1. Initial program 60.4%

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

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

      2. lower-+.f64 61.1

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

   5. Applied rewrites 61.1%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{+49} \lor \neg \left(t \leq 2.5 \cdot 10^{+55}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t + y}, a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t + y}, a\right)\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+55}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - b}{t}, y, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -2.8e+49)
   (fma y (/ z (+ t y)) a)
   (if (<= t 2.5e+55) (- (+ a z) b) (fma (/ (- z b) t) y a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -2.8e+49) {
		tmp = fma(y, (z / (t + y)), a);
	} else if (t <= 2.5e+55) {
		tmp = (a + z) - b;
	} else {
		tmp = fma(((z - b) / t), y, a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -2.8e+49)
		tmp = fma(y, Float64(z / Float64(t + y)), a);
	elseif (t <= 2.5e+55)
		tmp = Float64(Float64(a + z) - b);
	else
		tmp = fma(Float64(Float64(z - b) / t), y, a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -2.8e+49], N[(y * N[(z / N[(t + y), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision], If[LessEqual[t, 2.5e+55], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision], N[(N[(N[(z - b), $MachinePrecision] / t), $MachinePrecision] * y + a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.7999999999999998e49

   1. Initial program 68.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. distribute-lft-out-- N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-+.f64 62.7

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

   5. Applied rewrites 62.7%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 91.4%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 79.3%

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

   if -2.7999999999999998e49 < t < 2.50000000000000023e55

   1. Initial program 60.4%

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

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

      2. lower-+.f64 61.1

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

   5. Applied rewrites 61.1%

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

   if 2.50000000000000023e55 < t

   1. Initial program 54.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. distribute-lft-out-- N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-+.f64 42.9

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

   5. Applied rewrites 42.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 64.7%

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

Alternative 9: 61.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t + y}, a\right)\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+55}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z - b, \frac{y}{t}, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -2.8e+49)
   (fma y (/ z (+ t y)) a)
   (if (<= t 2.5e+55) (- (+ a z) b) (fma (- z b) (/ y t) a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -2.8e+49) {
		tmp = fma(y, (z / (t + y)), a);
	} else if (t <= 2.5e+55) {
		tmp = (a + z) - b;
	} else {
		tmp = fma((z - b), (y / t), a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -2.8e+49)
		tmp = fma(y, Float64(z / Float64(t + y)), a);
	elseif (t <= 2.5e+55)
		tmp = Float64(Float64(a + z) - b);
	else
		tmp = fma(Float64(z - b), Float64(y / t), a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -2.8e+49], N[(y * N[(z / N[(t + y), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision], If[LessEqual[t, 2.5e+55], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision], N[(N[(z - b), $MachinePrecision] * N[(y / t), $MachinePrecision] + a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.7999999999999998e49

   1. Initial program 68.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. distribute-lft-out-- N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-+.f64 62.7

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

   5. Applied rewrites 62.7%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 91.4%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 79.3%

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

   if -2.7999999999999998e49 < t < 2.50000000000000023e55

   1. Initial program 60.4%

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

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

      2. lower-+.f64 61.1

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

   5. Applied rewrites 61.1%

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

   if 2.50000000000000023e55 < t

   1. Initial program 54.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. distribute-lft-out-- N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-+.f64 42.9

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

   5. Applied rewrites 42.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 74.1%

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

   10. Applied rewrites 75.9%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 64.7%

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

Alternative 10: 59.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{+117} \lor \neg \left(t \leq 9.5 \cdot 10^{+52}\right):\\ \;\;\;\;a \cdot \frac{t}{t + x}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -1.25e+117) (not (<= t 9.5e+52)))
   (* a (/ t (+ t x)))
   (- (+ a z) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -1.25e+117) || !(t <= 9.5e+52)) {
		tmp = a * (t / (t + x));
	} else {
		tmp = (a + 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 ((t <= (-1.25d+117)) .or. (.not. (t <= 9.5d+52))) then
        tmp = a * (t / (t + x))
    else
        tmp = (a + 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 ((t <= -1.25e+117) || !(t <= 9.5e+52)) {
		tmp = a * (t / (t + x));
	} else {
		tmp = (a + z) - b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -1.25e+117) or not (t <= 9.5e+52):
		tmp = a * (t / (t + x))
	else:
		tmp = (a + z) - b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -1.25e+117) || !(t <= 9.5e+52))
		tmp = Float64(a * Float64(t / Float64(t + x)));
	else
		tmp = Float64(Float64(a + z) - b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -1.25e+117) || ~((t <= 9.5e+52)))
		tmp = a * (t / (t + x));
	else
		tmp = (a + z) - b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -1.25e+117], N[Not[LessEqual[t, 9.5e+52]], $MachinePrecision]], N[(a * N[(t / N[(t + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.24999999999999996e117 or 9.49999999999999994e52 < t

   1. Initial program 58.7%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 46.1

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

   5. Applied rewrites 46.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 58.3%

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

   if -1.24999999999999996e117 < t < 9.49999999999999994e52

   1. Initial program 61.5%

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

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

      2. lower-+.f64 60.9

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

   5. Applied rewrites 60.9%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{+117} \lor \neg \left(t \leq 9.5 \cdot 10^{+52}\right):\\ \;\;\;\;a \cdot \frac{t}{t + x}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -235 \lor \neg \left(y \leq 2.7 \cdot 10^{+44}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;a + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -235.0) (not (<= y 2.7e+44))) (- (+ a z) b) (+ a z)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -235.0) || !(y <= 2.7e+44)) {
		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) :: tmp
    if ((y <= (-235.0d0)) .or. (.not. (y <= 2.7d+44))) 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 tmp;
	if ((y <= -235.0) || !(y <= 2.7e+44)) {
		tmp = (a + z) - b;
	} else {
		tmp = a + z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -235.0) or not (y <= 2.7e+44):
		tmp = (a + z) - b
	else:
		tmp = a + z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -235.0) || !(y <= 2.7e+44))
		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)
	tmp = 0.0;
	if ((y <= -235.0) || ~((y <= 2.7e+44)))
		tmp = (a + z) - b;
	else
		tmp = a + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -235.0], N[Not[LessEqual[y, 2.7e+44]], $MachinePrecision]], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision], N[(a + z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -235 or 2.7e44 < y

   1. Initial program 38.9%

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

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

      2. lower-+.f64 76.5

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

   5. Applied rewrites 76.5%

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

   if -235 < y < 2.7e44

   1. Initial program 81.5%

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

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

      2. lower-+.f64 32.5

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

   5. Applied rewrites 32.5%

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

   6. Taylor expanded in b around inf

      \[\leadsto -1 \cdot \color{blue}{b} \]
   7. Step-by-step derivation

   8. Applied rewrites 6.1%

      \[\leadsto -b \]

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 41.3%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -235 \lor \neg \left(y \leq 2.7 \cdot 10^{+44}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;a + z\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 53.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{-82} \lor \neg \left(z \leq 1.7 \cdot 10^{-59}\right):\\ \;\;\;\;a + z\\ \mathbf{else}:\\ \;\;\;\;a - b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -5.6e-82) (not (<= z 1.7e-59))) (+ a z) (- a b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -5.6e-82) || !(z <= 1.7e-59)) {
		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 ((z <= (-5.6d-82)) .or. (.not. (z <= 1.7d-59))) 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 ((z <= -5.6e-82) || !(z <= 1.7e-59)) {
		tmp = a + z;
	} else {
		tmp = a - b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z <= -5.6e-82) or not (z <= 1.7e-59):
		tmp = a + z
	else:
		tmp = a - b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -5.6e-82) || !(z <= 1.7e-59))
		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 ((z <= -5.6e-82) || ~((z <= 1.7e-59)))
		tmp = a + z;
	else
		tmp = a - b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -5.6e-82], N[Not[LessEqual[z, 1.7e-59]], $MachinePrecision]], N[(a + z), $MachinePrecision], N[(a - b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.60000000000000049e-82 or 1.70000000000000009e-59 < z

   1. Initial program 49.4%

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

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

      2. lower-+.f64 56.4

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

   5. Applied rewrites 56.4%

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

   6. Taylor expanded in b around inf

      \[\leadsto -1 \cdot \color{blue}{b} \]
   7. Step-by-step derivation

   8. Applied rewrites 8.6%

      \[\leadsto -b \]

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 56.2%

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

   if -5.60000000000000049e-82 < z < 1.70000000000000009e-59

   1. Initial program 77.6%

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

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

      2. lower-+.f64 50.8

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

   5. Applied rewrites 50.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 51.4%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{-82} \lor \neg \left(z \leq 1.7 \cdot 10^{-59}\right):\\ \;\;\;\;a + z\\ \mathbf{else}:\\ \;\;\;\;a - b\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 50.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+20}:\\ \;\;\;\;z - b\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-59}:\\ \;\;\;\;a - b\\ \mathbf{else}:\\ \;\;\;\;a + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -3.5e+20) (- z b) (if (<= z 1.7e-59) (- a b) (+ a z))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -3.5e+20) {
		tmp = z - b;
	} else if (z <= 1.7e-59) {
		tmp = a - 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) :: tmp
    if (z <= (-3.5d+20)) then
        tmp = z - b
    else if (z <= 1.7d-59) then
        tmp = a - 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 tmp;
	if (z <= -3.5e+20) {
		tmp = z - b;
	} else if (z <= 1.7e-59) {
		tmp = a - b;
	} else {
		tmp = a + z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -3.5e+20:
		tmp = z - b
	elif z <= 1.7e-59:
		tmp = a - b
	else:
		tmp = a + z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -3.5e+20)
		tmp = Float64(z - b);
	elseif (z <= 1.7e-59)
		tmp = Float64(a - b);
	else
		tmp = Float64(a + z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -3.5e+20)
		tmp = z - b;
	elseif (z <= 1.7e-59)
		tmp = a - b;
	else
		tmp = a + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -3.5e+20], N[(z - b), $MachinePrecision], If[LessEqual[z, 1.7e-59], N[(a - b), $MachinePrecision], N[(a + z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.5e20

   1. Initial program 43.7%

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

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

      2. lower-+.f64 61.2

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

   5. Applied rewrites 61.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 56.9%

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

   if -3.5e20 < z < 1.70000000000000009e-59

   1. Initial program 76.2%

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

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

      2. lower-+.f64 49.5

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

   5. Applied rewrites 49.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 49.5%

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

   if 1.70000000000000009e-59 < z

   1. Initial program 49.9%

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

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

      2. lower-+.f64 55.6

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

   5. Applied rewrites 55.6%

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

   6. Taylor expanded in b around inf

      \[\leadsto -1 \cdot \color{blue}{b} \]
   7. Step-by-step derivation

   8. Applied rewrites 7.1%

      \[\leadsto -b \]

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 60.5%

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

Alternative 14: 51.8% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7 \cdot 10^{+210}:\\ \;\;\;\;a + z\\ \mathbf{else}:\\ \;\;\;\;-b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (if (<= b 7e+210) (+ a z) (- b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= 7e+210) {
		tmp = a + z;
	} else {
		tmp = -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 <= 7d+210) then
        tmp = a + z
    else
        tmp = -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 <= 7e+210) {
		tmp = a + z;
	} else {
		tmp = -b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= 7e+210:
		tmp = a + z
	else:
		tmp = -b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= 7e+210)
		tmp = Float64(a + z);
	else
		tmp = Float64(-b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= 7e+210)
		tmp = a + z;
	else
		tmp = -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, 7e+210], N[(a + z), $MachinePrecision], (-b)]

Derivation

1. Split input into 2 regimes

2. if b < 6.9999999999999999e210

   1. Initial program 61.2%

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

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

      2. lower-+.f64 55.5

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

   5. Applied rewrites 55.5%

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

   6. Taylor expanded in b around inf

      \[\leadsto -1 \cdot \color{blue}{b} \]
   7. Step-by-step derivation

   8. Applied rewrites 11.5%

      \[\leadsto -b \]

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 53.0%

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

   if 6.9999999999999999e210 < b

   1. Initial program 53.9%

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

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

      2. lower-+.f64 40.4

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

   5. Applied rewrites 40.4%

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

   6. Taylor expanded in b around inf

      \[\leadsto -1 \cdot \color{blue}{b} \]
   7. Step-by-step derivation

   8. Applied rewrites 35.7%

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

Alternative 15: 13.5% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ -b \end{array} \]

FPCore

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

C

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

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 = -b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := (-b)

Derivation

1. Initial program 60.5%

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

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

   2. lower-+.f64 54.2

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

5. Applied rewrites 54.2%

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

6. Taylor expanded in b around inf

   \[\leadsto -1 \cdot \color{blue}{b} \]
7. Step-by-step derivation

8. Applied rewrites 13.6%

   \[\leadsto -b \]
9. Add Preprocessing

Developer Target 1: 81.9% 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 2024352 
(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)))