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

Percentage Accurate: 60.7% → 93.8%

Time: 7.2s

Alternatives: 16

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 60.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y + x\right) + t\\ t_2 := t + \left(x + y\right)\\ t_3 := \frac{t + y}{t\_1}\\ \mathbf{if}\;z \leq -1.75 \cdot 10^{-118} \lor \neg \left(z \leq 1.5 \cdot 10^{-16}\right):\\ \;\;\;\;\mathsf{fma}\left(t\_3, a, z \cdot \mathsf{fma}\left(-1, \frac{b}{z} \cdot \frac{y}{t\_2}, \frac{x + y}{t\_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_3, a, \frac{\mathsf{fma}\left(y + x, z, \left(-b\right) \cdot y\right)}{t\_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ y x) t)) (t_2 (+ t (+ x y))) (t_3 (/ (+ t y) t_1)))
   (if (or (<= z -1.75e-118) (not (<= z 1.5e-16)))
     (fma t_3 a (* z (fma -1.0 (* (/ b z) (/ y t_2)) (/ (+ x y) t_2))))
     (fma t_3 a (/ (fma (+ y x) z (* (- b) y)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y + x) + t;
	double t_2 = t + (x + y);
	double t_3 = (t + y) / t_1;
	double tmp;
	if ((z <= -1.75e-118) || !(z <= 1.5e-16)) {
		tmp = fma(t_3, a, (z * fma(-1.0, ((b / z) * (y / t_2)), ((x + y) / t_2))));
	} else {
		tmp = fma(t_3, a, (fma((y + x), z, (-b * y)) / t_1));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y + x) + t)
	t_2 = Float64(t + Float64(x + y))
	t_3 = Float64(Float64(t + y) / t_1)
	tmp = 0.0
	if ((z <= -1.75e-118) || !(z <= 1.5e-16))
		tmp = fma(t_3, a, Float64(z * fma(-1.0, Float64(Float64(b / z) * Float64(y / t_2)), Float64(Float64(x + y) / t_2))));
	else
		tmp = fma(t_3, a, Float64(fma(Float64(y + x), z, Float64(Float64(-b) * y)) / t_1));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$2 = N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t + y), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[Or[LessEqual[z, -1.75e-118], N[Not[LessEqual[z, 1.5e-16]], $MachinePrecision]], N[(t$95$3 * a + N[(z * N[(-1.0 * N[(N[(b / z), $MachinePrecision] * N[(y / t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(x + y), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 * a + N[(N[(N[(y + x), $MachinePrecision] * z + N[((-b) * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.75e-118 or 1.49999999999999997e-16 < z

   1. Initial program 53.2%

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

   4. Applied rewrites 63.2%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. div-add-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-+.f64 97.9

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

   7. Applied rewrites 97.9%

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

   if -1.75e-118 < z < 1.49999999999999997e-16

   1. Initial program 72.6%

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

   4. Applied rewrites 87.3%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{-118} \lor \neg \left(z \leq 1.5 \cdot 10^{-16}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{t + y}{\left(y + x\right) + t}, a, z \cdot \mathsf{fma}\left(-1, \frac{b}{z} \cdot \frac{y}{t + \left(x + y\right)}, \frac{x + y}{t + \left(x + y\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t + y}{\left(y + x\right) + t}, a, \frac{\mathsf{fma}\left(y + x, z, \left(-b\right) \cdot y\right)}{\left(y + x\right) + t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y + x\right) + t\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ \mathbf{if}\;t\_2 \leq -\infty \lor \neg \left(t\_2 \leq 10^{+216}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{t + y}{t\_1}, a, z \cdot \mathsf{fma}\left(-1, \frac{b}{z} \cdot \frac{y}{t + \left(x + y\right)}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{t\_1}, z, \frac{\mathsf{fma}\left(t + y, a, \left(-b\right) \cdot y\right)}{t\_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ y x) t))
        (t_2 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y))))
   (if (or (<= t_2 (- INFINITY)) (not (<= t_2 1e+216)))
     (fma
      (/ (+ t y) t_1)
      a
      (* z (fma -1.0 (* (/ b z) (/ y (+ t (+ x y)))) 1.0)))
     (fma (/ (+ y x) t_1) z (/ (fma (+ t y) a (* (- b) y)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y + x) + t;
	double t_2 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
	double tmp;
	if ((t_2 <= -((double) INFINITY)) || !(t_2 <= 1e+216)) {
		tmp = fma(((t + y) / t_1), a, (z * fma(-1.0, ((b / z) * (y / (t + (x + y)))), 1.0)));
	} else {
		tmp = fma(((y + x) / t_1), z, (fma((t + y), a, (-b * y)) / t_1));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y + x) + t)
	t_2 = 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_2 <= Float64(-Inf)) || !(t_2 <= 1e+216))
		tmp = fma(Float64(Float64(t + y) / t_1), a, Float64(z * fma(-1.0, Float64(Float64(b / z) * Float64(y / Float64(t + Float64(x + y)))), 1.0)));
	else
		tmp = fma(Float64(Float64(y + x) / t_1), z, Float64(fma(Float64(t + y), a, Float64(Float64(-b) * y)) / t_1));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$2, (-Infinity)], N[Not[LessEqual[t$95$2, 1e+216]], $MachinePrecision]], N[(N[(N[(t + y), $MachinePrecision] / t$95$1), $MachinePrecision] * a + N[(z * N[(-1.0 * N[(N[(b / z), $MachinePrecision] * N[(y / N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y + x), $MachinePrecision] / t$95$1), $MachinePrecision] * z + N[(N[(N[(t + y), $MachinePrecision] * a + N[((-b) * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $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)) < -inf.0 or 1e216 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 11.4%

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

   4. Applied rewrites 38.8%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. div-add-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-+.f64 87.5

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

   7. Applied rewrites 87.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 78.3%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1e216

   1. Initial program 99.4%

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

   4. Applied rewrites 99.2%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq -\infty \lor \neg \left(\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq 10^{+216}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{t + y}{\left(y + x\right) + t}, a, z \cdot \mathsf{fma}\left(-1, \frac{b}{z} \cdot \frac{y}{t + \left(x + y\right)}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{\left(y + x\right) + t}, z, \frac{\mathsf{fma}\left(t + y, a, \left(-b\right) \cdot y\right)}{\left(y + x\right) + t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ t_2 := \left(y + x\right) + t\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{t + y}{t\_2}, a, z\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+244}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{t\_2}, z, \frac{\mathsf{fma}\left(t + y, a, \left(-b\right) \cdot y\right)}{t\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))
        (t_2 (+ (+ y x) t)))
   (if (<= t_1 (- INFINITY))
     (fma (/ (+ t y) t_2) a z)
     (if (<= t_1 2e+244)
       (fma (/ (+ y x) t_2) z (/ (fma (+ t y) a (* (- b) y)) t_2))
       (- (+ a z) b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
	double t_2 = (y + x) + t;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(((t + y) / t_2), a, z);
	} else if (t_1 <= 2e+244) {
		tmp = fma(((y + x) / t_2), z, (fma((t + y), a, (-b * y)) / t_2));
	} else {
		tmp = (a + z) - b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / Float64(Float64(x + t) + y))
	t_2 = Float64(Float64(y + x) + t)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(Float64(Float64(t + y) / t_2), a, z);
	elseif (t_1 <= 2e+244)
		tmp = fma(Float64(Float64(y + x) / t_2), z, Float64(fma(Float64(t + y), a, Float64(Float64(-b) * y)) / t_2));
	else
		tmp = Float64(Float64(a + z) - b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(t + y), $MachinePrecision] / t$95$2), $MachinePrecision] * a + z), $MachinePrecision], If[LessEqual[t$95$1, 2e+244], N[(N[(N[(y + x), $MachinePrecision] / t$95$2), $MachinePrecision] * z + N[(N[(N[(t + y), $MachinePrecision] * a + N[((-b) * y), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 6.4%

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

   4. Applied rewrites 39.9%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 74.1%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 2.00000000000000015e244

   1. Initial program 99.4%

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

   4. Applied rewrites 99.2%

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

   if 2.00000000000000015e244 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 11.0%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 71.2

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

   5. Applied rewrites 71.2%

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

Alternative 4: 87.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ t_2 := \left(y + x\right) + t\\ t_3 := \frac{t + y}{t\_2}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(t\_3, a, z\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+244}:\\ \;\;\;\;\mathsf{fma}\left(t\_3, a, \frac{\mathsf{fma}\left(y + x, z, \left(-b\right) \cdot y\right)}{t\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))
        (t_2 (+ (+ y x) t))
        (t_3 (/ (+ t y) t_2)))
   (if (<= t_1 (- INFINITY))
     (fma t_3 a z)
     (if (<= t_1 2e+244)
       (fma t_3 a (/ (fma (+ y x) z (* (- b) y)) t_2))
       (- (+ a z) b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
	double t_2 = (y + x) + t;
	double t_3 = (t + y) / t_2;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(t_3, a, z);
	} else if (t_1 <= 2e+244) {
		tmp = fma(t_3, a, (fma((y + x), z, (-b * y)) / t_2));
	} else {
		tmp = (a + z) - b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / Float64(Float64(x + t) + y))
	t_2 = Float64(Float64(y + x) + t)
	t_3 = Float64(Float64(t + y) / t_2)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(t_3, a, z);
	elseif (t_1 <= 2e+244)
		tmp = fma(t_3, a, Float64(fma(Float64(y + x), z, Float64(Float64(-b) * y)) / t_2));
	else
		tmp = Float64(Float64(a + z) - b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t + y), $MachinePrecision] / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(t$95$3 * a + z), $MachinePrecision], If[LessEqual[t$95$1, 2e+244], N[(t$95$3 * a + N[(N[(N[(y + x), $MachinePrecision] * z + N[((-b) * y), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 6.4%

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

   4. Applied rewrites 39.9%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 74.1%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 2.00000000000000015e244

   1. Initial program 99.4%

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

   4. Applied rewrites 99.1%

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

   if 2.00000000000000015e244 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 11.0%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 71.2

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

   5. Applied rewrites 71.2%

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

Alternative 5: 87.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{t + y}{\left(y + x\right) + t}, a, z\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+244}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y))))
   (if (<= t_1 (- INFINITY))
     (fma (/ (+ t y) (+ (+ y x) t)) a z)
     (if (<= t_1 2e+244) t_1 (- (+ a z) b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(((t + y) / ((y + x) + t)), a, z);
	} else if (t_1 <= 2e+244) {
		tmp = t_1;
	} else {
		tmp = (a + z) - b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / Float64(Float64(x + t) + y))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(Float64(Float64(t + y) / Float64(Float64(y + x) + t)), a, z);
	elseif (t_1 <= 2e+244)
		tmp = t_1;
	else
		tmp = Float64(Float64(a + z) - b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(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[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(t + y), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] * a + z), $MachinePrecision], If[LessEqual[t$95$1, 2e+244], t$95$1, N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 6.4%

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

   4. Applied rewrites 39.9%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 74.1%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 2.00000000000000015e244

   1. Initial program 99.4%

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

   if 2.00000000000000015e244 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 11.0%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 71.2

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

   5. Applied rewrites 71.2%

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

Alternative 6: 75.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{t + y}{\left(y + x\right) + t}, a, z\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+201}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t + y, a, \left(y + x\right) \cdot z\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ x t) y))
        (t_2 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) t_1)))
   (if (<= t_2 (- INFINITY))
     (fma (/ (+ t y) (+ (+ y x) t)) a z)
     (if (<= t_2 1e+201)
       (/ (fma (+ t y) a (* (+ y x) z)) t_1)
       (- (+ a z) b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + t) + y;
	double t_2 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma(((t + y) / ((y + x) + t)), a, z);
	} else if (t_2 <= 1e+201) {
		tmp = fma((t + y), a, ((y + x) * z)) / t_1;
	} else {
		tmp = (a + z) - b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + t) + y)
	t_2 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = fma(Float64(Float64(t + y) / Float64(Float64(y + x) + t)), a, z);
	elseif (t_2 <= 1e+201)
		tmp = Float64(fma(Float64(t + y), a, Float64(Float64(y + x) * z)) / t_1);
	else
		tmp = Float64(Float64(a + z) - b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[(t + y), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] * a + z), $MachinePrecision], If[LessEqual[t$95$2, 1e+201], N[(N[(N[(t + y), $MachinePrecision] * a + N[(N[(y + x), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 6.4%

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

   4. Applied rewrites 39.9%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 74.1%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1.00000000000000004e201

   1. Initial program 99.4%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 77.6

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

   5. Applied rewrites 77.6%

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

   if 1.00000000000000004e201 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 16.9%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 71.1

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

   5. Applied rewrites 71.1%

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

Alternative 7: 65.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+90} \lor \neg \left(t\_1 \leq 4 \cdot 10^{+56}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, z \cdot x\right)}{t + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y))))
   (if (or (<= t_1 -5e+90) (not (<= t_1 4e+56)))
     (- (+ a z) b)
     (/ (fma a t (* z x)) (+ t x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
	double tmp;
	if ((t_1 <= -5e+90) || !(t_1 <= 4e+56)) {
		tmp = (a + z) - b;
	} else {
		tmp = fma(a, t, (z * x)) / (t + x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / Float64(Float64(x + t) + y))
	tmp = 0.0
	if ((t_1 <= -5e+90) || !(t_1 <= 4e+56))
		tmp = Float64(Float64(a + z) - b);
	else
		tmp = Float64(fma(a, t, Float64(z * x)) / Float64(t + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e+90], N[Not[LessEqual[t$95$1, 4e+56]], $MachinePrecision]], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision], N[(N[(a * t + N[(z * x), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -5.0000000000000004e90 or 4.00000000000000037e56 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 38.2%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 67.7

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

   5. Applied rewrites 67.7%

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

   if -5.0000000000000004e90 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4.00000000000000037e56

   1. Initial program 99.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 62.1

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

   5. Applied rewrites 62.1%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq -5 \cdot 10^{+90} \lor \neg \left(\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \leq 4 \cdot 10^{+56}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, z \cdot x\right)}{t + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + z\right) - b\\ t_2 := t + \left(x + y\right)\\ t_3 := z \cdot \frac{x + y}{t\_2}\\ \mathbf{if}\;z \leq -2 \cdot 10^{+34}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-117}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-132}:\\ \;\;\;\;\frac{t + y}{t\_2} \cdot a\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+141}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (+ a z) b)) (t_2 (+ t (+ x y))) (t_3 (* z (/ (+ x y) t_2))))
   (if (<= z -2e+34)
     t_3
     (if (<= z -2e-117)
       t_1
       (if (<= z 6.6e-132)
         (* (/ (+ t y) t_2) a)
         (if (<= z 7.2e+141) t_1 t_3))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a + z) - b;
	double t_2 = t + (x + y);
	double t_3 = z * ((x + y) / t_2);
	double tmp;
	if (z <= -2e+34) {
		tmp = t_3;
	} else if (z <= -2e-117) {
		tmp = t_1;
	} else if (z <= 6.6e-132) {
		tmp = ((t + y) / t_2) * a;
	} else if (z <= 7.2e+141) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (a + z) - b
    t_2 = t + (x + y)
    t_3 = z * ((x + y) / t_2)
    if (z <= (-2d+34)) then
        tmp = t_3
    else if (z <= (-2d-117)) then
        tmp = t_1
    else if (z <= 6.6d-132) then
        tmp = ((t + y) / t_2) * a
    else if (z <= 7.2d+141) then
        tmp = t_1
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a + z) - b;
	double t_2 = t + (x + y);
	double t_3 = z * ((x + y) / t_2);
	double tmp;
	if (z <= -2e+34) {
		tmp = t_3;
	} else if (z <= -2e-117) {
		tmp = t_1;
	} else if (z <= 6.6e-132) {
		tmp = ((t + y) / t_2) * a;
	} else if (z <= 7.2e+141) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a + z) - b
	t_2 = t + (x + y)
	t_3 = z * ((x + y) / t_2)
	tmp = 0
	if z <= -2e+34:
		tmp = t_3
	elif z <= -2e-117:
		tmp = t_1
	elif z <= 6.6e-132:
		tmp = ((t + y) / t_2) * a
	elif z <= 7.2e+141:
		tmp = t_1
	else:
		tmp = t_3
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a + z) - b)
	t_2 = Float64(t + Float64(x + y))
	t_3 = Float64(z * Float64(Float64(x + y) / t_2))
	tmp = 0.0
	if (z <= -2e+34)
		tmp = t_3;
	elseif (z <= -2e-117)
		tmp = t_1;
	elseif (z <= 6.6e-132)
		tmp = Float64(Float64(Float64(t + y) / t_2) * a);
	elseif (z <= 7.2e+141)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a + z) - b;
	t_2 = t + (x + y);
	t_3 = z * ((x + y) / t_2);
	tmp = 0.0;
	if (z <= -2e+34)
		tmp = t_3;
	elseif (z <= -2e-117)
		tmp = t_1;
	elseif (z <= 6.6e-132)
		tmp = ((t + y) / t_2) * a;
	elseif (z <= 7.2e+141)
		tmp = t_1;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, Block[{t$95$2 = N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(z * N[(N[(x + y), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2e+34], t$95$3, If[LessEqual[z, -2e-117], t$95$1, If[LessEqual[z, 6.6e-132], N[(N[(N[(t + y), $MachinePrecision] / t$95$2), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[z, 7.2e+141], t$95$1, t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.99999999999999989e34 or 7.2000000000000003e141 < z

   1. Initial program 42.8%

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

   4. Applied rewrites 49.7%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 68.5

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

   7. Applied rewrites 68.5%

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

   if -1.99999999999999989e34 < z < -2.00000000000000006e-117 or 6.5999999999999994e-132 < z < 7.2000000000000003e141

   1. Initial program 69.3%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 54.3

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

   5. Applied rewrites 54.3%

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

   if -2.00000000000000006e-117 < z < 6.5999999999999994e-132

   1. Initial program 71.8%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 85.7%

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

   6. Taylor expanded in a around inf

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

      1. div-add-rev N/A

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

      2. lower-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 57.4

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

   8. Applied rewrites 57.4%

      \[\leadsto \frac{t + y}{t + \left(x + y\right)} \cdot a \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+34}:\\ \;\;\;\;z \cdot \frac{x + y}{t + \left(x + y\right)}\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-117}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-132}:\\ \;\;\;\;\frac{t + y}{t + \left(x + y\right)} \cdot a\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+141}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{x + y}{t + \left(x + y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+82} \lor \neg \left(y \leq 2 \cdot 10^{+63}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t + y}{\left(y + x\right) + t}, a, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -5.2e+82) (not (<= y 2e+63)))
   (- (+ a z) b)
   (fma (/ (+ t y) (+ (+ y x) t)) a z)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -5.2e+82) || !(y <= 2e+63)) {
		tmp = (a + z) - b;
	} else {
		tmp = fma(((t + y) / ((y + x) + t)), a, z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -5.2e+82) || !(y <= 2e+63))
		tmp = Float64(Float64(a + z) - b);
	else
		tmp = fma(Float64(Float64(t + y) / Float64(Float64(y + x) + t)), a, z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -5.2e+82], N[Not[LessEqual[y, 2e+63]], $MachinePrecision]], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision], N[(N[(N[(t + y), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] * a + z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.1999999999999997e82 or 2.00000000000000012e63 < y

   1. Initial program 36.6%

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

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

      2. lower-+.f64 76.6

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

   5. Applied rewrites 76.6%

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

   if -5.1999999999999997e82 < y < 2.00000000000000012e63

   1. Initial program 76.1%

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

   4. Applied rewrites 86.7%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 65.1%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+82} \lor \neg \left(y \leq 2 \cdot 10^{+63}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t + y}{\left(y + x\right) + t}, a, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{+149} \lor \neg \left(x \leq 5 \cdot 10^{+167}\right):\\ \;\;\;\;z + t \cdot \frac{a}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= x -5.4e+149) (not (<= x 5e+167)))
   (+ z (* t (/ a x)))
   (- (+ a z) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x <= -5.4e+149) || !(x <= 5e+167)) {
		tmp = z + (t * (a / 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 ((x <= (-5.4d+149)) .or. (.not. (x <= 5d+167))) then
        tmp = z + (t * (a / 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 ((x <= -5.4e+149) || !(x <= 5e+167)) {
		tmp = z + (t * (a / x));
	} else {
		tmp = (a + z) - b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (x <= -5.4e+149) or not (x <= 5e+167):
		tmp = z + (t * (a / x))
	else:
		tmp = (a + z) - b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((x <= -5.4e+149) || !(x <= 5e+167))
		tmp = Float64(z + Float64(t * Float64(a / 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 ((x <= -5.4e+149) || ~((x <= 5e+167)))
		tmp = z + (t * (a / x));
	else
		tmp = (a + z) - b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[x, -5.4e+149], N[Not[LessEqual[x, 5e+167]], $MachinePrecision]], N[(z + N[(t * N[(a / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.4000000000000002e149 or 4.9999999999999997e167 < x

   1. Initial program 45.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 34.2

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

   5. Applied rewrites 34.2%

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

   6. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. sub-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 60.1

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

   8. Applied rewrites 60.1%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 60.4%

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

   if -5.4000000000000002e149 < x < 4.9999999999999997e167

   1. Initial program 65.6%

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

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

      2. lower-+.f64 60.4

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

   5. Applied rewrites 60.4%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{+149} \lor \neg \left(x \leq 5 \cdot 10^{+167}\right):\\ \;\;\;\;z + t \cdot \frac{a}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.9 \cdot 10^{+149}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+178}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x -6.9e+149) z (if (<= x 3.3e+178) (- (+ a z) b) z)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -6.9e+149) {
		tmp = z;
	} else if (x <= 3.3e+178) {
		tmp = (a + z) - b;
	} else {
		tmp = z;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (x <= (-6.9d+149)) then
        tmp = z
    else if (x <= 3.3d+178) then
        tmp = (a + z) - b
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -6.9e+149) {
		tmp = z;
	} else if (x <= 3.3e+178) {
		tmp = (a + z) - b;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= -6.9e+149:
		tmp = z
	elif x <= 3.3e+178:
		tmp = (a + z) - b
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= -6.9e+149)
		tmp = z;
	elseif (x <= 3.3e+178)
		tmp = Float64(Float64(a + z) - b);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= -6.9e+149)
		tmp = z;
	elseif (x <= 3.3e+178)
		tmp = (a + z) - b;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, -6.9e+149], z, If[LessEqual[x, 3.3e+178], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision], z]]

Derivation

1. Split input into 2 regimes

2. if x < -6.9000000000000004e149 or 3.2999999999999998e178 < x

   1. Initial program 45.3%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 52.3%

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

   if -6.9000000000000004e149 < x < 3.2999999999999998e178

   1. Initial program 65.5%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 60.1

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

   5. Applied rewrites 60.1%

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

Alternative 12: 53.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.6 \cdot 10^{-99} \lor \neg \left(a \leq 175000\right):\\ \;\;\;\;a + z\\ \mathbf{else}:\\ \;\;\;\;z - b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -1.6e-99) (not (<= a 175000.0))) (+ a z) (- z b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -1.6e-99) || !(a <= 175000.0)) {
		tmp = a + z;
	} else {
		tmp = z - b;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-1.6d-99)) .or. (.not. (a <= 175000.0d0))) then
        tmp = a + z
    else
        tmp = z - b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -1.6e-99) || !(a <= 175000.0)) {
		tmp = a + z;
	} else {
		tmp = z - b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a <= -1.6e-99) or not (a <= 175000.0):
		tmp = a + z
	else:
		tmp = z - b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -1.6e-99) || !(a <= 175000.0))
		tmp = Float64(a + z);
	else
		tmp = Float64(z - b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a <= -1.6e-99) || ~((a <= 175000.0)))
		tmp = a + z;
	else
		tmp = z - b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -1.6e-99], N[Not[LessEqual[a, 175000.0]], $MachinePrecision]], N[(a + z), $MachinePrecision], N[(z - b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.6e-99 or 175000 < a

   1. Initial program 52.4%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 59.2

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

   5. Applied rewrites 59.2%

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

   6. Taylor expanded in b around 0

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

      1. lift-+.f64 57.7

         \[\leadsto a + z \]

   8. Applied rewrites 57.7%

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

   if -1.6e-99 < a < 175000

   1. Initial program 72.2%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 50.5

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

   5. Applied rewrites 50.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 47.0%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.6 \cdot 10^{-99} \lor \neg \left(a \leq 175000\right):\\ \;\;\;\;a + z\\ \mathbf{else}:\\ \;\;\;\;z - b\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 49.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{-279}:\\ \;\;\;\;a + z\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;a - b\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x 2.6e-279) (+ a z) (if (<= x 2.5e-5) (- a b) z)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= 2.6e-279) {
		tmp = a + z;
	} else if (x <= 2.5e-5) {
		tmp = a - b;
	} else {
		tmp = z;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (x <= 2.6d-279) then
        tmp = a + z
    else if (x <= 2.5d-5) then
        tmp = a - b
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= 2.6e-279) {
		tmp = a + z;
	} else if (x <= 2.5e-5) {
		tmp = a - b;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= 2.6e-279:
		tmp = a + z
	elif x <= 2.5e-5:
		tmp = a - b
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= 2.6e-279)
		tmp = Float64(a + z);
	elseif (x <= 2.5e-5)
		tmp = Float64(a - b);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= 2.6e-279)
		tmp = a + z;
	elseif (x <= 2.5e-5)
		tmp = a - b;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, 2.6e-279], N[(a + z), $MachinePrecision], If[LessEqual[x, 2.5e-5], N[(a - b), $MachinePrecision], z]]

Derivation

1. Split input into 3 regimes

2. if x < 2.6000000000000002e-279

   1. Initial program 61.3%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 56.6

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

   5. Applied rewrites 56.6%

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

   6. Taylor expanded in b around 0

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

      1. lift-+.f64 52.4

         \[\leadsto a + z \]

   8. Applied rewrites 52.4%

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

   if 2.6000000000000002e-279 < x < 2.50000000000000012e-5

   1. Initial program 65.8%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 63.7

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

   5. Applied rewrites 63.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 51.4%

      \[\leadsto a - b \]

   if 2.50000000000000012e-5 < x

   1. Initial program 55.1%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 42.6%

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

Alternative 14: 42.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-91}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-35}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x -5e-91) z (if (<= x 1.7e-35) a z)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -5e-91) {
		tmp = z;
	} else if (x <= 1.7e-35) {
		tmp = a;
	} else {
		tmp = z;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (x <= (-5d-91)) then
        tmp = z
    else if (x <= 1.7d-35) then
        tmp = a
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -5e-91) {
		tmp = z;
	} else if (x <= 1.7e-35) {
		tmp = a;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= -5e-91:
		tmp = z
	elif x <= 1.7e-35:
		tmp = a
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= -5e-91)
		tmp = z;
	elseif (x <= 1.7e-35)
		tmp = a;
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= -5e-91)
		tmp = z;
	elseif (x <= 1.7e-35)
		tmp = a;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, -5e-91], z, If[LessEqual[x, 1.7e-35], a, z]]

Derivation

1. Split input into 2 regimes

2. if x < -4.99999999999999997e-91 or 1.7000000000000001e-35 < x

   1. Initial program 56.7%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 39.6%

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

   if -4.99999999999999997e-91 < x < 1.7000000000000001e-35

   1. Initial program 66.8%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 46.5%

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

Alternative 15: 53.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.1 \cdot 10^{+178}:\\ \;\;\;\;a + z\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (if (<= x 4.1e+178) (+ a z) z))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= 4.1e+178) {
		tmp = a + z;
	} else {
		tmp = z;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (x <= 4.1d+178) then
        tmp = a + z
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= 4.1e+178) {
		tmp = a + z;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= 4.1e+178:
		tmp = a + z
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= 4.1e+178)
		tmp = Float64(a + z);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= 4.1e+178)
		tmp = a + z;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, 4.1e+178], N[(a + z), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if x < 4.09999999999999996e178

   1. Initial program 62.7%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 57.4

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

   5. Applied rewrites 57.4%

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

   6. Taylor expanded in b around 0

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

      1. lift-+.f64 52.6

         \[\leadsto a + z \]

   8. Applied rewrites 52.6%

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

   if 4.09999999999999996e178 < x

   1. Initial program 44.3%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 55.9%

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

Alternative 16: 32.8% accurate, 45.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.7%

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

3. Taylor expanded in t around inf

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

5. Applied rewrites 32.8%

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

Developer Target 1: 82.0% 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 2025086 
(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)))