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

Percentage Accurate: 60.0% → 98.9%

Time: 7.0s

Alternatives: 21

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: 60.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ t_2 := \left(y + x\right) + t\\ t_3 := \mathsf{fma}\left(\frac{y + x}{t\_2}, z, \frac{t + y}{\frac{t\_2}{a}} - \frac{y}{t\_2} \cdot b\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 20000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{t + y}{t\_2}, a, \frac{\left(y + x\right) \cdot z - b \cdot y}{t\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))
        (t_2 (+ (+ y x) t))
        (t_3
         (fma (/ (+ y x) t_2) z (- (/ (+ t y) (/ t_2 a)) (* (/ y t_2) b)))))
   (if (<= t_1 (- INFINITY))
     t_3
     (if (<= t_1 20000000000.0)
       (fma (/ (+ t y) t_2) a (/ (- (* (+ y x) z) (* b y)) t_2))
       t_3))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
	double t_2 = (y + x) + t;
	double t_3 = fma(((y + x) / t_2), z, (((t + y) / (t_2 / a)) - ((y / t_2) * b)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_1 <= 20000000000.0) {
		tmp = fma(((t + y) / t_2), a, ((((y + x) * z) - (b * y)) / t_2));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / Float64(Float64(x + t) + y))
	t_2 = Float64(Float64(y + x) + t)
	t_3 = fma(Float64(Float64(y + x) / t_2), z, Float64(Float64(Float64(t + y) / Float64(t_2 / a)) - Float64(Float64(y / t_2) * b)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_1 <= 20000000000.0)
		tmp = fma(Float64(Float64(t + y) / t_2), a, Float64(Float64(Float64(Float64(y + x) * z) - Float64(b * y)) / t_2));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(y + x), $MachinePrecision] / t$95$2), $MachinePrecision] * z + N[(N[(N[(t + y), $MachinePrecision] / N[(t$95$2 / a), $MachinePrecision]), $MachinePrecision] - N[(N[(y / t$95$2), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$3, If[LessEqual[t$95$1, 20000000000.0], N[(N[(N[(t + y), $MachinePrecision] / t$95$2), $MachinePrecision] * a + N[(N[(N[(N[(y + x), $MachinePrecision] * z), $MachinePrecision] - N[(b * y), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -inf.0 or 2e10 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 60.0%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 71.1%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 94.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-flip N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 94.2

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

   8. Applied rewrites 94.2%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 2e10

   1. Initial program 60.0%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 71.1%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   6. Applied rewrites 71.1%

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

Alternative 2: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ t_2 := \left(y + x\right) + t\\ t_3 := \mathsf{fma}\left(\frac{y + x}{t\_2}, z, \left(t + y\right) \cdot \frac{a}{t\_2} - \frac{y}{t\_2} \cdot b\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 10^{+217}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t + y}{t\_2}, a, \frac{\left(y + x\right) \cdot z - b \cdot y}{t\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))
        (t_2 (+ (+ y x) t))
        (t_3
         (fma (/ (+ y x) t_2) z (- (* (+ t y) (/ a t_2)) (* (/ y t_2) b)))))
   (if (<= t_1 (- INFINITY))
     t_3
     (if (<= t_1 1e+217)
       (fma (/ (+ t y) t_2) a (/ (- (* (+ y x) z) (* b y)) t_2))
       t_3))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
	double t_2 = (y + x) + t;
	double t_3 = fma(((y + x) / t_2), z, (((t + y) * (a / t_2)) - ((y / t_2) * b)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_1 <= 1e+217) {
		tmp = fma(((t + y) / t_2), a, ((((y + x) * z) - (b * y)) / t_2));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / Float64(Float64(x + t) + y))
	t_2 = Float64(Float64(y + x) + t)
	t_3 = fma(Float64(Float64(y + x) / t_2), z, Float64(Float64(Float64(t + y) * Float64(a / t_2)) - Float64(Float64(y / t_2) * b)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_1 <= 1e+217)
		tmp = fma(Float64(Float64(t + y) / t_2), a, Float64(Float64(Float64(Float64(y + x) * z) - Float64(b * y)) / t_2));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(y + x), $MachinePrecision] / t$95$2), $MachinePrecision] * z + N[(N[(N[(t + y), $MachinePrecision] * N[(a / t$95$2), $MachinePrecision]), $MachinePrecision] - N[(N[(y / t$95$2), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$3, If[LessEqual[t$95$1, 1e+217], N[(N[(N[(t + y), $MachinePrecision] / t$95$2), $MachinePrecision] * a + N[(N[(N[(N[(y + x), $MachinePrecision] * z), $MachinePrecision] - N[(b * y), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -inf.0 or 9.9999999999999996e216 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 60.0%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 71.1%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 94.4%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 9.9999999999999996e216

   1. Initial program 60.0%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 71.1%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   6. Applied rewrites 71.1%

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

Alternative 3: 95.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ t_2 := \left(y + x\right) + t\\ t_3 := \frac{y}{t\_2} \cdot b\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{t\_2}, z, a - t\_3\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+217}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t + y}{t\_2}, a, \frac{\left(y + x\right) \cdot z - b \cdot y}{t\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \frac{t + y}{\frac{t\_2}{a}} - t\_3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))
        (t_2 (+ (+ y x) t))
        (t_3 (* (/ y t_2) b)))
   (if (<= t_1 (- INFINITY))
     (fma (/ (+ y x) t_2) z (- a t_3))
     (if (<= t_1 1e+217)
       (fma (/ (+ t y) t_2) a (/ (- (* (+ y x) z) (* b y)) t_2))
       (fma 1.0 z (- (/ (+ t y) (/ t_2 a)) t_3))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
	double t_2 = (y + x) + t;
	double t_3 = (y / t_2) * b;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(((y + x) / t_2), z, (a - t_3));
	} else if (t_1 <= 1e+217) {
		tmp = fma(((t + y) / t_2), a, ((((y + x) * z) - (b * y)) / t_2));
	} else {
		tmp = fma(1.0, z, (((t + y) / (t_2 / a)) - t_3));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / Float64(Float64(x + t) + y))
	t_2 = Float64(Float64(y + x) + t)
	t_3 = Float64(Float64(y / t_2) * b)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(Float64(Float64(y + x) / t_2), z, Float64(a - t_3));
	elseif (t_1 <= 1e+217)
		tmp = fma(Float64(Float64(t + y) / t_2), a, Float64(Float64(Float64(Float64(y + x) * z) - Float64(b * y)) / t_2));
	else
		tmp = fma(1.0, z, Float64(Float64(Float64(t + y) / Float64(t_2 / a)) - t_3));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y / t$95$2), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(y + x), $MachinePrecision] / t$95$2), $MachinePrecision] * z + N[(a - t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+217], N[(N[(N[(t + y), $MachinePrecision] / t$95$2), $MachinePrecision] * a + N[(N[(N[(N[(y + x), $MachinePrecision] * z), $MachinePrecision] - N[(b * y), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], N[(1.0 * z + N[(N[(N[(t + y), $MachinePrecision] / N[(t$95$2 / a), $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 60.0%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 71.1%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 94.4%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 84.0%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 9.9999999999999996e216

   1. Initial program 60.0%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 71.1%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   6. Applied rewrites 71.1%

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

   if 9.9999999999999996e216 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 60.0%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 71.1%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 94.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-flip N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 94.2

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

   8. Applied rewrites 94.2%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 81.5%

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

Alternative 4: 95.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ t_3 := \left(y + x\right) + t\\ t_4 := \frac{y}{t\_3} \cdot b\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{t\_3}, z, a - t\_4\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+217}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, \mathsf{fma}\left(x, z, y \cdot \left(\left(a + z\right) - b\right)\right)\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \frac{t + y}{\frac{t\_3}{a}} - t\_4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ x t) y))
        (t_2 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) t_1))
        (t_3 (+ (+ y x) t))
        (t_4 (* (/ y t_3) b)))
   (if (<= t_2 (- INFINITY))
     (fma (/ (+ y x) t_3) z (- a t_4))
     (if (<= t_2 1e+217)
       (/ (fma a t (fma x z (* y (- (+ a z) b)))) t_1)
       (fma 1.0 z (- (/ (+ t y) (/ t_3 a)) t_4))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + t) + y;
	double t_2 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / t_1;
	double t_3 = (y + x) + t;
	double t_4 = (y / t_3) * b;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma(((y + x) / t_3), z, (a - t_4));
	} else if (t_2 <= 1e+217) {
		tmp = fma(a, t, fma(x, z, (y * ((a + z) - b)))) / t_1;
	} else {
		tmp = fma(1.0, z, (((t + y) / (t_3 / a)) - t_4));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + t) + y)
	t_2 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / t_1)
	t_3 = Float64(Float64(y + x) + t)
	t_4 = Float64(Float64(y / t_3) * b)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = fma(Float64(Float64(y + x) / t_3), z, Float64(a - t_4));
	elseif (t_2 <= 1e+217)
		tmp = Float64(fma(a, t, fma(x, z, Float64(y * Float64(Float64(a + z) - b)))) / t_1);
	else
		tmp = fma(1.0, z, Float64(Float64(Float64(t + y) / Float64(t_3 / a)) - t_4));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$4 = N[(N[(y / t$95$3), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[(y + x), $MachinePrecision] / t$95$3), $MachinePrecision] * z + N[(a - t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+217], N[(N[(a * t + N[(x * z + N[(y * N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(1.0 * z + N[(N[(N[(t + y), $MachinePrecision] / N[(t$95$3 / a), $MachinePrecision]), $MachinePrecision] - t$95$4), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 60.0%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 71.1%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 94.4%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 84.0%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 9.9999999999999996e216

   1. Initial program 60.0%

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

   2. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-+.f64 60.3

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

   4. Applied rewrites 60.3%

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

   if 9.9999999999999996e216 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 60.0%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 71.1%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 94.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-flip N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 94.2

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

   8. Applied rewrites 94.2%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 81.5%

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

Alternative 5: 95.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ t_3 := \left(y + x\right) + t\\ t_4 := \frac{y}{t\_3} \cdot b\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y + x}{t\_3}, z, a - t\_4\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+217}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, \mathsf{fma}\left(x, z, y \cdot \left(\left(a + z\right) - b\right)\right)\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \left(t + y\right) \cdot \frac{a}{t\_3} - t\_4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ x t) y))
        (t_2 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) t_1))
        (t_3 (+ (+ y x) t))
        (t_4 (* (/ y t_3) b)))
   (if (<= t_2 (- INFINITY))
     (fma (/ (+ y x) t_3) z (- a t_4))
     (if (<= t_2 1e+217)
       (/ (fma a t (fma x z (* y (- (+ a z) b)))) t_1)
       (fma 1.0 z (- (* (+ t y) (/ a t_3)) t_4))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + t) + y;
	double t_2 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / t_1;
	double t_3 = (y + x) + t;
	double t_4 = (y / t_3) * b;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma(((y + x) / t_3), z, (a - t_4));
	} else if (t_2 <= 1e+217) {
		tmp = fma(a, t, fma(x, z, (y * ((a + z) - b)))) / t_1;
	} else {
		tmp = fma(1.0, z, (((t + y) * (a / t_3)) - t_4));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + t) + y)
	t_2 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / t_1)
	t_3 = Float64(Float64(y + x) + t)
	t_4 = Float64(Float64(y / t_3) * b)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = fma(Float64(Float64(y + x) / t_3), z, Float64(a - t_4));
	elseif (t_2 <= 1e+217)
		tmp = Float64(fma(a, t, fma(x, z, Float64(y * Float64(Float64(a + z) - b)))) / t_1);
	else
		tmp = fma(1.0, z, Float64(Float64(Float64(t + y) * Float64(a / t_3)) - t_4));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$4 = N[(N[(y / t$95$3), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[(y + x), $MachinePrecision] / t$95$3), $MachinePrecision] * z + N[(a - t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+217], N[(N[(a * t + N[(x * z + N[(y * N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(1.0 * z + N[(N[(N[(t + y), $MachinePrecision] * N[(a / t$95$3), $MachinePrecision]), $MachinePrecision] - t$95$4), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 60.0%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 71.1%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 94.4%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 84.0%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 9.9999999999999996e216

   1. Initial program 60.0%

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

   2. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-+.f64 60.3

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

   4. Applied rewrites 60.3%

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

   if 9.9999999999999996e216 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 60.0%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 71.1%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 94.4%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 81.6%

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

Alternative 6: 94.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ t_3 := \left(y + x\right) + t\\ t_4 := \mathsf{fma}\left(\frac{y + x}{t\_3}, z, a - \frac{y}{t\_3} \cdot b\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq 10^{+217}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, \mathsf{fma}\left(x, z, y \cdot \left(\left(a + z\right) - b\right)\right)\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ x t) y))
        (t_2 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) t_1))
        (t_3 (+ (+ y x) t))
        (t_4 (fma (/ (+ y x) t_3) z (- a (* (/ y t_3) b)))))
   (if (<= t_2 (- INFINITY))
     t_4
     (if (<= t_2 1e+217)
       (/ (fma a t (fma x z (* y (- (+ a z) b)))) t_1)
       t_4))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + t) + y;
	double t_2 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / t_1;
	double t_3 = (y + x) + t;
	double t_4 = fma(((y + x) / t_3), z, (a - ((y / t_3) * b)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_2 <= 1e+217) {
		tmp = fma(a, t, fma(x, z, (y * ((a + z) - b)))) / t_1;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + t) + y)
	t_2 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / t_1)
	t_3 = Float64(Float64(y + x) + t)
	t_4 = fma(Float64(Float64(y + x) / t_3), z, Float64(a - Float64(Float64(y / t_3) * b)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_2 <= 1e+217)
		tmp = Float64(fma(a, t, fma(x, z, Float64(y * Float64(Float64(a + z) - b)))) / t_1);
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y + x), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(y + x), $MachinePrecision] / t$95$3), $MachinePrecision] * z + N[(a - N[(N[(y / t$95$3), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$4, If[LessEqual[t$95$2, 1e+217], N[(N[(a * t + N[(x * z + N[(y * N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$4]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -inf.0 or 9.9999999999999996e216 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 60.0%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 71.1%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 94.4%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 84.0%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 9.9999999999999996e216

   1. Initial program 60.0%

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

   2. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-+.f64 60.3

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

   4. Applied rewrites 60.3%

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

Alternative 7: 87.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ t_3 := \left(a + z\right) - b\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+240}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, \mathsf{fma}\left(x, z, y \cdot t\_3\right)\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ x t) y))
        (t_2 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) t_1))
        (t_3 (- (+ a z) b)))
   (if (<= t_2 (- INFINITY))
     t_3
     (if (<= t_2 2e+240) (/ (fma a t (fma x z (* y t_3))) t_1) t_3))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + t) + y;
	double t_2 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / t_1;
	double t_3 = (a + z) - b;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= 2e+240) {
		tmp = fma(a, t, fma(x, z, (y * t_3))) / t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + t) + y)
	t_2 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / t_1)
	t_3 = Float64(Float64(a + z) - b)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= 2e+240)
		tmp = Float64(fma(a, t, fma(x, z, Float64(y * t_3))) / t_1);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, 2e+240], N[(N[(a * t + N[(x * z + N[(y * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -inf.0 or 2.00000000000000003e240 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 60.0%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 55.5

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

   4. Applied rewrites 55.5%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 2.00000000000000003e240

   1. Initial program 60.0%

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

   2. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-+.f64 60.3

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

   4. Applied rewrites 60.3%

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

Alternative 8: 80.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \left(a + z\right) - b\\ t_3 := \left(x + y\right) \cdot z\\ t_4 := \frac{\left(t\_3 + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{+251}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+240}:\\ \;\;\;\;\frac{\left(t\_3 + a \cdot t\right) - y \cdot b}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ x t) y))
        (t_2 (- (+ a z) b))
        (t_3 (* (+ x y) z))
        (t_4 (/ (- (+ t_3 (* (+ t y) a)) (* y b)) t_1)))
   (if (<= t_4 -5e+251)
     t_2
     (if (<= t_4 2e+240) (/ (- (+ t_3 (* a t)) (* y b)) t_1) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + t) + y;
	double t_2 = (a + z) - b;
	double t_3 = (x + y) * z;
	double t_4 = ((t_3 + ((t + y) * a)) - (y * b)) / t_1;
	double tmp;
	if (t_4 <= -5e+251) {
		tmp = t_2;
	} else if (t_4 <= 2e+240) {
		tmp = ((t_3 + (a * t)) - (y * b)) / t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = (x + t) + y
    t_2 = (a + z) - b
    t_3 = (x + y) * z
    t_4 = ((t_3 + ((t + y) * a)) - (y * b)) / t_1
    if (t_4 <= (-5d+251)) then
        tmp = t_2
    else if (t_4 <= 2d+240) then
        tmp = ((t_3 + (a * t)) - (y * b)) / t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + t) + y;
	double t_2 = (a + z) - b;
	double t_3 = (x + y) * z;
	double t_4 = ((t_3 + ((t + y) * a)) - (y * b)) / t_1;
	double tmp;
	if (t_4 <= -5e+251) {
		tmp = t_2;
	} else if (t_4 <= 2e+240) {
		tmp = ((t_3 + (a * t)) - (y * b)) / t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x + t) + y
	t_2 = (a + z) - b
	t_3 = (x + y) * z
	t_4 = ((t_3 + ((t + y) * a)) - (y * b)) / t_1
	tmp = 0
	if t_4 <= -5e+251:
		tmp = t_2
	elif t_4 <= 2e+240:
		tmp = ((t_3 + (a * t)) - (y * b)) / t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + t) + y)
	t_2 = Float64(Float64(a + z) - b)
	t_3 = Float64(Float64(x + y) * z)
	t_4 = Float64(Float64(Float64(t_3 + Float64(Float64(t + y) * a)) - Float64(y * b)) / t_1)
	tmp = 0.0
	if (t_4 <= -5e+251)
		tmp = t_2;
	elseif (t_4 <= 2e+240)
		tmp = Float64(Float64(Float64(t_3 + Float64(a * t)) - Float64(y * b)) / t_1);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (x + t) + y;
	t_2 = (a + z) - b;
	t_3 = (x + y) * z;
	t_4 = ((t_3 + ((t + y) * a)) - (y * b)) / t_1;
	tmp = 0.0;
	if (t_4 <= -5e+251)
		tmp = t_2;
	elseif (t_4 <= 2e+240)
		tmp = ((t_3 + (a * t)) - (y * b)) / t_1;
	else
		tmp = t_2;
	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[(a + z), $MachinePrecision] - b), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(t$95$3 + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$4, -5e+251], t$95$2, If[LessEqual[t$95$4, 2e+240], N[(N[(N[(t$95$3 + N[(a * t), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -5.0000000000000005e251 or 2.00000000000000003e240 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 60.0%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 55.5

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

   4. Applied rewrites 55.5%

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

   if -5.0000000000000005e251 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 2.00000000000000003e240

   1. Initial program 60.0%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 52.6

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

   4. Applied rewrites 52.6%

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

Alternative 9: 79.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ t_3 := \left(a + z\right) - b\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{+75}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, y \cdot t\_3\right)}{t\_1}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+240}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, \mathsf{fma}\left(x, z, y \cdot \left(a - b\right)\right)\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ x t) y))
        (t_2 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) t_1))
        (t_3 (- (+ a z) b)))
   (if (<= t_2 (- INFINITY))
     t_3
     (if (<= t_2 -1e+75)
       (/ (fma a t (* y t_3)) t_1)
       (if (<= t_2 2e+240) (/ (fma a t (fma x z (* y (- a b)))) t_1) t_3)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + t) + y;
	double t_2 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / t_1;
	double t_3 = (a + z) - b;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= -1e+75) {
		tmp = fma(a, t, (y * t_3)) / t_1;
	} else if (t_2 <= 2e+240) {
		tmp = fma(a, t, fma(x, z, (y * (a - b)))) / t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + t) + y)
	t_2 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / t_1)
	t_3 = Float64(Float64(a + z) - b)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= -1e+75)
		tmp = Float64(fma(a, t, Float64(y * t_3)) / t_1);
	elseif (t_2 <= 2e+240)
		tmp = Float64(fma(a, t, fma(x, z, Float64(y * Float64(a - b)))) / t_1);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, -1e+75], N[(N[(a * t + N[(y * t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 2e+240], N[(N[(a * t + N[(x * z + N[(y * N[(a - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$3]]]]]]

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 or 2.00000000000000003e240 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 60.0%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 55.5

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

   4. Applied rewrites 55.5%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -9.99999999999999927e74

   1. Initial program 60.0%

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

   2. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-+.f64 60.3

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

   4. Applied rewrites 60.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 45.0

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

   7. Applied rewrites 45.0%

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

   if -9.99999999999999927e74 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 2.00000000000000003e240

   1. Initial program 60.0%

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

   2. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-+.f64 60.3

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

   4. Applied rewrites 60.3%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 52.2%

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

Alternative 10: 75.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ t_3 := \left(a + z\right) - b\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-40}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, y \cdot t\_3\right)}{t\_1}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+240}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t + y, z \cdot \left(x + y\right)\right)}{t + \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ x t) y))
        (t_2 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) t_1))
        (t_3 (- (+ a z) b)))
   (if (<= t_2 (- INFINITY))
     t_3
     (if (<= t_2 -1e-40)
       (/ (fma a t (* y t_3)) t_1)
       (if (<= t_2 2e+240)
         (/ (fma a (+ t y) (* z (+ x y))) (+ t (+ x y)))
         t_3)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + t) + y;
	double t_2 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / t_1;
	double t_3 = (a + z) - b;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= -1e-40) {
		tmp = fma(a, t, (y * t_3)) / t_1;
	} else if (t_2 <= 2e+240) {
		tmp = fma(a, (t + y), (z * (x + y))) / (t + (x + y));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + t) + y)
	t_2 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / t_1)
	t_3 = Float64(Float64(a + z) - b)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= -1e-40)
		tmp = Float64(fma(a, t, Float64(y * t_3)) / t_1);
	elseif (t_2 <= 2e+240)
		tmp = Float64(fma(a, Float64(t + y), Float64(z * Float64(x + y))) / Float64(t + Float64(x + y)));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, -1e-40], N[(N[(a * t + N[(y * t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 2e+240], N[(N[(a * N[(t + y), $MachinePrecision] + N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

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 or 2.00000000000000003e240 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 60.0%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 55.5

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

   4. Applied rewrites 55.5%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -9.9999999999999993e-41

   1. Initial program 60.0%

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

   2. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-+.f64 60.3

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

   4. Applied rewrites 60.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 45.0

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

   7. Applied rewrites 45.0%

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

   if -9.9999999999999993e-41 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 2.00000000000000003e240

   1. Initial program 60.0%

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

   2. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 47.5

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

   4. Applied rewrites 47.5%

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

Alternative 11: 72.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ t_3 := \left(a + z\right) - b\\ t_4 := \frac{\mathsf{fma}\left(a, t, y \cdot t\_3\right)}{t\_1}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 10^{-79}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, z, y \cdot \left(z - b\right)\right)}{t\_1}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+240}:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ x t) y))
        (t_2 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) t_1))
        (t_3 (- (+ a z) b))
        (t_4 (/ (fma a t (* y t_3)) t_1)))
   (if (<= t_2 (- INFINITY))
     t_3
     (if (<= t_2 1e-79)
       t_4
       (if (<= t_2 5e-7)
         (/ (fma x z (* y (- z b))) t_1)
         (if (<= t_2 2e+240) t_4 t_3))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + t) + y;
	double t_2 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / t_1;
	double t_3 = (a + z) - b;
	double t_4 = fma(a, t, (y * t_3)) / t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= 1e-79) {
		tmp = t_4;
	} else if (t_2 <= 5e-7) {
		tmp = fma(x, z, (y * (z - b))) / t_1;
	} else if (t_2 <= 2e+240) {
		tmp = t_4;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + t) + y)
	t_2 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / t_1)
	t_3 = Float64(Float64(a + z) - b)
	t_4 = Float64(fma(a, t, Float64(y * t_3)) / t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= 1e-79)
		tmp = t_4;
	elseif (t_2 <= 5e-7)
		tmp = Float64(fma(x, z, Float64(y * Float64(z - b))) / t_1);
	elseif (t_2 <= 2e+240)
		tmp = t_4;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, Block[{t$95$4 = N[(N[(a * t + N[(y * t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, 1e-79], t$95$4, If[LessEqual[t$95$2, 5e-7], N[(N[(x * z + N[(y * N[(z - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 2e+240], t$95$4, t$95$3]]]]]]]]

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 or 2.00000000000000003e240 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 60.0%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 55.5

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

   4. Applied rewrites 55.5%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1e-79 or 4.99999999999999977e-7 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 2.00000000000000003e240

   1. Initial program 60.0%

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

   2. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-+.f64 60.3

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

   4. Applied rewrites 60.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 45.0

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

   7. Applied rewrites 45.0%

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

   if 1e-79 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4.99999999999999977e-7

   1. Initial program 60.0%

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

   2. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-+.f64 60.3

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

   4. Applied rewrites 60.3%

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

   5. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 38.0

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

   7. Applied rewrites 38.0%

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

Alternative 12: 64.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + z\right) - b\\ \mathbf{if}\;y \leq -5.7 \cdot 10^{+119}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-161}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{t + y}, a\right)\\ \mathbf{elif}\;y \leq 10^{-92}:\\ \;\;\;\;\frac{a \cdot t}{t + x} + \frac{x \cdot z}{t + x}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{y \cdot t\_1}{\left(x + t\right) + y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (+ a z) b)))
   (if (<= y -5.7e+119)
     t_1
     (if (<= y -1.6e-161)
       (fma z (/ y (+ t y)) a)
       (if (<= y 1e-92)
         (+ (/ (* a t) (+ t x)) (/ (* x z) (+ t x)))
         (if (<= y 1.6e-7) (/ (* y t_1) (+ (+ x t) y)) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a + z) - b;
	double tmp;
	if (y <= -5.7e+119) {
		tmp = t_1;
	} else if (y <= -1.6e-161) {
		tmp = fma(z, (y / (t + y)), a);
	} else if (y <= 1e-92) {
		tmp = ((a * t) / (t + x)) + ((x * z) / (t + x));
	} else if (y <= 1.6e-7) {
		tmp = (y * t_1) / ((x + t) + y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a + z) - b)
	tmp = 0.0
	if (y <= -5.7e+119)
		tmp = t_1;
	elseif (y <= -1.6e-161)
		tmp = fma(z, Float64(y / Float64(t + y)), a);
	elseif (y <= 1e-92)
		tmp = Float64(Float64(Float64(a * t) / Float64(t + x)) + Float64(Float64(x * z) / Float64(t + x)));
	elseif (y <= 1.6e-7)
		tmp = Float64(Float64(y * t_1) / Float64(Float64(x + t) + y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[y, -5.7e+119], t$95$1, If[LessEqual[y, -1.6e-161], N[(z * N[(y / N[(t + y), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision], If[LessEqual[y, 1e-92], N[(N[(N[(a * t), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision] + N[(N[(x * z), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.6e-7], N[(N[(y * t$95$1), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -5.7000000000000002e119 or 1.6e-7 < y

   1. Initial program 60.0%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 55.5

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

   4. Applied rewrites 55.5%

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

   if -5.7000000000000002e119 < y < -1.59999999999999993e-161

   1. Initial program 60.0%

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

   2. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 47.5

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

   4. Applied rewrites 47.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 31.9

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

   7. Applied rewrites 31.9%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 52.5

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

   9. Applied rewrites 52.5%

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

   if -1.59999999999999993e-161 < y < 9.99999999999999988e-93

   1. Initial program 60.0%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 71.1%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 40.0

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

   7. Applied rewrites 40.0%

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

   if 9.99999999999999988e-93 < y < 1.6e-7

   1. Initial program 60.0%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 30.2

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

   4. Applied rewrites 30.2%

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

Alternative 13: 63.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + z\right) - b\\ \mathbf{if}\;y \leq -5.7 \cdot 10^{+119}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-161}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{t + y}, a\right)\\ \mathbf{elif}\;y \leq 10^{-92}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, x \cdot z\right)}{t + x}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{y \cdot t\_1}{\left(x + t\right) + y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (+ a z) b)))
   (if (<= y -5.7e+119)
     t_1
     (if (<= y -1.6e-161)
       (fma z (/ y (+ t y)) a)
       (if (<= y 1e-92)
         (/ (fma a t (* x z)) (+ t x))
         (if (<= y 1.6e-7) (/ (* y t_1) (+ (+ x t) y)) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a + z) - b;
	double tmp;
	if (y <= -5.7e+119) {
		tmp = t_1;
	} else if (y <= -1.6e-161) {
		tmp = fma(z, (y / (t + y)), a);
	} else if (y <= 1e-92) {
		tmp = fma(a, t, (x * z)) / (t + x);
	} else if (y <= 1.6e-7) {
		tmp = (y * t_1) / ((x + t) + y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a + z) - b)
	tmp = 0.0
	if (y <= -5.7e+119)
		tmp = t_1;
	elseif (y <= -1.6e-161)
		tmp = fma(z, Float64(y / Float64(t + y)), a);
	elseif (y <= 1e-92)
		tmp = Float64(fma(a, t, Float64(x * z)) / Float64(t + x));
	elseif (y <= 1.6e-7)
		tmp = Float64(Float64(y * t_1) / Float64(Float64(x + t) + y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[y, -5.7e+119], t$95$1, If[LessEqual[y, -1.6e-161], N[(z * N[(y / N[(t + y), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision], If[LessEqual[y, 1e-92], N[(N[(a * t + N[(x * z), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.6e-7], N[(N[(y * t$95$1), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -5.7000000000000002e119 or 1.6e-7 < y

   1. Initial program 60.0%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 55.5

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

   4. Applied rewrites 55.5%

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

   if -5.7000000000000002e119 < y < -1.59999999999999993e-161

   1. Initial program 60.0%

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

   2. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 47.5

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

   4. Applied rewrites 47.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 31.9

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

   7. Applied rewrites 31.9%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 52.5

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

   9. Applied rewrites 52.5%

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

   if -1.59999999999999993e-161 < y < 9.99999999999999988e-93

   1. Initial program 60.0%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 40.0

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

   4. Applied rewrites 40.0%

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

   if 9.99999999999999988e-93 < y < 1.6e-7

   1. Initial program 60.0%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 30.2

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

   4. Applied rewrites 30.2%

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

Alternative 14: 63.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{y}{t + y}, a\right)\\ t_2 := \left(a + z\right) - b\\ \mathbf{if}\;y \leq -5.7 \cdot 10^{+119}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-161}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{-103}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, x \cdot z\right)}{t + x}\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{+76}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z (/ y (+ t y)) a)) (t_2 (- (+ a z) b)))
   (if (<= y -5.7e+119)
     t_2
     (if (<= y -1.6e-161)
       t_1
       (if (<= y 6.4e-103)
         (/ (fma a t (* x z)) (+ t x))
         (if (<= y 4.3e+76) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(z, (y / (t + y)), a);
	double t_2 = (a + z) - b;
	double tmp;
	if (y <= -5.7e+119) {
		tmp = t_2;
	} else if (y <= -1.6e-161) {
		tmp = t_1;
	} else if (y <= 6.4e-103) {
		tmp = fma(a, t, (x * z)) / (t + x);
	} else if (y <= 4.3e+76) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(z, Float64(y / Float64(t + y)), a)
	t_2 = Float64(Float64(a + z) - b)
	tmp = 0.0
	if (y <= -5.7e+119)
		tmp = t_2;
	elseif (y <= -1.6e-161)
		tmp = t_1;
	elseif (y <= 6.4e-103)
		tmp = Float64(fma(a, t, Float64(x * z)) / Float64(t + x));
	elseif (y <= 4.3e+76)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(y / N[(t + y), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[y, -5.7e+119], t$95$2, If[LessEqual[y, -1.6e-161], t$95$1, If[LessEqual[y, 6.4e-103], N[(N[(a * t + N[(x * z), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.3e+76], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.7000000000000002e119 or 4.29999999999999978e76 < y

   1. Initial program 60.0%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 55.5

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

   4. Applied rewrites 55.5%

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

   if -5.7000000000000002e119 < y < -1.59999999999999993e-161 or 6.39999999999999953e-103 < y < 4.29999999999999978e76

   1. Initial program 60.0%

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

   2. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 47.5

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

   4. Applied rewrites 47.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 31.9

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

   7. Applied rewrites 31.9%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 52.5

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

   9. Applied rewrites 52.5%

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

   if -1.59999999999999993e-161 < y < 6.39999999999999953e-103

   1. Initial program 60.0%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 40.0

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

   4. Applied rewrites 40.0%

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

Alternative 15: 61.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + z\right) - b\\ \mathbf{if}\;y \leq -6.6 \cdot 10^{+49}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{-236}:\\ \;\;\;\;a + z\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{-103}:\\ \;\;\;\;\frac{x \cdot z}{t + x}\\ \mathbf{elif}\;y \leq 9.4 \cdot 10^{+21}:\\ \;\;\;\;a + \frac{y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (+ a z) b)))
   (if (<= y -6.6e+49)
     t_1
     (if (<= y -2.1e-236)
       (+ a z)
       (if (<= y 5.4e-103)
         (/ (* x z) (+ t x))
         (if (<= y 9.4e+21) (+ a (/ (* y z) t)) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a + z) - b;
	double tmp;
	if (y <= -6.6e+49) {
		tmp = t_1;
	} else if (y <= -2.1e-236) {
		tmp = a + z;
	} else if (y <= 5.4e-103) {
		tmp = (x * z) / (t + x);
	} else if (y <= 9.4e+21) {
		tmp = a + ((y * z) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a + z) - b
    if (y <= (-6.6d+49)) then
        tmp = t_1
    else if (y <= (-2.1d-236)) then
        tmp = a + z
    else if (y <= 5.4d-103) then
        tmp = (x * z) / (t + x)
    else if (y <= 9.4d+21) then
        tmp = a + ((y * z) / t)
    else
        tmp = t_1
    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 tmp;
	if (y <= -6.6e+49) {
		tmp = t_1;
	} else if (y <= -2.1e-236) {
		tmp = a + z;
	} else if (y <= 5.4e-103) {
		tmp = (x * z) / (t + x);
	} else if (y <= 9.4e+21) {
		tmp = a + ((y * z) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a + z) - b
	tmp = 0
	if y <= -6.6e+49:
		tmp = t_1
	elif y <= -2.1e-236:
		tmp = a + z
	elif y <= 5.4e-103:
		tmp = (x * z) / (t + x)
	elif y <= 9.4e+21:
		tmp = a + ((y * z) / t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a + z) - b)
	tmp = 0.0
	if (y <= -6.6e+49)
		tmp = t_1;
	elseif (y <= -2.1e-236)
		tmp = Float64(a + z);
	elseif (y <= 5.4e-103)
		tmp = Float64(Float64(x * z) / Float64(t + x));
	elseif (y <= 9.4e+21)
		tmp = Float64(a + Float64(Float64(y * z) / t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a + z) - b;
	tmp = 0.0;
	if (y <= -6.6e+49)
		tmp = t_1;
	elseif (y <= -2.1e-236)
		tmp = a + z;
	elseif (y <= 5.4e-103)
		tmp = (x * z) / (t + x);
	elseif (y <= 9.4e+21)
		tmp = a + ((y * z) / t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[y, -6.6e+49], t$95$1, If[LessEqual[y, -2.1e-236], N[(a + z), $MachinePrecision], If[LessEqual[y, 5.4e-103], N[(N[(x * z), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.4e+21], N[(a + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -6.5999999999999997e49 or 9.4e21 < y

   1. Initial program 60.0%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 55.5

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

   4. Applied rewrites 55.5%

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

   if -6.5999999999999997e49 < y < -2.09999999999999979e-236

   1. Initial program 60.0%

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

   2. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 47.5

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

   4. Applied rewrites 47.5%

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

   5. Taylor expanded in y around inf

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

      1. lower-+.f64 51.2

         \[\leadsto a + z \]

   7. Applied rewrites 51.2%

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

   if -2.09999999999999979e-236 < y < 5.40000000000000019e-103

   1. Initial program 60.0%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 25.4

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

   4. Applied rewrites 25.4%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 21.8

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

   7. Applied rewrites 21.8%

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

   if 5.40000000000000019e-103 < y < 9.4e21

   1. Initial program 60.0%

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

   2. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 47.5

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

   4. Applied rewrites 47.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 31.9

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

   7. Applied rewrites 31.9%

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

   8. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 28.4

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

   10. Applied rewrites 28.4%

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

Alternative 16: 59.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{z}{t + y}, a\right)\\ \mathbf{if}\;t \leq -1.4 \cdot 10^{+47}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+69}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma y (/ z (+ t y)) a)))
   (if (<= t -1.4e+47) t_1 (if (<= t 1.25e+69) (- (+ a z) b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(y, (z / (t + y)), a);
	double tmp;
	if (t <= -1.4e+47) {
		tmp = t_1;
	} else if (t <= 1.25e+69) {
		tmp = (a + z) - b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(y, Float64(z / Float64(t + y)), a)
	tmp = 0.0
	if (t <= -1.4e+47)
		tmp = t_1;
	elseif (t <= 1.25e+69)
		tmp = Float64(Float64(a + z) - b);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y * N[(z / N[(t + y), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]}, If[LessEqual[t, -1.4e+47], t$95$1, If[LessEqual[t, 1.25e+69], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.39999999999999994e47 or 1.25000000000000009e69 < t

   1. Initial program 60.0%

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

   2. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 47.5

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

   4. Applied rewrites 47.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 31.9

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

   7. Applied rewrites 31.9%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 49.9

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

   9. Applied rewrites 49.9%

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

   if -1.39999999999999994e47 < t < 1.25000000000000009e69

   1. Initial program 60.0%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 55.5

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

   4. Applied rewrites 55.5%

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

Alternative 17: 56.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + z\right) - b\\ \mathbf{if}\;y \leq -6.6 \cdot 10^{+49}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{-236}:\\ \;\;\;\;a + z\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{-103}:\\ \;\;\;\;\frac{x \cdot z}{t + x}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (+ a z) b)))
   (if (<= y -6.6e+49)
     t_1
     (if (<= y -2.1e-236)
       (+ a z)
       (if (<= y 5.4e-103) (/ (* x z) (+ t x)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a + z) - b;
	double tmp;
	if (y <= -6.6e+49) {
		tmp = t_1;
	} else if (y <= -2.1e-236) {
		tmp = a + z;
	} else if (y <= 5.4e-103) {
		tmp = (x * z) / (t + x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a + z) - b
    if (y <= (-6.6d+49)) then
        tmp = t_1
    else if (y <= (-2.1d-236)) then
        tmp = a + z
    else if (y <= 5.4d-103) then
        tmp = (x * z) / (t + x)
    else
        tmp = t_1
    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 tmp;
	if (y <= -6.6e+49) {
		tmp = t_1;
	} else if (y <= -2.1e-236) {
		tmp = a + z;
	} else if (y <= 5.4e-103) {
		tmp = (x * z) / (t + x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a + z) - b
	tmp = 0
	if y <= -6.6e+49:
		tmp = t_1
	elif y <= -2.1e-236:
		tmp = a + z
	elif y <= 5.4e-103:
		tmp = (x * z) / (t + x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a + z) - b)
	tmp = 0.0
	if (y <= -6.6e+49)
		tmp = t_1;
	elseif (y <= -2.1e-236)
		tmp = Float64(a + z);
	elseif (y <= 5.4e-103)
		tmp = Float64(Float64(x * z) / Float64(t + x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a + z) - b;
	tmp = 0.0;
	if (y <= -6.6e+49)
		tmp = t_1;
	elseif (y <= -2.1e-236)
		tmp = a + z;
	elseif (y <= 5.4e-103)
		tmp = (x * z) / (t + x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[y, -6.6e+49], t$95$1, If[LessEqual[y, -2.1e-236], N[(a + z), $MachinePrecision], If[LessEqual[y, 5.4e-103], N[(N[(x * z), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.5999999999999997e49 or 5.40000000000000019e-103 < y

   1. Initial program 60.0%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 55.5

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

   4. Applied rewrites 55.5%

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

   if -6.5999999999999997e49 < y < -2.09999999999999979e-236

   1. Initial program 60.0%

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

   2. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 47.5

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

   4. Applied rewrites 47.5%

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

   5. Taylor expanded in y around inf

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

      1. lower-+.f64 51.2

         \[\leadsto a + z \]

   7. Applied rewrites 51.2%

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

   if -2.09999999999999979e-236 < y < 5.40000000000000019e-103

   1. Initial program 60.0%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 25.4

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

   4. Applied rewrites 25.4%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 21.8

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

   7. Applied rewrites 21.8%

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

Alternative 18: 54.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + z\right) - b\\ \mathbf{if}\;y \leq -6.6 \cdot 10^{+49}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{-85}:\\ \;\;\;\;a + z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (+ a z) b)))
   (if (<= y -6.6e+49) t_1 (if (<= y 4.7e-85) (+ a z) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a + z) - b;
	double tmp;
	if (y <= -6.6e+49) {
		tmp = t_1;
	} else if (y <= 4.7e-85) {
		tmp = a + z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a + z) - b
    if (y <= (-6.6d+49)) then
        tmp = t_1
    else if (y <= 4.7d-85) then
        tmp = a + z
    else
        tmp = t_1
    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 tmp;
	if (y <= -6.6e+49) {
		tmp = t_1;
	} else if (y <= 4.7e-85) {
		tmp = a + z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a + z) - b
	tmp = 0
	if y <= -6.6e+49:
		tmp = t_1
	elif y <= 4.7e-85:
		tmp = a + z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a + z) - b)
	tmp = 0.0
	if (y <= -6.6e+49)
		tmp = t_1;
	elseif (y <= 4.7e-85)
		tmp = Float64(a + z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a + z) - b;
	tmp = 0.0;
	if (y <= -6.6e+49)
		tmp = t_1;
	elseif (y <= 4.7e-85)
		tmp = a + z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[y, -6.6e+49], t$95$1, If[LessEqual[y, 4.7e-85], N[(a + z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.5999999999999997e49 or 4.70000000000000009e-85 < y

   1. Initial program 60.0%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 55.5

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

   4. Applied rewrites 55.5%

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

   if -6.5999999999999997e49 < y < 4.70000000000000009e-85

   1. Initial program 60.0%

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

   2. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 47.5

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

   4. Applied rewrites 47.5%

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

   5. Taylor expanded in y around inf

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

      1. lower-+.f64 51.2

         \[\leadsto a + z \]

   7. Applied rewrites 51.2%

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

Alternative 19: 50.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 6.8 \cdot 10^{+69}:\\ \;\;\;\;a + z\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (if (<= t 6.8e+69) (+ a z) a))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 6.8e+69) {
		tmp = a + z;
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (t <= 6.8d+69) then
        tmp = a + z
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 6.8e+69) {
		tmp = a + z;
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= 6.8e+69:
		tmp = a + z
	else:
		tmp = a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= 6.8e+69)
		tmp = Float64(a + z);
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= 6.8e+69)
		tmp = a + z;
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, 6.8e+69], N[(a + z), $MachinePrecision], a]

Derivation

1. Split input into 2 regimes

2. if t < 6.79999999999999973e69

   1. Initial program 60.0%

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

   2. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 47.5

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

   4. Applied rewrites 47.5%

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

   5. Taylor expanded in y around inf

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

      1. lower-+.f64 51.2

         \[\leadsto a + z \]

   7. Applied rewrites 51.2%

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

   if 6.79999999999999973e69 < t

   1. Initial program 60.0%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 31.1%

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

Alternative 20: 42.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.8 \cdot 10^{-11}:\\ \;\;\;\;a\\ \mathbf{elif}\;a \leq 8.5 \cdot 10^{-135}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -4.8e-11) a (if (<= a 8.5e-135) z a)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -4.8e-11) {
		tmp = a;
	} else if (a <= 8.5e-135) {
		tmp = z;
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-4.8d-11)) then
        tmp = a
    else if (a <= 8.5d-135) then
        tmp = z
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -4.8e-11) {
		tmp = a;
	} else if (a <= 8.5e-135) {
		tmp = z;
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -4.8e-11:
		tmp = a
	elif a <= 8.5e-135:
		tmp = z
	else:
		tmp = a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -4.8e-11)
		tmp = a;
	elseif (a <= 8.5e-135)
		tmp = z;
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -4.8e-11)
		tmp = a;
	elseif (a <= 8.5e-135)
		tmp = z;
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -4.8e-11], a, If[LessEqual[a, 8.5e-135], z, a]]

Derivation

1. Split input into 2 regimes

2. if a < -4.8000000000000002e-11 or 8.49999999999999942e-135 < a

   1. Initial program 60.0%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 31.1%

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

   if -4.8000000000000002e-11 < a < 8.49999999999999942e-135

   1. Initial program 60.0%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 33.4%

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

Alternative 21: 31.1% accurate, 29.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.0%

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

2. Taylor expanded in t around inf

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

4. Applied rewrites 31.1%

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

Reproduce

herbie shell --seed 2025150 
(FPCore (x y z t a b)
  :name "AI.Clustering.Hierarchical.Internal:ward from clustering-0.2.1"
  :precision binary64
  (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))